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Page 1: The Mathematics of Computerized Tomography
Page 2: The Mathematics of Computerized Tomography

The Mathematicsof Computerized

Tomography

Page 3: The Mathematics of Computerized Tomography

SIAM's Classics in Applied Mathematics series consists of books that were previouslyallowed to go out of print. These books are republished by SIAM as a professionalservice because they continue to be important resources for mathematical scientists.

Editor-in-ChiefRobert E. O'Malley, Jr., University of Washington

Editorial BoardRichard A. Brualdi, University of Wisconsin-MadisonHerbert B. Keller, California Institute of TechnologyAndrzej Z. Manitius, George Mason UniversityIngram Olkin, Stanford UniversityStanley Richardson, University of EdinburghFerdinand Verhulst, Mathematisch Instituut, University of Utrecht

Classics in Applied MathematicsC. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in theNatural SciencesJohan G. F. Belinfante and Bernard Kolman, A Survey of Lie Groups and Lie Algebraswith Applications and Computational MethodsJames M. Ortega, Numerical Analysis: A Second CourseAnthony V. Fiacco and Garth P. McCormick, Nonlinear Programming: SequentialUnconstrained Minimization TechniquesF. H. Clarke, Optimization and Nonsmooth AnalysisGeorge F. Carrier and Carl E. Pearson, Ordinary Differential EquationsLeo Breiman, ProbabilityR. Bellman and G. M. Wing, An Introduction to Invariant ImbeddingAbraham Berman and Robert J. Plemmons, Nonnegative Matrices in the Mathemat-ical SciencesOlvi L. Mangasarian, Nonlinear Programming*Carl Friedrich Gauss, Theory of the Combination of Observations Least Subjectto Errors: Part One, Part Two, Supplement. Translated by G. W. StewartRichard Bellman, Introduction to Matrix AnalysisU. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of BoundaryValue Problems for Ordinary Differential EquationsK. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-ValueProblems in Differential-Algebraic EquationsCharles L. Lawson and Richard J. Hanson, Solving Least Squares ProblemsJ. E. Dennis, Jr. and Robert B. Schnabel, Numerical Methods for UnconstrainedOptimisation and Nonlinear EquationsRichard E. Barlow and Frank Proschan, Mathematical Theory of ReliabilityCornelius Lanczos, Linear Differential OperatorsRichard Bellman, Introduction to Matrix Analysis, Second EditionBeresford N. Parlett, The Symmetric Eigenvalue Problem

*First time in print.

Page 4: The Mathematics of Computerized Tomography

Classics in Applied Mathematics (continued)

Richard Haberman, Mathematical Models: Mechanical Vibrations, PopulationDynamics, and Traffic FlowPeter W. M. John, Statistical Design and Analysis of ExperimentsTamer Basar and Geert Jan Olsder, Dynamic Noncooperative Game Theory, SecondEditionEmanuel Parzen, Stochastic ProcessesPetar Kokotovic, Hassan K. Khalil, and John O'Reilly, Singular Perturbation Methodsin Control: Analysis and DesignJean Dickinson Gibbons, Ingram Olkin, and Milton Sobel, Selecting and OrderingPopulations: A New Statistical MethodologyJames A. Murdock, Perturbations: Theory and MethodsIvar Ekeland and Roger Temam, Convex: Analysis and Variational ProblemsIvar Stakgold, Boundary Value Problems of Mathematical Physics, Volumes I and IIJ. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations inSeveral VariablesDavid Kinderlehrer and Guido Stampacchia, An Introduction to VariationalInequalities and Their ApplicationsF. Natterer, The Mathematics of Computerized TomographyAvinash C. Kak and Malcolm Slaney, Principles of Computerized Tomographic ImagingR. Wong, Asymptotic Approximations of IntegralsO. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems:Theory and ComputationDavid R. Brillinger, Time Series: Data Analysis and TheoryJoel N. Franklin, Methods of Mathematical Economics: Linear and NonlinearProgramming, Fixed-Point TheoremsPhilip Hartman, Ordinary Differential Equations, Second EditionMichael D. Intriligator, Mathematical Optimization and Economic TheoryPhilippe G. Ciarlet, The Finite Element Method for Elliptic ProblemsJane K. Cullum and Ralph A. Willoughby, Lanczos Algorithms for Large SymmetricEigenvalue Computations, Vol. I: TheoryM. Vidyasagar, Nonlinear Systems Analysis, Second EditionRobert Mattheij and Jaap Molenaar, Ordinary Differential Equations in Theory andPracticeShanti S. Gupta and S. Panchapakesan, Multiple Decision Procedures: Theory andMethodology of Selecting and Ranking PopulationsEugene L. Allgower and Kurt Georg, Introduction to Numerical Continuation MethodsHeinz-Otto Kreiss and Jens Lorenz, Initial-Boundary Value Problems and the Navier-Stokes Equations

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The Mathematicsof Computerized

Tomography

F NattererUniversitat Munster

Minister, Germany

siamSociety for Industrial and Applied MathematicsPhiladelphia

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Copyright © 2001 by the Society for Industrial and Applied Mathematics.

This S1AM edition is an unabridged republication of the work first published byB. G. Teubner, Stuttgart, and John Wiley & Sons, Chichester and New York,1986.

1 0 9 8 7 6 5 4 3 2

All rights reserved. Printed in the United States of America. No part of thisbook may be reproduced, stored, or transmitted in any manner without the writtenpermission of the publisher. For information, write to the Society for Industrialand Applied Mathematics, 3600 University City Science Center, Philadelphia,PA 19104-2688.

Library of Congress Cataloging-in-Publication Data

Natterer, F. (Frank), 1941-The mathematics of computerized tomography / F. Natterer.

p. cm.-- (Classics in applied mathematics ; 32)Reprint. Published by Wiley in 1986.Includes bibliographical references and index.ISBN 0-89871-493-l(pbk.)

1. Tomography—Mathematics. I. Title. II. Series.

RC78.7.T6N372001616.07'572'0151--dc21

2001020478

siam is a registered trademark.

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ContentsPreface to the Classics Edition ixPreface xiGlossary of Symbols xiiiErrata xv

I. Computerized Tomography 1I.1 The basic example: transmission computerized tomography 1I.2 Other applications 3I.3 Bibliographical notes 8

II. The Radon Transform and Related Transforms 9II. 1 Definition and elementary properties of some integral operators 9II.2 Inversion formulas 18II.3 Uniqueness 30II.4 The ranges 36II.5 Sobolev space estimates 42II.6 The attenuated Radon transform 46II.7 Bibliographical notes 52

III. Sampling and Resolution 54III.1 The sampling theorem 54III.2 Resolution 64III.3 Some two-dimensional sampling schemes 71III.4 Bibliographical notes 84

IV. Ill-posedness and Accuracy 85IV.1 Ill-posed problems 85IV.2 Error estimates 92IV.3 The singular value decomposition of the Radon transform 95IV.4 Bibliographical notes 101

V. Reconstruction Algorithms 102V.1 Filtered backprojection 102V.2 Fourier reconstruction 119V.3 Kaczmarz's method 128

vii

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viii

V.4 Algebraic reconstruction technique (ART) 137V.5 Direct algebraic methods 146V.6 Other reconstruction methods 150V.7 Bibliographical notes 155

VI. Incomplete Data 158VI.1 General remarks 158VI.2 The limited angle problem 160VI.3 The exterior problem 166VI.4 The interior problem 169VI.5 The restricted source problem 174VI.6 Reconstruction of homogeneous objects 176VI.7 Bibliographical notes 178

VII. Mathematical Tools 180VII.1 Fourier analysis 180VII.2 Integration over spheres 186VII.3 Special functions 193VII.4 Sobolev spaces 200VII.5 The discrete Fourier transform 206

References 213

Index 221

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Preface to the Classics Edition

This book was originally published by Wiley and Teubner in 1986. It went outof print in the late nineties. The Russian edition appeared in 1990.

In the eighties, the mathematical theory of tomography was a specialty of a fewmathematicians mainly in Russia and the United States. Since then it has becomean industry. New imaging modalities have come into being, some of them onlyremotely related to the original problem of tomography. Conferences andpublications in the field are abundant in both the mathematical and theengineering communities. The range of applications has become considerablybroader, now touching almost every branch of science and technology. Since everynew application requires new mathematics, the supply of mathematical problemsin this field seems inexhaustible.

In spite of this lively development and the growing sophistication of the field, thebasic topics treated in the book are still relevant. For instance, sampling theory andregularization are indispensible tools also in the more recent imaging modalities, andthe algorithmic paradigms, such as filtered and iterative—additive andmultiplicative—backprojection, are still valid. Integral geometry still plays a majorrole, even though the modeling has become much more demanding and complicated.

Thus I am very happy that SIAM is publishing a reprint of the 1986 book in theirClassics in Applied Mathematics series. The only addition I have made is a list oferrata. I gratefully acknowledge the contributions of readers all over the world whopointed out typographic and other errors. I am confident that the reprinted versionwill serve as an introduction into the mathematical theory of imaging and as areference in the same way as the 1986 edition did.

ix

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Preface

By computerized tomography (CT) we mean the reconstruction of a functionfrom its line or plane integrals, irrespective of the field where this technique isapplied. In the early 1970s CT was introduced in diagnostic radiology and sincethen, many other applications of CT have become known, some of thempreceding the application in radiology by many years.

In this book I have made an attempt to collect some mathematics which is ofpossible interest both to the research mathematician who wants to understandthe theory and algorithms of CT and to the practitioner who wants to apply CT inhis special field of interest. I also want to present the state of the art of themathematical theory of CT as it has developed from 1970 on. It seems thatessential parts of the theory are now well understood.

In the selection of the material 1 restricted myself—with very fewexceptions — to the original problem of CT, even though extensions to otherproblems of integral geometry, such as reconstruction from integrals overarbitrary manifolds are possible in some cases. This is because the field ispresently developing rapidly and its final shape is not yet visible. Another glaringomission is the statistical side of CT which is very important in practice and whichwe touch on only occasionally.

The book is intended to be self-contained and the necessary mathematicalbackground is briefly reviewed in an appendix (Chapter VII). A familiarity withthe material of that chapter is required throughout the book. In the main text Ihave tried to be mathematically rigorous in the statement and proof of thetheorems, but I do not hesitate in giving a loose interpretation of mathematicalfacts when this helps to understand its practical relevance.

The book arose from courses on the mathematics of CT I taught at theUniversities of Saarbriicken and Munster. I owe much to the enthusiasm anddiligence of my students, many of whom did their diploma thesis with me. Thanksare due to D. C. Solmon and E. T. Quinto, who, during their stay in Munsterwhich has been made possible by the Humboldt-Stiftung, not only read criticallyparts of the manuscript and suggested major improvements but also gave theiradvice in the preparation of the book. I gratefully acknowledge the help of A.Faridani, U. Heike and H. Kruse without whose support the book would neverhave been finished. Last but not least I want to thank Mrs I. Berg for her excellenttyping.

Miinster, July 1985 Frank Natterer

xi

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Glossary of Symbols

Symbol Explanation References

R" n-dimensional euclidean spaceunit ball of R"

Sn-l unit sphere in R"Z unit cylinder in Rn + 1 II. 1T tangent bundle to Sn-1 II.1

subspace or unit vector perpendicular to 6D1 derivative of order / = ( l 1 , . . . , l n )x • 0 inner product|x| euclidean normC" complex n-dimensional space7,7 Fourier transform and its inverse VII.1

Gegenbauer polynomials, normed VII.3by 1

Y1 spherical harmonics of degree / VII.3N (n, l) number of linearly independent VII.3

spherical harmonics of degree /Jk Bessel function of the first kind VII.3Uk, Tk Chebyshev polynomials VII.3

Dirac's -function VII.1sincb sincfunction III.ln( , b) exponentially decaying function III.2

Gamma function0(M) Quantity of order M

Schwartz space on R" VII.1Schwartz space on Z, T II.1tempered distributions VII.1

Cm m times continuously differentiablefunctions

C infinitely differentiable functionsfunctions in C with compact support

, Sobolev spaces of order a on VII.4Sobolev spaces of order a on Z, T 11.5

xiii

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xiv

LP( , w)

<ul , • • • , um>

R, R

P,PDa

R* etc.I

A*AT

Zn

HM

same as Lp ( ) but with weight wspan of u1, . . . , um

Radon transformX-ray transformdivergent beam transformdual of R, etc.Riesz potentialattenuated Radon transformexponential Radon transformadjoint of operator Atranspose of matrix An-tupels of integersn-tupels of non-negative integers/ perpendicular to gHilbert transformMellin transformend of proof

II.lII.lII.lII.lII.2II.6II.6

VII.lVII.3

space with norm

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Errata

page6

7

8

12

13

13

25

28

48

48

50

53

57

59

59

60

62

65

69

74

76

79

82

85

92

line fromtop/bottom

7t

l0t

14 b

4 t

1 b

4 b

3, 4 t

4, 5t

9 t

13 t

7b

8b

13 t

4 t

9t

15 t

14 t

3b

9 t

Fig. III.2

Fig. III.4

9b

7 b

6 b

5b

should read

f is given

Pf(x)g(x)

dx

(f(r)-f(0)

g(n-l)

9 = Tf

Theorem 1.3

Theorem 1.3

{Wl = l Zn}

0(m)

f

horizontal

K

f is given

pf( ,x)9( ,x)d0

Theorem 1.2

Theorem 1.2

{Wl : l Zn}

O(l/m)

vertical

H

xv

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xvi

page

95

96

100

100

105

106

106

107

111

112

112

112

113

114

115

116

117

122

130

133

136

136

140

140

142

line fromtop/bottom

10t10t4 t

1, 5, 10, 11 b

2, 6 t

7,8t

7t

8t

1 b

1 b

Fig. V.2

Fig. V.2

8, 13 t

8t

15 t

4 t

13 t

12 t

8b

5b

8t

11b

3t

l1t

2 t

should readII.5.3

Y( )

N(m,l)

C(n,l)

C(n,l)

N(n,l)

Fb

Fb

L(a ,b)

Fb

Fb

cos , sini

X

(3.23)

Theorem (3.6)

(4.10)

II.5.1

N(n, l )

C(n, 2l)

C(n, 21)

N(n,2l)

wb

wb

L(B,a)

wb

wb

sin , cost

w

X

(3.20)

Theorem 3.6

(4.9)

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xvii

page142

143

143

146

148

149

152

153

156

156

161

162

163

162

166

169

173

174

181

195

206

212

213

214

216

line fromtop/bottom

3b

3t

1 b

1b

9b

14 t

10t9t

6t

8t

10t1 t

4t

5t

12 t

7, 8b

1 b

1b

14 b

2t

15 t

15 t

12 b

15 t

3t

should read

Horn (1973) Horn (1979)

Wb, Wb Wb, Wb

From For

k = 0, l = 0,

and Riddle, A.C.

195-207

1973

2908-2913

1979

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IComputerized TomographyIn this chapter we describe a few typical applications of CT. The purpose is to givean idea of the scope and limitations of CT and to motivate the mathematicalapparatus we are going to develop in the following chapters.

In Section I.1 we give a short description of CT in diagnostic radiology. Thiswill serve as a standard example throughout the book. In Section 1.2 we considermore examples and discuss very briefly some physical principles which lead to CT.

I.1 The Basic Example: Transmission Computerized Tomography

The most prominent example of CT is still transmission CT in diagnosticradiology. Here, a cross-section of the human body is scanned by a thin X-raybeam whose intensity loss is recorded by a detector and processed by a computerto produce a two-dimensional image which in turn is displayed on a screen. Werecommend Herman (1979, 1980), Brooks and Di Chiro (1976), Scudder (1978),Shepp and Kruskal (1978), Deans (1983) as introductory reading and Gambarelliet a1, (1977), Koritke and Sick (1982) for the medical background.

A simple physical model is as follows. Let f(x) be the X-ray attenuationcoefficient of the tissue at the point x, i.e. X-rays traversing a small distance Ax atx suffer the relative intensity loss

Let Io be the initial intensity of the beam L which we think of as a straight line,and let I1 be its intensity after having passed the body. It follows from (1.1) that

i.e. the scanning process provides us with the line integral of the function f alongeach of the lines L. From all these integrals we have to reconstruct /.

The transform which maps a function on R2 into the set of its line integrals iscalled the (two-dimensional) Radon transform. Thus the reconstruction problemof CT simply calls for the inversion of the Radon transform in R2. In principle,

1

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2

this has been done as early as 1917 by Radon who gave an explicit inversionformula, cf. II, Section 2. However, we shall see that Radon's formula is of limitedvalue for practical calculations and solves only part of the problem.

In practice the integrals can be measured only for a finite number of lines L.Their arrangement, which we refer to as scanning geometry, is determined by thedesign of the scanner. There are basically two scanning geometries in use(Fig. I.1). In the parallel scanning geometry a set of equally spaced parallel linesare taken for a number of equally distributed directions. It requires a single sourceand a single detector which move in parallel and rotate during the scanningprocess. In the fan-beam scanning geometry the source runs on a circle around thebody, firing a whole fan of X-rays which are recorded by a linear detector arraysimultaneously for each source position. Parallel scanning has been used in A.Cormack's 1963 scanner and in the first commercial scanner developed by G.Hounsfield from EMI. It has been given up in favour of fan-beam scanning tospeed up the scanning process.

FIG. I.1 Principle of transmission CT. Cross-section of the abdomen showsspine, liver and spleen. Left: parallel scanning. Right: fan-beam scanning.

So the real problem in CT is to reconstruct / from a finite number of its lineintegrals, and the reconstruction procedure has to be adapted to the scanninggeometry. The impact of finite sampling and scanning geometry on resolutionand accuracy is one of the major themes in CT.

Sometimes it is not possible or not desirable to scan the whole cross-section. Ifone is interested only in a small part (the region of interest) of the body it seemsreasonable not to expose the rest of the body to radiation. This means that onlylines which hit the region of interest or which pass close by are measured. Theopposite case occurs if an opaque implant is present. We then have to reconstructfoutside the implant with the line integrals through the implant missing. Finally itmay happen that only directions in an angular range less than 180° are available.In all these cases we speak of incomplete data problems. A close investigation of

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uniqueness and stability and the development of reconstruction methods forincomplete data problems is indispensable for a serious study of CT.

So far we have considered only the two-dimensional problem. In order toobtain a three-dimensional image one scans the body layer by layer. If this is tootime consuming (e.g. in an examination of the beating heart) this reduction to aseries of two-dimensional problems is no longer possible. One then has to dealwith the fully three-dimensional problem in which line integrals through all partsof the body have to be processed simultaneously. Since it is virtually impossible toscan a patient from head to toe, incomplete data problems are the rule rather thanthe exception in three-dimensional CT. As an example we mention only the conebeam scanning geometry, which is the three-dimensional analog of fan-beamscanning. In cone beam scanning, the source spins around the object on a circle,sending out a cone of X-rays which are recorded by a two-dimensional detectorarray. Even in the hypothetical case of an infinite number of source positions,only those lines which meet the source curve are measured in this way. Since theselines form only a small fraction of the set of all lines meeting the object the data setis extremely incomplete.

The problems we have mentioned so far are clearly mathematical in nature, andthey can be solved or clarified by mathematical tools. This is what this book istrying to do. It will be concerned with questions such as uniqueness, stability,accuracy, resolution, and, of course, with reconstruction algorithms.

There is a bulk of other problems resulting from insufficient modelling whichwe shall not deal with. We mention only beam hardening: in reality, the functionfdoes not only depend on x but also on the energy £ of the X-rays. Assuming T (E)to be the energy spectrum of the X-ray source, (1.2) has to be replaced by

Using (1.2) instead of (1.3) causes artefacts in the reconstructed image. Morespecifically we have approximately (see Stonestrom et al, 1981)

with C the Klein-Nishina function which varies only little in the relevant energyrange. Therefore the beam hardening effect is more pronounced at low energies.

1.2 Other Applications

From the foregoing it should be clear that CT occurs whenever the internalstructure of an object is examined by exposing it to some kind of radiation whichpropagates along straight lines, the intensity loss being determined by (1.1). Thereare many applications of this type. We mention only electron microscopy since itis a typical example for incomplete data of the limited angle type. In transmissionelectron microscopy (Hoppe and Hegerl, 1980) an electron beam passes through aplanar specimen under several incidence angles. Since the beam has to traverse the

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specimen more or less transversally, the incidence angle is restricted to an intervalless than 180°, typically 120°.

In other applications the radiating sources are inside the object and it is thedistribution of sources which is sought for. An example is emission CT in nuclearmedicine (Budinger et al., 1979). Here one wants to find the distributionfof aradiopharmaceutical in a cross-section of the body from measuring the radiationoutside the body. If is the attenuation of the body ( now plays the role ofdinSection I.1) and assuming the same law as in (1.1) to be valid, the intensity /outside the body measured by a detector which is collimated so as to pick up onlyradiation along the straight line L (single particle emission computerizedtomography: SPECT) is given by

where L(x) is the section of L between x and the detector.If is negligible, then / is essentially the line integral offand we end up with

standard CT. However, in practice is not small. That means that we have toreconstructffrom weighted line integrals, the weight function being determinedby the attenuation . The relevant integral transform is now a generalization ofthe Radon transform which we call attenuated Radon transform.

An important special case arises if the sources eject the particles pairwise inopposite directions and if the radiation in opposite directions is measured incoincidence, i.e. only events with two particles arriving at opposite detectors at thesame time are counted (positron emission tomography: PET). Then, (2.1) has tobe replaced by

where L+(x), L_(x) are the two half-lines of L with endpoint x. Since theexponent adds up to the integral over L we obtain

which does not lead to a new transform.In SPECT as well as in PET we are interested inf, not in . Nevertheless, since

enters the integral equation for / we have to determine it anyway, be it byadditional measurements (e.g. a transmission scan) or by mathematical tools.

Another source of CT-type problems is ultrasound tomography in thegeometrical acoustics approximation. Here the relevant quantity is the refractiveindex n of the object under examination. In the simplest case the travel timeT (x, y) of a sound signal travelling between two points x and y lying on the surface

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of the object is measured. The path of the signal is the geodesic n (x, y) withrespect to the metric joining x and y. We have

Knowing T(x,y) for many emitter-receiver pairs x, y we want to compute n.Equation (2.3) is a nonlinear integral equation for n. Schomberg(1978) developedan algorithm for the nonlinear problem. Linearizing by assuming n = n0 +fwithn0 known and / small we obtain approximately

and we have to reconstruct/from its integrals over the curves no (x, y). This is theproblem of integral geometry in the sense of Gelfand et al. (1965). Inversionformulas for special families of curves have been derived by Cormack (1981),Cormack and Quinto (1980) and Helgason (1980). In seismology the curves arecircles, see Anderson (1984). If n0 is constant, the geodesies are straight lines andwe end up with the problem of CT.

It is well known that inverse problems for hyperbolic differential equationssometimes can be reduced to a problem in integral geometry, see Romanov(1974). As an example we derive the basic equation of diffraction tomography andshow that CT is a limiting case of diffraction tomography.

Consider a scattering object in R2 which we describe by its refractive index n(x)= (1 +/(x))1/2 where fvanishes outside the unit ball of R2, i.e. the object iscontained in the unit circle. This object is exposed to a time harmonic 'incomingwave'

with frequency k. We consider only plane waves, i.e.

the unit vector being the direction of propagation.In (direct) scattering (Courant and Hilbert (1962), ch. IV, Section 5) one finds

for / given, a scattered wave

such that u = u1 + us satisfies the reduced wave equation

A the Laplacian, and a boundary condition at infinity. In order to solve (2.5)within the Rytov approximation (Tatarski, 1961, ch. 7.2) we put in (2.5)

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6

where the factor k has been introduced for convenience. We obtain

with V the gradient. The Rytov approximation UR to u is obtained by neglecting

where WR satisfies

or

This differential equation can be solved using an appropriate Green's functionwhich in our case happens to be H0(k ) with 4iH0 the zero order Hankelfunction of the first kind. All we need to know about H0 is the integralrepresentation

where (Morse and Feshbach, 1953, p. 823). The solution of(2.7) can now be written as

This solves—within the Rytov approximation—the direct scattering problem.In inverse scattering (see e.g. Sleeman (1982) for a survey) we have to computef

from a knowledge of us outside the scattering object. If the Rytov approximationis used, the inverse scattering problem can be solved simply by solving (2.9) forf,using as data

where r is a fixed number > 1, i.e. WR is measured outside a circle of radius r.Inserting ui, from (2.4) and H0 from (2.8) into (2.9) yields

Since r > 1 and ) = 0 for > 1 we can drop the absolute value in. Interchanging the order of integration we get

i.e.

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7

The integration with respect to is a Fourier transform in R2, hence

The integration with respect to a is an inverse Fourier transform in R1. From theFourier inversion theorem we obtain

where is the one-dimensional Fourier transform with respect to the secondargument. If a runs over the interval [ — k,.k], then

runs over the half-circle around — k6 with top at the origin, see Fig. 1.2. Hence, ifvaries over S1,fis given by (2.10) in a circle of radius at least k. Assuming that weare not interested in frequencies beyond k (i.e. that we are satisfied with a finiteresolution in the sense of the sampling theorem, cf. Chapter III.l) and that theRytov approximation is valid, (2.10) solves in principle the inverse scatteringproblem. In fact (2.10) is the starting point of diffraction tomography assuggested by Mueller et al (1979), see also Devaney (1982), Ball et al. (1980).

FIG. 1.2. Half-circle on which is given by (2.10).

If we let then hence (2.10) becomes

i.e. the circles on whichfis given turn into lines. Later we shall see (cf. TheoremII. 1.1) that

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8

where

is the Radon transform. Comparing this with (2.11) we find that

or

for r > 1. Thus, for large frequencies and within the Rytov approximation, theline integrals offare given by the scattered field outside the scattering object, i.e.CT is a limiting case of the inverse scattering problem of the wave equation. Ofcourse (2.12) follows also directly from (2.6) by letting

A similar treatment of the inverse problem in electromagnetic scattering showsthat the target ramp response in radar theory provides the radar silhouette area ofthe target along the radar line of incidence, see Das and Boerner (1978). Thismeans that identifying the shape of the target is equivalent to reconstructing thecharacteristic function of the target from its integrals along planes in R3. This isan instance of CT in R3 involving the three-dimensional Radon transform,though with very incomplete data. Complete sampling of the three-dimensionalRadon transform occurs in NMR imaging (Hinshaw and Lent, 1983). Eventhough the formulation in terms of the Radon transform is possible and has beenused in practice, (Marr et at, 1981), the present trend is to deal with the NMRproblem entirely in terms of Fourier transforms.

1.3 Bibliographical Notes

Tomography' is derived from the greek word = slice. It stands for severaldifferent techniques in diagnostic radiology permitting the imaging of cross-sections of the human body. Simple focusing techniques for singling out a specificlayer have been used before the advent of CT, which sometimes has been calledreconstructive tomography to distinguish it from other kinds of tomography. SeeLittleton (1976) for the early history of tomography and Klotz et al. (1974) forrefined versions employing coded apertures.

The literature on CT has grown tremendously in the last few years, and we donot try to give an exhaustive list of references. We rather refer to the reasonablycomplete bibliography in Deans (1983); this book also gives an excellent surveyon the applications of CT. A state-of-the-art review is contained in the March1983 issue of the Proceedings of the IEEE which is devoted to CT. The conferenceproceedings Marr (1974a), Gordon (1975), Herman and Natterer (1981) give animpression of the exciting period of CT in the 1970s.

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IIThe Radon Transform andRelated TransformsIn this chapter we introduce and study various integral transforms from atheoretical point of view. As general reference we recommend Helgason (1980).The material of this chapter serves as the theoretical basis for the rest of the book.It contains complete proofs, referring only to Chapter VII.

II.1 Definition and Elementary Properties of some Integral Operators

The (n-dimensional) Radon transform R maps a function on R" into the set of itsintegrals over the hyperplanes of R". More specifically, if and , then

is the integral of (Rn), the Schwartz space, over the hyperplane perpendicularto 0 with (signed) distance s from the origin. For the Schwartz space and othernotions not explained here see Chapter VII. Obviously, Rfis an even function onthe unit cylinder Z = Sn-1 x R1 of Rn + 1 , i.e. Rf( , -s) = Rf( ,s). We alsowrite

The (n-dimensional) X-ray transform P maps a function on R" into the set of itsline integrals. More specifically, if and , then

is the integral of over the straight line through x with direction 6.Obviously, Pf( , x) does not change if x is moved in the direction . We thereforenormally restrict x to which makes Pfa function on the tangent bundle

9

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to Sn-1. We also write

P fis sometimes called the projection of fon to . For n = 2, P and R coincideexcept for the notation of the arguments. Of course it is possible to expressRf(w, s) as an integral over Pf: for any with we have

The divergent beam (in two dimensions also fan-beam) transform is defined by

This is the integral of falong the half-line with endpoint and direction. We also write

If then R f, P f, Rf P/are in the Schwartz spaces on R1, Z, Trespectively, the latter ones being defined either by local coordinates or simply byrestricting the functions in (Rn+1) to Z, those in (R2n) to T.

Many important properties of the integral transforms introduced above followfrom formulas involving convolutions and Fourier transforms. Wheneverconvolutions or Fourier transforms of functions on Z or Tare used they are to betaken with respect to the second variable, i.e.

for (Z)and

for (T).We begin with the so-called 'projection theorem' or 'Fourier slice theorem'

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THEOREM 1.1 we have

Proof We have

With x = s + y the new variable of integration we have s = .x, dx = dy ds,hence

Similarly,

As a simple application we derive the formula

where D acts on the second variable of Rf. For the proof we use Theorem 1.1, R3in VII.1 and the fact that the Fourier transform is invertible on (R1), obtaining

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In order to derive a formula for the derivatives of Rfwith respect to 0 we give asecond equivalent definition of the Radon transform in terms of the one-dimensional function. Putting

we know from VII. 1 that pointwise in Hence, forf ,

In this sense we write

(1.3) provides a natural extension of Rfto (Rn — {0}) x R1 since, for r > 0,

i.e. Rfis extended as a function homogeneous of degree — 1. This function can bedifferentiated with respect to its first variable and we obtain

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This formal procedure as well as the derivation of (1.4) can be justified by usingthe approximate -function . For k a multi-index we get

where D denotes the derivative with respect to 9.As an immediate consequence from R4 of VII.1 and Theorem 1.1, or by direct

calculation as in the preceding proof, we obtain

THEOREM 1.2 For f, we have

In the following we want to define dual operators We start outfrom

Defining

we thus have

Integrating over Sn-l we obtain

Similarly,

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where

with E the orthogonal projection on .Note that R, R# form a dual pair in the sense of integral geometry: while R

integrates over all points in a plane, R# integrates over all planes through a point.This is the starting point of a far-reaching generalization of the Radon transform,see Helgason (1980). The same applies to P, P#.

THEOREM 1.3 For (Rn) and (Z) (T) respectively we have

Proof We have

Making the substitution in the inner integral we obtain

P is dealt with in the same way.

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THEOREM 1.4 For we have

Proof For we (Rn) we obtain from the definition of R# as dual of R

where we have made use of Rule 5 of VII.1 in the inner integral. By Theorem 1.1,

hence

where we have put = separately for a > 0 and a < 0. This shows that thedistribution (R*g)* is in fact represented by the function

THEOREM 1.5 For f (Rn) we have

is the surface of the unit sphere sm-1 in Rm, see VII.2.

Proof We have

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since . Using (VII.2.8) with f(x) replaced by f(x+y) we obtain

hence

Similarly,

since x — E x = (x • 0)0 is a multiple of . Breaking the inner integral into positiveand negative part we obtain

upon setting y = t0. This proves the second formula.

So far we have considered the integral transforms only on (Rn). Since

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the operators R , P are easily extended to L1(Rn), Da to L 1 R n , .Whenever we apply this operators to functions other than we tacitly assumethat the extended operators are meant. The duality relations (1.6), (1.7) holdwhenever either of the functions is integrable, and cor-respondingly for (1.8), (1.9).

The Hilbert spaces L 2 (Z) L2(T) are defined quite naturally by the innerproducts

We have the following continuity result.

THEOREM 1.6 Let be the unit ball in Rn. Then, the operators

are continuous.

Proof For with support in we have from the Cauchy-Schwarzinequality

with the volume of the unit ball of Rn, hence

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This settles the case of R . Integrating over Sn-l gives the continuity of R. P , Pare dealt with in the same way. For Da we have

hence

From Theorem 1.6 it follows that the operators R , P , R, P, Da havecontinuous adjoints. For instance

is given by

II.2 Inversion Formulas

Explicit inversion formulas for the integral transforms are not only of obvioussignificance for the development of inversion algorithms, but play also animportant role in the study of the local dependence of the solution on the data. Inthe following we derive inversion formulas for R and P.

For a < n we define the linear operator I by

is called the Riesz potential. If I is applied to functions on Z or T it acts onthe second variable. For , hence I f makes sense and

THEOREM 2.1 Let Then, for any a < n, we have

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Proof We start out from the Fourier inversion formula

Introducing polar coordinates , = yields

Here we express fby (Rf) using Theorem 1.1. We obtain

Replacing 6 by — 6 and a by — a and using that (Rf) is even yields the sameformula with the integral over (0, ) replaced by the integral over (— , 0).Adding both formulas leads to

The inner integral can be expressed by the Riesz potential, hence

and the inversion formula for R follows by applying I .For the second inversion formula we also start out from (2.3). Using the

integral formula (VII.2.8).

for the integral in (2.3) we obtain

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Here we express by (Pf) using Theorem 1.1. We obtain

The inner integral can be expressed by the Riesz potential, hence

and the inversion formula for P follows.

We want to make a few remarks.

(1) Putting = 0 in (2.1) yields

where I1-n acts on a function in R1. Since for h (R1)

and since the Hilbert transform H can be defined by

see (VII.1.11), we can write

hence

where the (n — l)st derivative is taken with respect to the second argument. Using

and the explicit form of R* from Section II. 1, (2.4) can be written as

(2) The fact that H shows up in (2.5) only for n even has an important practicalconsequence. Remember that g( ,x ) is the integral of f over the hyperplaneperpendicular to 0 containing x. For n odd, (2.5) can be evaluated if isknown for 0 Sn-l and y in a neighbourhood of x, i.e. if the integrals offalong all

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hyperplanes meeting a neighbourhood of x are known. Thus we see that theproblem of reconstructing a function from its integrals over hyperplanes is, forodd dimension, local in the following sense: the function is determined at somepoint by the integrals along the hyperplanes through a neighbourhood of thatpoint. This is not true for even dimension, since the Hilbert transform

(see VII.1.10) is not local. Thus the problem of reconstructing a function from itsintegrals over hyperplanes is, in even dimension, not local in the sense thatcomputing the function at some point requires the integrals along allhyperplanes meeting the support of the function. Anticipating later discussions(see VI.4) we mention that locality can be restored only to a very limited extent.

(3) For n even we want to give an alternative form of (2.5). Expressing H in (2.5)by (2.6) we obtain

where we have made the substitution t = q + x • . The inner integral, which is aCauchy principal value integral, can be expressed as an ordinary integral as in(VII.1.9). We obtain

Now interchanging the order of integration is permitted and yields

Since g is even and n — 1 is odd, g ( n - l ) is odd. Making the substitution forthe second term in the inner integral we see that the integrals are the same.

Thus (2.8) simplifies to

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Fx is even because g is. Hence F ( n - 1 ) is odd and we obtain finally Radon's originalinversion formula (Radon, 1917)

which, for n = 2, he wrote as

This formula can be found on the cover of the Journal for Computer AssistedTomography.

This is the basis for the -filtered layergram algorithm, see V.6. For n odd, I1-n issimply a differential operator

with A the Laplacian. Specifically, for n = 3,

where A acts on the variable x. This has also been derived by Radon.

(5) Iffis a radial function, i.e.f(x) =fo(|x|) with some functionf0 on R1, theng(s) = Rf( , s) is independent of 0 and

(4) Putting n — 1 in (2.1) we obtain

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Thus Radon's integral equation reduces for radial functions to an Abel typeintegral equation, which can be solved by the Mellin transform (what we will do inTheorem 2.3 below in a more general context) or by the following elementaryprocedure: integrating (2.11) against s(s2 — +2) (n -3 ) /2 from r to yields

For the inner integral we obtain by the substitution

the value

hence

We have

Applying the operator (l/2r) (d/dr) n — 2 more times yields

and this is clearly an inversion formula for the Radon transform in the case ofradial symmetry.

(6) We cannot resist sketching Radon's original direct and elegant proof of (2.9).It basically relies on two different evaluations of the integral

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Substitutingx = wehave(|x|2 - q 2 ) - l / 2 d x = dsd0, hence(2.12) becomes

with F0 as in remark 3.On the other hand, introducing x = r , Sl in (2.12) yields

where

Comparing the two expressions for (2.12) we obtain

This is an Abel integral equation of type (2.11) for n = 2 which we have solvedin the preceding remark by elementary means, but Radon decided to proceedmore directly. Computing

with F0 from (2.13) gives

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The first integral is , and the second integral tends to zero as . Hence

which is (2.9) for x = 0.(7) The inversion formula for P coincides—up to notation—with the inversionformula for R if n = 2. For n > 3 it is virtually useless since it requires theknowledge of all line integrals while only a small fraction of the line integrals canbe measured in practice. An inversion formula which meets the needs of practicemuch better will be derived in VI.5.

A completely different inversion formula for the Radon transform is derived byexpanding / and g — Rf in spherical harmonics, see VII.3, i.e.

The following theorem gives a relation between flk and gik.

THEOREM 2.2 Let Then we have for s > 0

where is the (normalized) Gegenbauer polynomial of degree /, see VII.3.

Proof We can express integrals over x • = s by integrals over the half-sphere

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by rotating the coordinate system in (VII.2.5). We obtain

We apply this to the function /(x) =fik(\x\)Ylk(x/\x\), obtaining

Here we can apply the Funk-Hecke theorem (VII.3.12). Defining

we get

We obtain (2.14) by putting r = s/t.

The integral equation (2.14) is an Abel type integral equation; in fact it reducesto an Abel equation for n = 2 and / = 0. It can be solved by the Mellin transform.This leads to

THEOREM 2.3 Let . Then we have for r > 0

(For n = 2 one has to take the limit , i.e. c(2) = — 1/ ).

Proof Using the convolution as defined in VII.3. in connection with the Mellintransform we may write (2.14) in the form

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Taking the Mellin transform and making use of the rules (VII.3.8), (2.16) becomes

Mb is obtained from (VII.3.9) with

Writing

we see from (VII.3.10) with = (n — 2)/2 that the last term is the Mellin transformof the function

Here, we have to interpret c2 as being 2 in the case n = 2, i.e. we have to take thelimit n 2. Hence,

With the rules (VII.3.8) we obtain from (2.17) with s replaced by s- 1

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This is equivalent to

where the convolution is to be understood in the sense of VII.3 again, i.e.

This is essentially (2.15).

The relevance of (2.15) comes from the fact that it gives an explicit solution tothe exterior problem, see VI.3. It is usually called Cormack's inversion formulasince it has been derived for n = 2 in Cormack's (1963) pioneering paper.We remark that the assumption in Theorems 2.2 and 2.3 has been made toensure that the integrals (2.14), (2.15) make sense. For functions which are notdecaying fast enough at infinity the exterior problem is not uniquely solvable as isseen from the example following Theorem 3.1 below.

Let us have a closer look at the two-dimensional case. The expansions inspherical harmonics are simply Fourier series which, in a slightly differentnotation, read

Formulas (2.14), (2.15) can now be written as

with T1 the Chebyshev polynomials of the first kind, see VII.3. We will give analternative form of (2.18). From Theorem 4.1 we know that

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is a polynomial of degree m — 1 in , i.e.

with some constants cmk. Then,

for m < |1| because of the orthogonality properties of the exponential functions.Since g is even, gl( — s) = ( — 1)1 + 1g1 (s). Thus, if is a polynomial of degree < |1|which is even for / odd and odd for / even, then

Therefore we can rewrite (2.18) as

Now we want to choose so as to make the factor of g1 in the first integral small,improving in this way the stability properties of the formula. From (VII.3.4) weknow that for x > 1

If we choose

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with U1, the Chebyshev polynomials of the second kind, then

If we use this in (2.19) with x = s/r we obtain

where we define U-1 =0. The practical usefulness of formulas (2.18) and (2.20)will be discussed in VI.3 and V.6.

II.3 Uniqueness

From the inversion formulas in Section 2 we know that is uniquelydetermined by either of the functions Rf, Pf. In many practical situations weknow these functions only on a subset of their domain. The question arises if thispartial information still determines / uniquely.

The following 'hole theorem' follows immediately from Theorem 2.3.

THEOREM 3.1 Let f y, and let K be a convex compact set ('the hole') in R". IfRf( , s) = 0 for every plane x - = s not meeting K, thenf= 0 outside K.

Proof For K a ball the theorem follows immediately from Theorem 2.3. In thegeneral case we can find for each x K a ball containing K but not x. Applying thetheorem to that ball we conclude that f(x) = 0, hence the theorem for K.

Since the integrals over planes can be expressed in terms of integrals over lineswe obtain immediately the corresponding theorem for P:

THEOREM 3.2 Let /, K be as in Theorem 3.1. If Pf ( , x) = 0 for each line x + tnot meeting K, then / = 0 outside K.

We see that the problem of recovering a function in the exterior of some ballfrom integrals over planes or lines outside that ball (exterior problem, see VI.3) isuniquely solvable, provided the function is decaying fast enough at infinity. Thisassumption cannot be dropped, as is seen from the following example in R2. Letz = x1 + ix2 and

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with p > 2 an integer. We show that Rf( , s) = 0 for 0 S1 and s > 0. We have

where L is the straight line X' = s. Putting w = 1/z we get

where the image C of L under w = l/z is a finite circle since L does not meet 0. ByCauchy's theorem, the integral over C vanishes for p > 2, hence Rf( , s) = 0 fors > 0 and p > 2.

Much more can be said if / is known to have compact support. The followingtheorem is given in terms of the divergent beam transform D.

THEOREM 3.3 Let S be an open set on Sn-1, and let A be a continuouslydifferentiable curve. Let Rn be bounded and open. Assume that for each Sthere is an a A such that the half-line a + t , t > 0 misses . If andDf(a, ) = 0 for a A and S, then f=0 in the 'measured region'{a + t0:a A, S}.

Proof Define for each integer k > 0 and a, x R"

We shall prove that, under the hypothesis of the theorem,

for all k > 0. Since / has compact support, (3.1) clearly implies the theorem.The proof of (3.1) is by induction. For k = 0, (3.1) follows from Df(a, ) = 0,

a A, S. Let (3.1) be satisfied for some k > 0. Then,

for x in the open cone C = {x 0: x/|x| S}. Let a = a(s) be a parametricrepresentation of A. Then,

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with fi the ith partial derivative of f,

for x C since D k f ( a ( s ) , x) vanishes in the open set C. Thus, Dk + 1 f ( a , x) isconstant along A for x C. Since, by the hypothesis of the theorem, Dk + 1 f ( a , x)= 0 for at least one a A, Dk+1 f(a, x) must be zero on A for x C. This is (3.1)with k replaced by k +1, hence the theorem.

The assumption on the half-line missing looks strange but cannot bedropped. This can be seen from the example in Fig. II. 1, where is the unit disk inR2 and f is a smoothed version of the function which is +1 in the upper strip and— 1 in the lower strip and zero elsewhere. With A the arc ab and 5 given by theangle the hypothesis of the theorem is not satisfied. In this case, / does notvanish in the measured region, even though Df is zero on A x S. However, if A ischosen to be ac, then the hypothesis of the theorem is satisfied, but Df is nolonger zero on A x S.

A situation to which Theorem 3.3 applies occurs in three-dimensionaltomography, see I.1. It tells us that, in principle, three-dimensional reconstructionof an object is possible if the X-ray sources run on a circle surrounding the object.

FIG. II.1 Example of non-uniqueness.

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However, we shall see in VI.5 that the reconstruction problem is seriously ill-posed unless the curve satisfies a restrictive condition.

In the next two theorems we exploit the analyticity of the Fourier transform offunctions with compact support.

THEOREM 3.4 Let A be a set of directions such that no non-trivial homogeneouspolynomial vanishes on A. If f ( R n ) and R0f= 0 for A, then f= 0.

Proof From Theorem 1.1 we have

for A. Since / has compact support, / is an analytic function whose powerseries expansion can be written as

with homogeneous polynomials ak of degree k. For = ad we obtain

for each a and e A. It follows that ak ( ) vanishes for A, hence ak = 0 and thetheorem is proved.

THEOREM 3.5 Let ( Rn), and let P0/=0 for an infinite number ofdirections. Then, / = 0.

Proof From Theorem 1.1 we have for infinitely many 's

if n • 9 = 0. Since / is an analytic function, f ( n ) must contain the factor n • 6 foreach of the direction 8. Hence fhas a zero of infinite order at 0 and hence vanishesidentically.

For the divergent beam transform we have the following uniqueness theorem,which turns into Theorem 3.5 as the sources tend to infinity.

THEOREM 3.6 Let ." be the unit ball in R" and let A be an infinite set outside .If and D af= 0 for a e A then /= 0.

Proof Let 0bea limit point of . Choose e > 0 such that f(x) = 0for |x| > 1 — 2e. After possibly removing some elements from A we may assumethat, for a suitable neighbourhood U( 0) of 0 on Sn-1,

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Now we compute

where we have put y = tco and x = a + y. Note that the last integral makes sensesince for f(x) 0

Putting x = s0 + y, y e 0 we obtain

R1

Since Daf= 0 for ae we have

The power series

converges uniformly in |s| < 1 — 2e for each aeA, eU ( 0). All we need to knowabout the coefficients ck is that they do not vanish. Hence

for aeA, 0eU(00). Anticipating Theorem4.1 we have

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where Pk is a homogeneous polynomial of degree k. It follows that

for a e A and 9 e U ( 0). Since the set of points a e A for which a - 6 assumes onlyfinitely many values is contained in a (n — l)-dimensional set we can find an opensubset U' U( 0) such that {a • : a e A} is infinite for 6 e U'. Then, varying a inA gives

for e V and each k, hence Pk = 0 for each k. Thus,

for each k. Since the monomials are dense in L2( — 1, + 1) we conclude Rf= 0,hence f=0.

It is not surprising that the last two theorems do not hold for finitely manydirections or sources. However, the extent of non-uniqueness in the finite case is,in view of the applications, embarrassing. We consider only the case of finitelymany directions in Theorem 3.4.

THEOREM 3.7 Let 01 , . . . , 0 p eSn - l , let K R" be compact, and let / be anarbitrary function in (K). Then, for each compact set K0 which lies in theinterior of K there is a functiof0 e (K) which coincides with/on K0 and forwhich P fo = 0, k = l, . . . , p .

Proof Let

and let Q be the differential operator obtained from q( ) by replacing , by— i . We compute a solution h of Qh = f. For p = 1 this differential equationreads

with V the gradient. Putting x = s +y, y , a solution is

For p > 1 we repeat this construction. Now let (K) be 1 on K0 and putfo = . We have (K), and on K0,f0 = Qh=f. Also, by R3 from VII. 1,

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and, by Theorem 1.1,

for It follows that P = 0 and the theorem is proved.

Loosely speaking Theorem 3.7 states that for each object there is another onediffering from the first one only in an arbitrary small neighbourhood of itsboundary and for which the projections in the directions vanish. At afirst glance this makes it appear impossible to recover a function from finitelymany projections. A closer look at the proof of Theorem 3.7 reveals thatf0 is ahighly oscillating function: Since q is a polynomial of degree p, f0 assumes largevalues for > 1 fixed and p large. This means that for p large the second objectbehaves quite erratically in the thin boundary layer where it is different from thefirst one. In order to make up for this indeterminacy in the case of finitely manyprojections we have to put restrictions on the variation of the object. This ispursued further in Chapters III and IV.

II.4 The Ranges

In this section we shall see that the ranges of the operators introduced in SectionII. 1 are highly structured. This structure will find applications in the study ofresolution, of algorithms and of problems in which the data are not fully specified.

THEOREM 4.1 Let . Then, for m = 0, 1, ...

with pm, qm homogeneous polynomials of degree m, qm being independent of 9.

Proof We compute

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where we have put x = s + y. Obviously, this is a homogeneous polynomial ofdegree m in 6. Likewise,

for y _ where we have put z = x + t . This is a homogeneous polynomial ofdegree m in y which is independent of 9.

The condition (4.1) for the Radon transform has the following consequencewhich we shall use in later chapters. we can expand Rfin terms ofthe products Ykj, where are the Gegenbauer polynomials and Ykj thespherical harmonics, see VII.3. These functions form a complete orthogonalsystem in L2 (Z, (1 — s2) ). The expansions reads

where j runs over all N (n, k) spherical harmonics of degree k. The Cf areorthogonal in [ — 1, +1] with respect to the weight function (1 — s2) 1/2, hence

According to (4.1), the left-hand side is a polynomial of degree J in 6 which, due tothe evenness of Rf, is even for / even and odd for / odd. Hence, cikj 0 only for k= /, / — 2, . . . , and Rf assumes the form

where h, is a linear combination of spherical harmonics of degree /, / — 2,. . . .The conditions (4.1) are called the Helgason-Ludwig consistency conditions.

They characterize the range of R, P resp. in the following sense.

THEOREM 4.2 Let g e (Z) be even (i.e. g ( , s) = g (— 6, — s)) and assume thatfor each m — 0, 1, . . .

is a homogeneous polynomial of degree m in 0. Then, there is e y (Rn) such thatg = Rf. If, in addition, g (0, s) = 0 for \s\ > a, then /(x) = 0 for |x| > a.

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Proof In view of Theorem 1.1 it appears natural to define

g is even since g is, hence this is true for a < 0, too. It suffices to show that/e y.For in that case,fe and it follows from Theorem 1.1 that g = R .

The crucial point in the proof that/e is smoothness at the origin. Here, theessential hypothesis of the theorem is needed. We start out from

For fdefined as above we obtain

with pm the polynomial of the theorem. Since pm is homogeneous we have for

Let Ap be the class of expressions of the form

where a and its derivatives are bounded C -functions and h e y (Z), or a finitelinear combination thereof. We shall show that for derivatives Da of order |a| < qwith respect to , we have

For the proof we have to express in terms of . We do thecalculation in the spherical cap

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From the chain rule we obtain

Since

we obtain for

with ij — 1 for i = j and = 0 otherwise. Hence,

where D stands for a linear differential operator with respect towhich is of first order and which has C -coefficients depending only on . Notethat

as is easily seen from the definition of Ap. Doing this calculation in an arbitraryspherical cap we obtain globally

The proof of (4.4) is by induction. The case | | = 0 is obvious sinceeq (as)g (9, s) E Aq. Assume (4.4) to be correct for all derivatives up to some orderp < q and let | | = p. Then, from (4.5) and (4.6)

hence (4.4) holds also for derivatives of order p + 1. This proves (4.4).Now let Da be a derivative of order | | = q. From (4.3) we have for | | 0

and this is a continuous and bounded function of in a punctured neighbour-hood of 0 because of (4.4). Since q is arbitrary, feC°°.

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It remains to show that each derivative offis decaying faster than any power of| | as | | -» . This follows immediately from repeated application of (4.5) to

(Z).The remark on the supports of g and / follows immediately from Theorem

3.1.

THEOREM 4.3 Let ge (T) and g(9, x) = 0 for |x| > a. Assume that form = 0, 1, . . .

is a homogeneous polynomial of degree m in y which does not depend on 6. Then,there is/e (Rn) withf(x) = 0 for |x| > a and g = Pf.

Proof We want to reduce Theorem 4.3 to Theorem 4.2. If we already knew thatg = Pfwe could express Rf(w, s) in terms of Pfby (1.1). Therefore we put forsome

and try to show that he does not depend on the choice of 0 and that h = Rfforsomefe.y (Rn) with support in |x| < a.

To begin with, we compute

where qm is the polynomial of the theorem. Since qm does not depend on 9, theintegral on the left-hand side of (4.7) does not depend on 9 either. he (w, s) vanishesfor \s\> a because of g (0, x) = 0 for |x| > a. Since the polynomials are densein L2 (— a, a) it follows that ho does not depend on 9, and from (4.7) we seethat h = hg satisfies the assumptions of Theorem 4.2. Hence h = Rf for somefe Sf (Rn) with support in |x| < a.

For n > 2 we have to show that g = Pf. For 9 fixed we shall show that theintegrals of g and Pfover arbitrary planes in coincide. If issuch a plane where w , the integral of g over this plane gives h (w, s), and the

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integral of Pf is

41

Since h = Rf, the integrals coincide. This means that the Radon transforms ofg ( , ), Pf(0, •) on 0 coincide. But the Radon transform on y (0 ) is injective,hence g — Pf.

Incidentally, Theorem 4.3 does not hold if the condition g ( , x) = 0 for |x| > a(and the conclusion thatf(x) = 0 for |x| > a) is dropped: let h be a non-trivialeven function in (R1) such that

let ue and let (0,x) = u(0)h(\x\). Then, ge ( T ) , and

where we have put x = sw in . Thus, g satisfies the consistency conditions of thetheorem.

For n > 3, let w . Then,

where h is considered as the radial function x -> h (|x|) and R is the (n — 1)-

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dimensional Radon transform. On the other hand, if g = Pffor somefe (R"),we have from (1.1)

and it follows that

for co . This is a contradiction unless u is constant since R h is independent of coand Rh 0 because of the injectivity of R.

II.5 Sobolev Space Estimates

In Theorem 1.6 we derived some simple continuity results in an L2-setting. Sinceall the transforms considered in that theorem have a certain smoothing property,these results cannot be optimal. In particular, the inverses are not continuous asoperators between L2 spaces. However, they are continuous as operators betweensuitable Sobolev spaces. Sobolev space estimates are the basic tool for thetreatment of our integral equations as ill-posed problems in IV.2.

The Sobolev spaces Ha ( ), H ( ) on s Rn are defined in VII.4. The Sobolevspaces on Z and T we are working with are defined by the norms

The Fourier transforms are to be understood as in Section 1. Because of R3 fromVIM an equivalent norm in H (Z) for a > 0 an integer is

where g(l) is the lth derivative with respect to s. For most of the results of thissection it is essential that the functions have compact support. For simplicity werestrict everything to

THEOREM 5.1 For each a there exist positive constants c (a, n), C (a, n) such thatfor

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Proof We start with the inequality for R. From Theorem 1.1 we have

hence

Substituting we get

This is the left-hand side of the inequality for R. For the right-hand side we startout from (5.1) and decompose the integral into an integral over | | > 1 and anintegral over | | < 1. In the first one we have hence

The integral over | | < 1 is estimated by

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In order to estimate the sup we choose a function x e (Rn) which is 1 on andput X • Then, with R5 from VII.l, and with the inverse Fouriertransform of x,

The H a (Rn) norm of is a continuous function of , hence

with some constant c2 (a, n). Combining (5.1)-(5.5) we get

and this is the right-hand side of the inequality for R.For the inequality for P we also start out from Theorem 1.1, applying to the

resulting integral formula (VII.2.8). We obtain

This corresponds to (5.1) in the case of R. From here on we proceed exactly inthe same way as above.

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In Theorem 5.1 we can put a stronger norm on Z which also includesderivatives with respect to . In order to do so we define the derivatives Dk

e as in(1.5), i.e. we extend functions on Z to all of (R" — {0}) x R1 by homogeneity ofdegree — 1. Then we define for a an integer

For a > 0 real we define H (Z) by interpolation, i.e. we put for 0 < < 1 and a,integers > 0

As in (VII.4.6) this definition is independent of the choice of a, because of thereiteration Lemma VII.4.3.

THEOREM 5.2 The norms are equivalent on range (R) fora > 0, R being considered as an operator on

Proof We first show the equivalence for a an integer > 0. From (1.5) we have

hence

From Theorem 5.1 we get with some constant C1 (a, n)

Now choose such that x = 1 on . Then, and sincemultiplication with the function xk x is a continuous operation in Sobolevspaces (compare Lemma VII.4.5) we find that

Combining this with (5.6) we get

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where c3 (a, n) is an other constant. Now we use Theorem 5.1 again to estimate thenorm of fby the Ha(Z)-norm of Rf, obtaining

On the other hand we obviously have

with some positive constant c5 (a, n). (5.7), (5.8) show that the normsHa (Z), Ha (Z) are equivalent on range (R) for a an integer > 0. In order toestablish the equivalence for arbitrary real non-negative order we use aninterpolation argument. Let 0 < a < be integers and let y = a(l — 5) 4- withsome [0,1]. From Theorems 5.1 and (5.7) we see that

continuously for = 0,1. Hence, by Lemma VII.4.2, R is continuous for0 < < 1, i.e.

with some constant c6 (y, n). Using Theorem 5.1 we can estimate the right-handside by , obtaining

The opposite inequality follows directly by interpolation since for a an integer>0,

on all of H (Z).As a consequence the first half of Theorem 5.1 holds also for the stronger norm

H (Z):

THEOREM 5.3 For a > 0 there exist positive constants c (a, n), C (a, n) such thatfor

II.6 The Attenuated Radon Transform

In R2 we define for functions with compact support the attenuated Radontransform

Here we denote by - the unit vector perpendicular to 0 S1 for which del= 4-1. For 6 = (cos , sin )T, = (— sin , cos )T. u is a real function on R2

which plays the role of a parameter. Ru is the integral transform in emission

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tomography, see 1.2. In this section we extend as far as possible the results about Rin the previous sections to Ru.

A special case occurs if the function u has a constant value u0 in a convexdomain containing the support of /.

If denotes that point in which the ray starting at s with directionhits the boundary of , then

for . Therefore,

Tuo is known as the exponential Radon transform.In the following we extend some of the results for R to Ru, Tu. The proofs are

omitted since they are obvious modifications of the proofs for R.The projection theorem for Tu reads

This formula is not as useful as Theorem 1.1 since it gives f on a two-dimensionalsurface in the space C2 of two complex variables while the Fourier inversionformula integrates over R2.

Theorems 1.2, 1.3 hold for T,, as well:

Here, the dual operators are

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Theorem 1.5 extends even to Ru in the form

For Tu this reduces to

This formula corresponds to Theorem 1.4 and is the starting point for areconstruction method of the p-filtered layergram type, see V.6. A Radon typeinversion formula for Tu is obtained in the next theorem.

THEOREM 6.1 Letf (R2). Then,

where is the generalized Riesz potential

Proof We start out from (6.3), trying to determine the function g in that formulasuch that is a constant multiple of the function.

For some b > \u\ we put

and we compute

In the inner integral we put 9 = (cos p, sin )T, x = r (cos , sin ) obtaining

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see (VII.3.17). Hence

where we have used (VII.3.25). Comparing this with (VII. 1.3) we see that

is an approximate -function in the sense that pointwise in as. From (6.3) we have

It follows that

where we have used R4 of VII. 1. This is our inversion formula for Tu.

Theorem 6.1 is an extension of Theorem 2.1 to Tu for a = 0. It is the basis for afiltered back-projection algorithm for the inversion of Tu, see V.I. As for R, theactual numerical implementation starts out from (6.6).

Next we consider the range of Ru.

THEOREM 6.2 Let f, u e (Rn). Then, for k > m > 0 integers, we have

where = (cos , sin )T.

Proof The theorem follows essentially by considering the Fourier expansions

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where h is a function on Z. We have

Putting y = — we obtain

We compute the Fourier transform of ,. With = p(cos sin )T we get

where we have used (VII.3.16). From (VII.3.27) we get

From (6.7) and R4 of VII. 1 we get

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For q we have

From Theorem 1.4 we get

The similarity between (6.8) and (6.9) is the core of the proof. We define twofunctions h+, h- on Z by

and compute the Fourier coefficients u1,(X) of the function

From (6.9), (6.8) we get

and this vanishes for l > 0. Hence the Fourier expansion of u+ contains onl>terms with l <, 0, and so does the Fourier expansion of exp {u +}. Since (x )m,m > 0 an integer, is a trigonometric polynomial of degree < m, the Fourieiexpansion of

contains for k > m only terms of positive order. Integrating over [0, 2 ] andinserting the explicit expression for u+ we obtain for k > m

But this can be written as

In the same way we see that

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for k> m. R* being the dual of Ru it follows that for fe (R 2)

for k > m.We give (6.10) a more explicit form. Making use of Theorem 1.1 we have

For the Hilbert transform H we have

see (VII.1.11). Combining the last two formulas with the definition of h± yields

or

Using this in (6.10) we obtain the theorem.

Theorem 6.2 generalizes Theorem 4.1 to the attenuated Radon transform: ifH = 0, then Theorem 6.2 states that

for k > m. This means that the inner integral is a trigonometric polynomial ofdegree < m.

II.7 Bibliographical Notes

Most of the results for R and P extend to the general k-plane transform whichintegrates over k-dimensional subspaces, see Solmon (1976). In this way a unifiedtreatment of R (k = n — 1) and P (k = 1) is possible.

The inversion formulas of Theorem 2.1 in the general form given here has firstbeen obtained by Smith et al (1977). Special cases have been known before, mostnotably Radon (1917) for n = 2, and also John (1934), Helgason (1965), Ludwig(1966). Theorem 2.2 has been obtained by Ludwig (1966), Theorem 2.3 by Deans(1979). The case n = 2 was been settled in 1963 by A. Cormack (1963,1964) whoreceived for this work the 1979 Nobel price for medicine, jointly with G.Hounsfield. However, it has also been used in a completely different context byKershaw (1962, 1970) as early as 1962.

Theorem 3.1, with a completely different proof, is due to Helgason (1965). Intwo dimensions it is due to Cormack (1963). The other uniqueness theorems inSection 3 have been obtained, even for non-smooth functions, in Hamaker et al.,(1980), Smith et al. (1977); see also Leahy et al (1979).

Theorem 4.2 is due to Helgason (1980). Different proofs have been given byLax and Phillips (1970), Ludwig (1966), Smith, et al. (1977) and Droste (1983).

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Theorem 4.3 has been obtained by Solmon (1976) in the L2 case and by Helgason(1980) in the case given here. Solmon also observed that the condition of compactsupport in this theorem cannot be dropped (oral communication).

Sobolev space estimates such as given in Theorem 5.1 have been obtained inNatterer (1977, 1980) for n = 2 and by Smith et al. (1977), Louis (1981), Hertle(1983). That paper contains also the estimate of Theorem 5.3, see Natterer (1980)for the case n = 2.

The reduction of Ru to Tu for constant u is due to Markoe (1986). Theinversion formula for the exponential Radon transform has been given by Tretiakand Metz (1980). An inversion procedure based on (6.1) and Cauchy's theoremhas been given in Natterer (1979). The consistency conditions for the range of theattenuated Radon transform given in Theorem 6.2 have been used to solve insimple cases the identification problem in ECT, see 1.2: given g = Ruf

, find uwithout knowing /. For this and similar problems see Natterer (1983a,b, 1984).Heike (1984) proved the analogue of Theorem 5.1 for Ru, provided Ru isinvertible. The invertibility of Ru has not yet been settled in general.

The attenuated Radon transform is a special case of the generalized Radontransform

where the Lebesgue measure dx has been replaced by the smooth positivemeasure (x, , s)dx. Quinto (1983) proved invertibility if R is rotationinvariant, Boman (1984b) if is real analytic and positive, Markoe and Quinto(1985) obtained local invertibility. Hertle (1984) showed that R is invertible if

(x, 0, s) = exp ( (x, )} with depending linearly on x. On the other hand,Boman (1984a) showed that R is not invertible in general even if is a Cofunction.

Another type of generalized Radon transforms integrates over manifoldsrather than over planes. As standard reference we mention Helgason (1980) andGelfand et al. (1965). Funk (1913) derived a Radon type inversion formula forrecovering a function on S2 from integrals over great circles. John (1934),Romanov (1974), Cormack (1981), Cormack and Quinto (1980), Boman (1984b)and Bukhgeim and Lavrent'ev (1973) considered special families of manifolds.

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I l lSampling and ResolutionIn this chapter we want to find out how to properly sample Pf, Rf for somefunction / We have seen in Theorem II.3.7 that / is basically undetermined byPf( j, x) for finitely many directions even in the semi-discrete case in which xruns over all of . Therefore we have to restrict / somehow. It turns out thatpositive and practically useful results are obtained for (essentially) band-limitedfunctions /. These functions and their sampling properties are summarized inSection III.l. In Section III.2 we study the possible resolution if the Radontransform is available for finitely many directions. In Section III.3. we find theresolution of some fully discrete sampling schemes in the plane.

III.l The Sampling Theorem

One of the basic problems in digital image processing is the sampling of imagesand the reconstruction of images from samples, see e.g. Pratt (1978). Since animage is described by its density function / the mathematical problem is todiscretize / and to compute / from the discrete values.

The fundamental theorem in sampling is Shannon's sampling theorem, seeJerry (1977) for a survey. It deals with band-limited functions. We give a very briefaccount of band-limited functions which play an important role in communi-cation theory.

A function in Rn is called band-limited with bandwidth b (or b-band-limited) ifits Fourier transform is locally integrable and vanishes a.e. outside the ball ofradius b. The simplest example of a band-limited function in R1 is the sine-function

54

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Since it is the inverse Fourier transform of a function vanishing outside[—1, 4-1] its bandwidth is 1. In Rn we define the sine-function by

sinc(x) = sinc(x1) . . . sinc(xn)

where x = (x1, . . . , Xn,)T. The function sincb(x) = sinc(bx) is another example of

a band-limited function, and we have with x the characteristic function of[-1,+1]n

In Fig. III.l we show the graph of sincb. It is positive in |x | < /b and decays in anoscillating way outside of that interval. In the jargon of digital image processing,sincb represents a detail of size 2 /b. Hence, a b-band-limited function containsno details smaller than 2 /b, and for representing details of this size one needsfunctions of bandwidth at least b. We also say that details of size 2 /b or less in animage with density / are described by the values of f(<!;) for | | > b while thevalues off( ) for ) | < b are responsible for the coarser features. Therefore thevariable is sometimes called spacial frequency.

FIG. III.l Graph of sincb

Since images are usually of finite extent, the density functions we have to dealwith have compact support. The Fourier transform of such a function is analyticand cannot vanish outside a ball unless it is identically 0. Thus, image densities areusually not band-limited in the strict sense. Therefore we call a function /essentially b-band-limited if f( ) is negligible for \ \>b in some sense.Essentially band-limited functions admit a similar interpretation as (strictly)band-limited functions.

The Shannon sampling theorem, in its simplest form, reads as follows.

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THEOREM 1.1 Let / be fe-band-limited, and let h < /b. Then, / is uniquelydetermined by the values f ( h k ) , k Z", and, in L2(Rn),

Moreover, we have

in - If 9 is another fe-band-limited function, we have

Proof Since f vanishes outside [ — , (1.2) is simply the Fourierexpansion of f in [ — ( , see VII. 1, and (1.3) is ParsevaFs relation.Multiplying (1.2) with the characteristic function Xh/ of yields

in all of R". Since this series converges in L2(R") we can take the inverse Fouriertransform term by term. The result is (1.1).

A few remarks are in order. The condition h < /b is called the Nyquistcondition. It requires that / be sampled with a sampling distance at most half ofthe smallest detail contained in /. If it is satisfied, / can be reconstructed from itssamples by means of the sine series (1.1), and this reconstruction process is stablein the sense that

see (1.3) for f= g. From (1.2), (1.3) we see that under the Nyquist condition,Fourier transforms and inner products can be computed exactly by thetrapezoidal rule.

If the Nyquist condition is over-satisfied, i.e. if h < /b, then / is called over-sampled. The next theorem deals with that case.

THEOREM 1.2 Let / be fc-band-limited, and let h < /b. Let y (Rn) vanishfor |x| > 1 and

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Then,

Proof Again we start out from (1.2) which holds in [ — ( ), ]n- The right-hand side of (1.2) is the periodic extension of f with period 2n/h, whose support is

Since supp (f) [-b,b]n, (1.2) holds in [-a,a]" where a = 2 /h-b> /h.Thus, if is 1 on and 0 outside [-a, a]n we have

in all of Rn. If y is as in the theorem, then

with ya satisfies the above conditions, and the theorem follows by aninverse Fourier transform.

The significance of Theorem 1.3 lies in the fact that (1.4) converges much fasterthan the sine series (1.1). The reason is that the function , being the inverseFourier transform of a -function, decays at infinity faster than any power of\x\. This means that for the computation of / from (1.4) one needs to evaluateonly a few terms of the series, i.e. an over-sampled function can be evaluated veryefficiently. For a more explicit version of Theorem 1.2 see Natterer (1986).

If, on the other hand, the Nyquist condition is not satisfied, i.e. if h > /b, then/ is called under-sampled. This is always the case if / is not band-limited at all, i.e.if b = . If f is nevertheless computed from its sinc series

we commit an error which we will now consider.

THEOREM 1.3 Let f . Then, there is a L -function xx with such that

We also have

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If f,ge then

where the discrete convolution is defined by

Proof (1.6) is simply (VII.1.4). To prove (1.7) we use R2 from VH.l to compute

where we have applied (VII.1.4) to g. Hence

and (1.7) follows by R4 of VH.l.Putting

we have from (VII.1.4)

in [ — ( /h), /h]n. Comparing this with the Fourier transform of Shf we find that

with Xh/x the characteristic function of [ — ( /h), /h]n. Now we decompose g intof+ a with

hence

Taking the inverse Fourier transform we obtain

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Inserting a and putting n = — (2 /h)l the second integral becomes

where

The theorem follows with

We want to make some remarks.

(1) If / is fr-band-limited and h < /b, thenf vanishes outsideand we obtain Shf = f, i.e. (1.1). If / is essentially b-band-limited in the sense that

and h < /b, then

i.e. we have an error estimate for the reconstruction of an essentially b-band-limited function by its sinc series.(2) The spectral composition of the error Shf—f is more interesting than theabove estimate. According to (1.8) it consists of two functions.

The Fourier transform of the first function is (xh/ — 1) which vanishes in, i.e. this function describes only details of size less than 2h.

According to our interpretation of Shannon's sampling theorem, this part of theerror is to be expected since for the proper sampling of such details a samplingdistance less than h is required.

The Fourier transform of the second function is a which vanishes outside[ — ( i.e. this function describes only features of size 2h and larger.

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Thus we see that under-sampling not only produces spurious details twice as bigor smaller than the sampling distance but also global artefacts perturbing, oraliasing, the image.

In order to prevent aliasing one has to band-limit the function prior tosampling. This can be done by filtering.

Let F be a b-band-limited function and put

f f = F * f

Then fF is also b-band-limited and can be properly sampled with step-sizeh < /b. In this context, F is called a low-pass filter with cut-off frequency b. Anexample is the ideal low-pass filter which is defined by

F has been computed in (VII. 1.3).The sinc series renders possible error-free evaluation of properly sampled

functions. However, the sinc series—even in the generalized form of Theorem1.3—is rather difficult to compute. Therefore we shall make use occasionally ofsimple B-spline interpolation.

Let x be the characteristic function of i.e. x is 1 in that interval and 0outside. Let

B = x * • • • * X (k factors).

B is called a B-spline of order k. Obviously, B is (k — 2) times continuouslydifferentiate, vanishes outside [ —(k/2), k/2] and reduces to a polynomial ofdegree k — 1 in each of the intervals [l,l + 1] for k even and for k odd,/ being an integer.

Let (R1) and h > 0. With B1/h (s) = B(s/h) we define

and consider IHg as an approximation to g. For k = 1(2), Ihg is the piecewiseconstant (linear) function interpolating g.

In the following theorem we shall see that the effect of approximating a band-limited function g by Ihg is equivalent to applying a low-pass filter and adding ahigh frequency function.

THEOREM 1.4 Let g be band-limited with bandwidth /h. Then,

where for \ \ ̂ /h, and for a, ^ 0

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The semi-norm defined by

Proof With R2 from VII. 1 we compute

where we have used (VII. 1.4). With Rl, R4 from VII. 1 we obtain

Hence, from (1.10),

Since g has bandwidth /h, ah ( ) = 0 for \ \ < /h. The estimate on ah is obtainedas follows. We have

Since the support of the functions in the sum do not overlap we can continue

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The sum is

where we have used that 1+3 2 + 5 2 + . . . = 2/8, hence

The theorem now follows from R4 of VII. 1.

So far the Cartesian grid h Z" and the cube [ — ( /h), /H] played a special role:if/vanishes outside that cube, reconstruction of/from its values on the grid ispossible. Below we will prove a sampling theorem for arbitrary grids, replacingthe cube by a suitable set. We need a fairly general version of that theorem whichgoes back to Peterson and Middleton (1962). Here, a functionfon R" is sampledon the grid

where Wis a real non-singular n x n matrix, and the reconstruction is done by theformula

where xK is the characteristic function of some open set K e Rn.

THEOREM 1.5 Assume that the sets K + 2 (W- 1)Tk, k e Zn are mutually dis-joint, and let g e . Then, there is a L function xx vanishing on K with \ x\ < 1such that

Proof we repeat the proof of Theorem 1.3 for the function

f(x) = g(Wx)

and replace the cube (- ( /h\ /h)n by K.

We shall apply Theorem 1.5 to functions g which vanish on the grid

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{Wl:leZn}. Then,

If g is a 2a-periodic C function, we get from (VII. 1.7)

and we expect from (1.11)

to hold. This is in fact the case. To prove (1.12) we choose a real valued functionwe and apply Theorem 1.5 to the function wge , obtaining

where xx is a L function which vanishes on K and \ x\ < 1. Proceeding formallywe obtain

where g is to be understood as the extension of g from to the bounded Cfunctions, i.e.

(1.13) is easily justified by explicit computation. It follows that

and (1.12) follows by letting w 1 in a suitable way: for a radialfunction with mean value 1 we put Weobtain for each > 0

and (1.12) follows by letting e -> 0.

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If (R2) has period 2 in its first variable only and is in (R1) as afunction of the second variable, then

This can be established in the same way as (1.12).

III.2 Resolution

In this section we consider the Radon transform Rf of a function / which issupported in the unit ball of R" and which is essentially fc-band-limited in asense to be made precise later. We want to find out for which discrete directions0 e Sn -1 the function Refmust be given iffis to be recovered reliably, i.e. if detailsof size 2 /b are to be resolved. Since sampling conditions for P in threedimensions can be obtained by applying the sampling conditions for R to planeswe consider only R.

Let H'm be the set of spherical harmonics of degree < m which are even for meven and odd for m odd. We show that

We proceed by induction. For m = 0,1, we have H'0 = < 1 >, H1 = < x1, . . .,xn >, hence

i.e. (2.1) is correct for m = 0,1. Assume that it is correct for some m > 1. Then,from (VII.3.11),

This is (2.1) with m replaced by m + 2, hence (2.1) is established.Now we make a definition which is crucial for questions of resolution.A set A Sn-1 is called m-resolving if no non-trivial heH'm vanishes on A.

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If A is m-resolving then the number \A\ of elements of A satisfies

Conversely, if \A\ satisfies (2.2) it usually will be m-resolving, except if A lies on acertain algebraic manifold of degree m on S n - l .

THEOREM 2.1 Let A be m-resolving, let , and let > - 1/2. If R fvanishes for e A, then

with C the Gegenbauer polynomials (see VII.3) and .

Proof According to (II.4.2) we have the expansion

with hi e Hi. If R f = 0 for some it follows that h1, ( ) = 0 for each /. Since A ism-resolving it follows that hi = 0 for l < m, hence the theorem.

Theorem 2.1 tells us that the expansion of R fin Gegenbauer polynomialsstarts with the (m + l)st term. For m large, R fis therefore a highly oscillatingfunction. In fact, we shall see in the next theorem that | (R f) | is negligible in aninterval only slightly smaller than [ —m, +m] for m large. In order to formulatestatements like this we introduce the following notation. For 0 < < 1 and b > 0we denote by n( , b) any quantity which admits an estimate of the form

provided b > B( ), where ( ), C( ), B( ) are positive numbers. Thus, for < 1fixed, n( , b) decays exponentially as b tends to infinity. We shall use the notationn in a generic way, i.e. different quantities are denoted by the same symbol n if theyonly satisfy (2.4). Specific examples of functions n are

as can be seen from (VII.3.20). In fact, a short calculation shows that

Since for A > 0, d > 0 and A — d/m > 0

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as can be as seen from the inequality

the functions

also satisfy (2.4). Of course, the functions

fulfil (2.4) if r\ does.In order to express what we mean by an essentially band-limited function we

introduce the notation

For 0 < 3 and b > 1 we have

This can be seen as follows. If h is a non-negative function, then

whenever the integrals make sense. To establish (2.10) we write

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which holds for b > 1. This proves (2.10). Applying (2.10) with

and integrating over Sn- 1 yields (2.9).

THEOREM 2.2 Let A be m-resolving and let/e . If R fvanishes for e A,then

forf leS n - 1 .

Proof We use Theorem 2.1 with = 0, obtaining

Taking the Fourier transform term by term we obtain from (VII.3.18)

Integrating it follows from (2.11) that

where we have used (2.5) and (2.7). This is the theorem with n( , m)

By means of Theorem II. 1.1, Theorem 2.2 translates into the correspondingtheorem forf

THEOREM 2.3 Let A be m-resolving, and let . If R fvanishes for e A,then

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Proof From Theorem II. 1.1 and Theorem 2.2,

withn from Theorem 2.2. Since mn- l n ( , m) satisfies (2.4) if n does, the theoremfollows.

An easy consequence is

THEOREM 2.4 Let A be m-resolving and let/e . If R fvanishes for 0 e A,then

Proof We have

where we have used Theorem 2.3. It follows that

and the theorem follows by solving for

The last two theorems clearly give an answer to the question of resolution.Theorem 2.3 says that the Fourier transform of a function on whose Radontransform vanishes on a m-resolving set is almost entirely concentrated outsidethe ball of radius dm around the origin, provided m is large. In the light of SectionIII.l this can be rephrased for close to 1 somewhat loosely as a statement onresolution:

A function on whose Radon transform vanishes on a m-resolving set doesnot contain details of size 2 /m or larger.

Similarly, Theorem 2.4 admits the following interpretation. If a function /on

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Q" is essentially b-band-limited in the sense that (f, b) is negligible, then/can berecovered reliably from the values of Rfon a m-resolving set, provided b < mand b large. Or, in a less exact but more practical fashion: a function on whichdoes not contain details of size 2 /b or smaller can be recovered reliably from thevalues of its Radon transform on an m-resolving set, provided that m > b.

To conclude this section we want to find out which sets A are m-resolving. Wehave found in (2.2) that, apart from exceptional cases, a set is m-resolving if andonly if it contains

directions. So, for m large, we may think of an m-resolving set as a collection ofm"~l/(n — 1)! directions. For n = 2, this is in fact correct. For, let

In order that A be m-

resolving we must have p > dim and must

not be the zeros of a function From (VII.3.14) we know that such afunction hm is a trigonometric polynomial of the form

where 0 = , and ' denotes summation over k + m even only. If hm

vanishes for p > m mutually different angles then hm = 0. Hence, forn = 2, A is m-resolving if and only if it contains p > m mutually differentdirections in the angular interval [0, ]. This means that we can reconstructreliably functions of essential bandwidth b < p from p mutually differentdirections in [0, ]. No assumption on the distribution of these directions isneeded. For instance, the directions could be concentrated in a small angularinterval. This is the discrete analogue to the uniqueness result of Theorem II.3.4.

Now let n = 3. Here, the condition that A be m-resolving means that

1 = (m + 2) (m + 1) and that no non-trivial function hm of

the form (see (VII.3.15))

vanishes on A, where are the spherical coordinates of 0. We show that the(m + I)2 directions 0jl with spherical coordinates (see VII.2)

form a m-resolving set. The proof depends on the fact that a function in H'm, being

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a trigonometric polynomial of degree m on a circle on S2, can have only 2m zeroson a circle, unless being there identically zero. For each i = 0, . . . , m consider thegreat circle Ci on which or If vanishes on (2.13) it has2m + 2 zeros on each Ci namely those with spherical coordinates ,j = 0, . . . , m, and its antipodals. Hence hm = 0 on each of the great circles Ci . Itfollows that hm has 2m + 2 zeros on each horizontal circle on S2, i.e. hm = 0 on S2.This proves that (2.13) is m-resolving. Since, for (m + l)J = 0, . . . , m, thefunction

which is in H'm + 1 vanishes on (2.13), (2.13) is certainly not (m+ l)-resolving ingeneral.

We conclude that / can be recovered reliably from the directions (2.13),provided / is essentially fc-band-limited with b < m.

From (2.12) we expect m-resolving sets with (m + 2) (m + 1 )/2 directions to exist.A possible candidate for such a set, for m even, is

For m = 2 we can show that (2.14) is m-resolving: if h vanishes on (2.14) it has fourzeros on each great circle Ci on which , namely those withspherical coordinates and its antipodals. If h 0 at the North Polethese are the only zeros of h on Ci i = 0,1, 2. Thus h has six sign changes alongeach circle with fixed in ( ) what is impossible. If h = 0 at the North Pole,then h has six zeros on Ci , hence h = 0 on Ci, hence h — 0 on S2.

So far we have assumed that Re f (s) is known for e A and for all values of s. Inpractice we know R f ( s ) for finitely many values of s only. In view of TheoremII.l.l, R fis essentially b-band-limited iffis. This suggests to apply Theorem 1.3to R /, with the result that the sampling is correct if the Nyquist condition issatisfied, i.e. if R f(sl) is known for

From (2.12) we know that the number p of directions we need to recover such afunction/is essentially p > b n - l /(n — 1)! Thus for the minimal numbers p, q wehave approximately

But we have to be cautious: R f(sl) = 0 implies only that R fis small, but we donot know how large Rfcan be if R fis small for 6 e A. In fact, we shall see in VI.2that there is no reasonable estimate of Rfin terms of Re f, e A if A is a propersubset of the half-sphere, even if A is infinite.

In order to obtain a positive statement we would have to put a stabilityrequirement on A which guarantees that heH'mis small on all of Sn -l if it is smallon A. However, we will not pursue this approach further. Rather we shall considerin the next section some specific fully discrete sampling schemes.

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III.3 Some Two-Dimensional Sampling Schemes

In this section we study the resolution of some sampling schemes—or, in thejargon of CT, scanning geometries—in the plane which are fully discrete andwhich are actually used in practice. We assumefto be supported in the unit disk

2 and to be essentially b-band-limited in an appropriate sense.We start with the (standard) parallel scanning geometry. Here, for each of the

directions 0l . . . , 0p uniformly distributed over the half-circle, R0if is sampledat 2q+ 1 equally spaced points st. We have

From the discussion at the end of Section III. 2 we guess that the correct samplingconditions are p > b and q > b/ . Below we shall see that this is in fact the case.Surprisingly enough, this scanning geometry is by no means optimal. We shall seethat the interlaced parallel geometry which samples the functions R fonly atpoints sl with l +j even has the same resolution as the standard parallel geometrywith only one-half of the data.

Our approach will be based on Theorem 1.5 applied to Rfconsidered as afunction on R2, the crucial point being the shape of the (essential) support of theFourier transform of that function viewed as a periodic distribution.

THEOREM 3.1 Let f e , and let

For 0 < 9 < 1 and b > 1, define the set

in R2, see Fig. III.2. Let W be a real non-singular 2x2 matrix such that the setsK + 2 ( W - 1 ) l, l Z2, are mutually disjoint. If g(Wl) = 0 for l Z2, then

Proof Let g be the one-dimensional Fourier transform of g with respect to s andgk the kth Fourier coefficient of g, i.e.

gk can be computed by Theorem II. 1.1 and the integral representation (VII.3.16)

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for the Bessel functions. We obtain

where we have put

We deduce the estimates

where we have assumed for convenience that |f/||Ll = 1.From Fig. III.2 we see that

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For we get from (3.4) and (2.7) immediately

In 2 we have no more than 2b/ terms. From (3.3) we get

In 3 we use (3.3) with b replaced by \k\ to obtain

where we have used (2.9). Combining the estimates for — we obtain

According to our hypothesis we can apply Theorem 1.5 in the form of (1.14) to g,K and W, obtaining

and this is (3.2).

Theorem 3.1 tells us that a function/which is essentially b-band-limited in thesense that (f,b) is negligible can be recovered reliably from the values of g = R/

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on the grid Wl, Z2 if the sets K + 2 (W- l)Tl, le Z2 are mutually disjoint. Thisholds for arbitrarily close to 1 and b large.

For the standard parallel geometry we have

and from Fig. III.2 we see that the sets K + 2 ( W - l ) T l , /e Z2 are mutuallydisjoint if

These are the sampling requirements for the standard parallel geometry. There isno point in over-satisfying either of these inequalities. Therefore it is reasonableto tie p to q by the relation p = q/9.

FIG. III.2 The set K defined in (3.1) and some of its translates (dashed) forthe interlaced parallel geometry. Crosses indicate the columns of 2 ( W - 1 ) T

(9 = 4/5, 9' = 3/(2 - 9) = 2/3).

In order to obtain the interlaced parallel geometry we choose

In Fig. III.2 we have drawn some of the sets K + 2n(W - 1)T / . We see that thesesets are also mutually disjoint and they cover the plane more densely than thecorresponding sets for the standard parallel geometry. The grid generated by

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FIG. III.3 The sampling points of the interlaced parallel geometry in the— s plane. The crosses indicate the columns of W ( ' = 4/5).

is shown in Fig. III.3. It is identical to the interlaced parallel geometry with

provided b/ ' is an even integer. The sampling conditions for the interlacedparallel geometry are therefore

Note that ' is an arbitrary number in (0, 1) as is 9 since for any ' e (0, 1) we canfind 9 e (0, 1) with 9' < 9/(2 - 9).

We remark that the first of these conditions which we stipulated for thestandard parallel geometry simply to avoid redundancy is mandatory here since itprevents the sets K + 2 (W- 1)Tl, le Z2 from overlapping.

Now we turn to fan-beam scanning geometries. Let r > 1 and let

be a point (the source) on the circle with radius r

around the origin. With L ( ) we denote the straight line through a whichmakes an angle a with the line joining a with the origin. L( ), which is the

dashed line in Fig. II 1.4, is given by where s, satisfy

The first relation is obvious since s6 is the orthogonal projection of a onto <0>.For the second one it suffices to remark that the angle — between a and 9 is

/2 —a. Note that for each line L meeting 2 there are precisely tworepresentations in terms of the coordinates a, as well as in the coordinates s, ,but the relation (3.7) between these coordinate systems is nevertheless one-to-one.

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FIG. III.4 Fan-beam scanning. The reconstruction region2 is the disk with radius 1, while the sources are on the

circle with radius r > 1 (r = 2 in the figure). CompareFigure I.I.

In fan-beam scanning, Da fis sampled for 2q + l equally spaced directions forp sources a1 . . . ,ap uniformly distributed over the circle of radius r > 1 aroundthe origin. This amounts to sampling the function g in (3.8) below at ( , a1),j = 1, . . . , p, l = - q, . . . , q where

Of course only the lines L meeting 2 need to be measured, i.e. those forwhich | | < a(r) = arcsin (1/r). Thus the number of data is practically

A special case arises for p = 2q. In this case, the scanning geometry is made up ofthe p(p— l)/2 lines joining the p sources, see Fig. III.5. Since this geometry arises,at least in principle, in positron emission tomography (PET), see 1.2, we speak ofthe PET geometry. It can be rearranged to become a parallel geometry with non-uniform interlaced lateral sampling, see Fig. III.5. The directions 0, for thisparallel geometry are, for p even,

and, for each j, the lateral sampling points s are

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FIG. III.5 PET geometry with p = 8. Left: Measured lines. Right:Rearrangement into a parallel geometry. Solid line: j even. Dashed line:

j odd.

Our analysis of fan-beam scanning is analogous to our treatment of the parallelcase but technically more complicated.

THEOREM 3.2 Let fe , and let

For 0 < < 1 and b > 1 define the set

in R2, see Fig. III. 6. Let W be a real non-singular 2x2 matrix such that the setsK + 2 ( W- l )Tl, le Z2 are mutually disjoint. Then, i f ( W l ) = for / e Z2, wehave

where b' = b min (1, In r}.

Proof We consider as a function on R2 with period 2n in and in a. ItsFourier coefficients are

The gkm will be computed by transforming the integral to the coordinates , s. In

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FIG. III.6 The set K of (3.9) for r = 7/5 and = 4/5. Thecrosses indicate the columns of 2 ( W - l ) i for the fan-beamgeometry. Dashed lines separate the ranges of the sums in (3.16).

the inner integral we substitute = + a — /2 for a. This yields with

where we have used that, due to the evenness of Rf, the integrand has period .Here we use Theorem II. 1.1 to obtain

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Inserting this and interchanging integrations we get

According to (VII.3.16) the inner integral is

hence

The integral is broken into an integral over \ \ < 0 with O chosen suitably lateron, and the rest whose absolute value can be estimated by 2 - 1 ( f , 0). Hence

In the inner integral we express / by the Fourier integral, obtaining

where x = Again with (VII.3.16) we get

Inserting this in (3.11) yields

We use (3.12) to estimate gkm for (k, 2m) K. For convenience we assume thatThe estimates depend on a judicious choice of 0.

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If r\k\ > \k-2m\ we put O = \k\, obtaining from (2.5)

On the other hand, if r \k \ < \k — 2m\ we choose O = \k — 2m\/r to obtain, againfrom (2.5),

If \k-2m\ < \k\ we put 0 = . From (VII.3.21, 23) we see that for |x| < 1

where we have used that C ( r) is decreasing, hence

for \k-2m\<\k\.Now consider the sum

the ranges of summation being

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see Fig. III.6. In we have \k — 2m\ > r\k\. Hence we get from (3.14) for fixed k

where we have used (2.7) and (2.9), the latter one with replaced by /r. Since atmost 2b/ values of k appear in we get for fixed k

In 2 we have |k — 2m\ < r\k\, and the m sum contains at most r\k\ terms. Hence,from (3.13), (2.7)

In 3 we have | k — 2m \ > r \ k \ as in . Proceeding exactly as in we get for the msum in 3

hence

see (2.7), (2.9). Finally, in 4, we have at most \k \ + 1 terms in the m sum, and all of

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these terms can be estimated by (3.15). Hence, with a = (1/ — \)rb

by (2.6). It follows that

where n satisfies (2.4).Combining the estimates for . . . , we get from (3.16)

Now we can apply Theorem 1.5 in the form of (1.12) to g, K and W, obtaining

and this is (3.10).

Theorem 3.2 admits an interpretation analogous to Theorem 3.1. For the fan-beam geometry we have

From Fig. III.6 we see that the sets K + 2(W - 1)T/, / e Z2 are mutually disjoint if

These are the sampling conditions for the fan-beam geometry. For the minimalvalues of p, q we have p = (4/r)q.

For r close to 1, a denser non-overlapping covering of the plane is possible. Weenlarge K by including the four tips which have been cut off by the horizontallines k = ± b/ . The resulting set K' is shown in Fig. III.7. The translates of K' bythe grid generated by

are mutually disjoint. The matrix

defines a grid in the plane which is identical to the PET geometry if p = rb/ .

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FIG. III.7 The set K' (enclosing K of (3.9)) for the PETgeometry. The crosses indicate the columns of 2 ( W - l )T.Some of the translates of K' are also drawn (dashed), r and

are chosen as in Fig. III.6.

Therefore the sampling condition for the PET geometry is

with some < 1.We conclude this section by a survey on the scanning geometries discussed.

Table III.l contains the sampling conditions, an optimality relation which holdsif angular and lateral sampling rates are both minimal for a certain bandwidth,and the number of sampled line integrals for the optimal choice p, q. Forsimplicity we have replaced the numbers , by 1.

We see that the interlaced parallel geometry needs less data than the others.Therefore its relative inefficiency is normalized to 1. Second best is the PETgeometry (provided r is close to 1). We find it very satisfactory that a naturalsampling scheme such as PET is also an efficient one and even the most efficientone among the fan-beam schemes. Third best is the standard parallel geometry,

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TABLE III. 1 Scanning geometries. The sampling conditions refer to the reconstruction ofan essentially b-band-limited function with support in 2.

Scanninggeometry

Standardparallel

Interlacedparallel

Fan-beam

PET(r close to 1 )

Sampling Optimalcondition relation

p > b, q > b/ p = q

p > b, q > b/ p = q

p > 2b, p = 4q/rq > br/2

p > b —

Numberof data

b2

b2

4ra(r)b2

2

Relativeinefficiency

2

1

4ra(r),a(r) = arcsin(l/r)

2

Formula

(3.5)

(3.6)

(3.17)

(3.18)

while the fan-beam geometry has relative inefficiency between 4(r large) and2 (r close to 1).

III.4 Bibliographical Notes

The material in Section III.2 is based on Louis (1981a, 1982, 1984a, 1984b).Theorem 2.1 is taken from Louis (1984b), while the proof of Theorem 2.3 ispatterned after Louis (1984a). The 3-dimensional examples are taken from Louis(1982). For n = 2 a much more elaborate version of Theorem 2.3, with an analysisof the transition region \ \ ~ m, has been obtained by Logan (1975).

The basic approach of Section III.3 goes back to Lindgren and Rattey (1981).They found that what we call the interlaced parallel geometry has the sameresolution as the standard parallel geometry, with only one-half of the data. Theinterlaced parallel geometry has originally been suggested by Cormack (1978)based on a very nice geometrical argument. In the same way Cormack (1980)found that the PET geometry is optimal among the fan-beam geometries. Theconditions (3.5) have been given by Bracewell and Riddle (1967) and by manyothers who have not been aware of that work. A heuristic derivation of therelation p = q has been given by Crowther et al. (1970), see V.2. The samplingconditions (3.17) for fan-beam scanning appear to be new. Joseph and Schulz(1980) gave the non-optimal condition p > 2b/(l —sin a(r)), while Rattey andLindgren (1981), based on an approximation of the fan-beam geometry by aparallel one, obtained the approximate condition p > b for 1 < r < 3 andp > 2b/(l + 3/r) for r > 3. A treatment of the PET geometry as an interlacedparallel scheme reveals that it is the fan-beam analog of the interlaced parallelscheme. This may serve as an explanation for its favourable efficiency.

Of course, one can derive from Theorem 3.2 other fan-beam geometries simplyby finding grids which lead to non-overlapping coverings of the plane by thetranslates of K. We have not found scanning geometries in this way whichpromise to be superior to the existing ones.

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IVIll-posedness and AccuracyAll problems in CT are ill-posed, even if to a very different degree. Therefore, anyserious mathematical treatment of CT requires some fundamental notions fromthe theory of ill-posed problems. We give in Section IV. 1 a short introduction intothis field. In Section IV.2 we study the accuracy we can expect for CT problems,given the accuracy of the data and the number of projections. In Section IV.3 wederive the singular value decomposition of the Radon transform. The subject ofill-posedness is taken up again in Chapter VI on incomplete data problems.

IV.l III-Posed Problems

We discuss ill-posed problems only in the framework of linear problems inHilbert spaces. Let H, K be Hilbert spaces and let A be a linear bounded operatorfrom H into K. The problem

is called well-posed by Hadamard (1932) if it is uniquely solvable for each g Kand if the solution depends continuously on g. Otherwise, (1.1) is called ill-posed.This means that for an ill-posed problem the operator A -l either does not exist,or is not defined on all of K, or is not continuous. The practical difficulty with anill-posed problem is that even if it is solvable, the solution of Af = g need not beclose to the solution of Af = g if ge is close to g.

In the sequel we shall remove step by step the deficiencies of an ill-posedproblem. First we define a substitute for the solution if there is none. We simplytake a minimizer of \\Af- g\\ as such a substitute. This makes sense if g range(A) + (range (A )) . Then we dispose of a possible non-uniqueness by choosingamong all those minimizers that one which has minimal norm. This well definedelement of K is called the (Moore-Penrose) generalized solution of (1.1) and isdenoted by A+ g. The linear operator A*:K H with domain range(A)+ ( range (A) ) is called the (Moore-Penrose) generalized inverse.

THEOREM 1.1 f= A+ g is the unique solution of

A* Af= A*g

in range (A*).

85

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Proof f minimizes \\Af-g\\ if and only if

(Af-g,Au) = 0

for all u H, i.e. if and only if A* Af= A* g. Among all solutions of this equationthe unique element with least norm is the one in (ker (A )) = range (A*) .

We remark that the system A* Af = A* g is usually called the system of normalequations.

In general, A + is not a continuous operator. To restore continuity we introducethe notion of a regularization of A+. This is a family (Ty )y > 0 of linear continuousoperators Ty: K H which are defined on all of K and for which

on the domain of A +. Obviously, || Ty \\ as y 0 if A + is not bounded. Withthe help of a regularization we can solve (1.1) approximately in the followingsense. Let g K be an approximation to g such that \\g — g \\ . Let y ( ) be suchthat, as 0,

Then, as

Hence, Ty( ) g is close to A + g if g is close to g.The number y is called a regularization parameter. Determining a good

regularization parameter is one of the crucial points in the application ofregularization methods. We do not discuss this matter. We rather assume that wecan find a good regularization parameter by trial and error.

There are several methods for constructing a regularization.

(1) The Truncated Singular Value Decomposition By a singular value de-composition we mean a representation of A in the form

where (fk), (gk) are normalized orthogonal systems in H, K respectively and k arepositive numbers, the singular values of A. We always assume the sequence { k}to be bounded. Then, A is a linear continuous operator from H into K with adjoint

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and the operators

are self-adjoint operators in H, K respectively. The spectrum of A* A consists ofthe eigenvalues with eigenelements fk and possibly of the eigenvalue 0 whosemultiplicity may be infinite. The same is true for A A* with eigenelements gk. Thetwo eigensystems are related by

Vice versa, if (fk), (gk) are normalized eigensystems of A* A, A A* respectivelysuch that (1.3) holds, then A has the singular value decomposition (1.2). Inparticular, compact operators always admit a singular value decomposition.

THEOREM 1.2 If A has the singular value decomposition (1.2), then

Proof First we show that the series converges if g is in the domain of A+, i.e. ifg range (A) + (range (A) ) . For g = Av + u with u (range ( A ) ) we have

By (1.3), gk range (A) and A*gk = k f k . Hence

and it follows that

converges. Applying A* A term by term we obtain

Since, by (1.3), f+ range (A*) it follows from Theorem 1.1 that in factf+=A+g.

We see from Theorem 1.2 that A + is unbounded if and only if k 0 as k -> .

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In that case we can construct a regularization of A + by the truncated singularvalue decomposition

It follows from Theorem 1.2 that Ty g A + g as y 0, and Ty is bounded with

More generally we can put

where Fy ( ) approximates for a large and tends to zero as a 0. Fy is called afilter and regularization methods based on (1.5) are referred to as digital filtering.

The singular value decomposition gives much insight into the character of theill-posedness. Let ge be an approximation to g such that \\g — g \\ . Knowingonly g , all we can say about the expansion coefficients for A + g in Theorem 1.2 is

We see that for k small, the contribution of gk to A + g cannot be computedreliably. Thus looking at the singular values and the corresponding elements gk

shows which features of the solution fof (1.1) can be determined from anapproximation g to g and which ones can not.

(2) The Method of Tikhonov-Phillips Here we put

Equivalently, fy = Tyg can be defined to be the minimizer of

We show that as y 0. From Theorem 1.1 we have

It follows that

Let (E ) be the resolution of the identity associated with the self-adjoint operatorA* A, see Yosida (1968), ch. XI. 5:

Note that E = 0 for < 0. Using this integral representation in (1.7) we obtain

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Hence,

With P the orthogonal projection on ker (A* A) = k e r ( A ) we have

since f+ ker (A). Now it follows from (1.8) that in fact fy f+ as y 0.If A has the singular value decomposition (1.2) it is readily seen that fy has the

form (1.5) with

Thus the method of Tikhonov-Phillips is a special case of digital filtering.

(3) Iterative Methods Suppose we have an iterative method for the solution ofAf= g which has the form

with linear continuous operators Bk,Ck. Assume that fk converges to A + g. Foreach y > 0 let k(y) be an index such that k(y) -> as y 0. Then,

is obviously a regularization of A+. Here, the determination of a goodregularization parameter y amounts to stopping the iteration after a suitablenumber of steps. In general, an iteration method (1.9) which converges for g in thedomain of A + does not converge for all g. It rather exhibits a semi-convergencephenomenon: even if it provides a satisfactory solution after a certain number ofsteps, it deteriorates if the iteration goes on.

By means of regularization methods we can solve an ill-posed problemapproximately. In order to find out what accuracy we can get we have to make

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specific assumptions on A as well as on the exact solution f. In the context of CT,the following setting will be useful: we take H = L2 (

n), the unit sphere in Rn,and we make two assumptions:

(i) There is a number a > 0 such that, with positive constants m, M,

(ii) The exact solution f is in H ( n) for some B > 0, and

From (1.10) it follows that A- l exists but is unbounded as an operator from Kinto L2( ). This means that the equation Af= g is ill-posed in the L2( ) — K.setting, the degree of the ill-posedness being measured by the number a. Equation(1.11) is a smoothness requirement on the exact solution and represents a typicalexample of a priori information in the theory of ill-posed problems. In Theorem2.2 we shall use (1.11) to restrict the variation of f so as to avoid the indeterminacyresult of Theorem II.3.7.

Now assume that instead of g only an approximation g with \\g — g || isavailable. Then, our information on the exact solution can be summarized in theinequalities

If f1, f2 are elements satisfying (1.12) it follows that

With

we obtain

Therefore d (2 , 2p) can be considered as the worst case error in the reconstructionof f from the erroneous data g , given that (1.10), (1.11) holds. The asymptoticbehaviour of d ( , p) as 0 is clarified in the following theorem.

THEOREM 1.3 If (1.10) holds, then there is a constant c(m,a,B) such that

Proof From (VII.4.9) with y = 0 and a replaced by — a we get

If \\Af\\ , then

from (1.10). Hence, if and then

and the theorem follows.

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We observe that the right-hand side of (1.10) is irrelevant for Theorem 1.3 tohold. However, it makes sure that the estimate of Theorem 1.3 is best possible asfar the exponents of , p are concerned.

We interpret Theorem 1.3 as follows. If the data error of an ill-posed problemsatisfying (1.10), (1.11) is , then the true solution can be recovered with anaccuracy 0( B/(a+B)). Since B/(a + B) < 1 we always have a loss in accuracy. We callthe problem

Severely ill-posed if B/(a + B) is closed to 0.In that case the loss of accuracy is catastrophic. Typically,a = , i.e. the estimate (1.10) does not hold for any finitea,

Mildly ill-posed if B/(a + B) is close to 1. In that case we have only little lossin accuracy,

Modestly ill-posed if B/(a + B) is neither close to zero nor close to 1.

We shall see that this classification of ill-posed problems provides a simple meansfor judging problems in CT. Note that the degree of ill-posedness not onlydepends on the properties of the operator A as expressed by (1.10) but also on thesmoothness of the true solution.

A completely different approach to ill-posed problems comes from statistics.Here we think of f,g as families of random variables which are jointly normallydistributed with mean values f, g and covariances F, G, respectively, i.e.

with E the mathematical expectation, and correspondingly for G. Assume thatthere is linear relation

between f and g where A is a linear operator and n is a family of random variableswhich is normally distributed with mean value 0 and covariance . We assume fand n to be independent, n represents measurement errors. Then, (f, g) has themean value (f, g) and the covariance

As solution fB of (1.10) we take the conditional expectation of f having observed g.In analogy to the case of finitely many variables, see Anderson (1958), we have

fB is known as the best linear mean square estimate for f, see Papoulis (1965),Ch. 11. Note that for f= 0, F = I and = yI, (1.16) is formally the same as themethod of Tikhonov-Phillips (1.6). The choice of = yI corresponds to 'whitenoise' and is an appropriate assumption for measurement errors. The choice of Freflects the properties of f For f an image density a model frequently used in

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digital image processing is the isotropic exponential model

with constants , > 0, see Pratt (1978). This model assumes that theinterrelation between the grey levels of the picture at the points x,x', as measuredby the covariance between the density distributions f(x),f(x'), depends only onthe distance between x,x' and decays exponentially as this distance increases. Inthe next section we shall see that (1.17) translates into a smoothness assumptionfor a deterministic description of pictures.

IV.2 Error Estimates

The purpose of this section is to apply Theorem 1.3 to R and to P. In view ofTheorem II.5.1 these transforms satisfy (1.10) with a = (n —1)/2, a = 1/2,respectively. It only remains to find out what Sobolev spaces we have to take forthe picture densities / we want to reconstruct.

In view of the applications we want to reconstruct at least pictures which aresmooth except for jumps across smooth (n — l)-dimensional manifolds. A typicalpicture of this kind is given by the characteristic function f of . As in (VII.1.3) weobtain

with Jn/2 the Bessel function, see VII.3. Hence

Because of (VII.3.24) this is finite if and only if B < 1/2. Hencef HB if and only ifB < 1/2. We conclude that for the densities of simple pictures such as describedabove Sobolev spaces of order less than 1/2 are appropriate.

On a more scientific level we come to the same conclusion. Adopting theisotropic exponential model (1.17) we think of a picture as a family (f(x))x R

2 ofrandom variables such that, with f = Ef,

where we have added the C function a to express that the picture is of finiteextent. We want to compute E\(f— ) |2 for a real functionf We have

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Applying £ we get from (2.1)

With k(x) = e- |xl we obtain with rules R2 and R4 of VII. 1 after some algebra

For £ we obtain

where we have put x = r , = p . . The integral over Sn-1 is the case / = 0 of(VII.3.19):

This leads to

The latter integral is the case v = n/2 -1 of (VII.3.26). We finally obtain

with some constant cl (n). From (2.2) we obtain

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In view of (2.3) this is finite if and only if B < 1/2.These observations suggest one thinks of picture densities as functions in the

Sobolev space H ( ) with B close to 1/2. Since we are working with Cfunctions only we assume a picture density to be a function f C ( ) such that

with B close to 1/2 and p not too large.Now let

Remember that dR(2 , 2p), dp(2 , 2p) can be considered as worst case errors forour reconstruction problems with data error and a priori information (2.4).

THEOREM 2.1 There is a constant c(B, n) such that

Proof Since R, P satisfy (1.10) with a = (n - l)/2, a = 1/2, respectively it sufficesto apply Theorem 1.3 for these values of a.

Of course the estimates for R, P are the same for n = 2. We are interested invalues for B close to 1/2. Putting B = 1/2 for simplicity we obtain

In the terminology of Section IV.1, both reconstruction problems are modestlyill-posed (for dimensions 2,3). Note that the degree of ill-posedness is dimensiondependent for R but not for P. While a data error of size leads to an error in thereconstruction from line integrals of order 1/2, the error in the reconstructionfrom plane integrals is of order 1/3.

Besides inaccurate measurements a further source of errors is discretesampling. Let us assume that Rfis known only for a finite number of directions, .., and real numbers s 1 . . ., sq. We assume that the ( ,s1) cover Sn-1

x [— 1, +1] uniformly in the sense that

where the factor of has been introduced in order to be in agreement with

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(V.2.14). In analogy with the definition of dR( ,p) we put

Then, dR(h, 2p) can be considered as worst case error for the reconstruction offfrom Rf given at the points , s1, satisfying (2.7) under the a priori information(2.4).

THEOREM 2.2 Let B > 1/2. Then there is a constant c((B,n) such that

Proof Let ||f||HB( ) p and let g = Rf vanish on ( ,si), j = 1, . . . , p,/ = 1,. . . , q. According to Theorem II.5.3 we have

with some constant c l(B,n). We now apply Lemma VII.4.8 to = Sn- l x(-1,+1) and = {( j,si):j = 1, . . . , p, l = 1, . . ., q}. This is possible bymeans of local coordinates, identifying the Sobolev spaces HB+(n-1)/2(Z) andHB+ (n-1)/2 ( ) Since B + (n-1)/ 2 > n/2 we obtain

with some constant c2 (B, n). Now we have

Applying Theorem 2.1 we obtain

hence the theorem.

It is interesting to compare Theorem 2.2 with Theorem II.3.7. While the lattertheorem tells us that a function is essentially undetermined by finitely manyprojections—even if it is C —, Theorem 2.2 states that this indeterminacydisappears as soon as the variation of the function—as measured by its norm inH ( )—is restricted.

We remark that the estimates of the last two theorems are sharp as far as theexponents are concerned. This is easily shown in the case of Theorem 2.1. Itfollows from Natterer (1980a), Louis (1981a, 1984a) for Theorem 2.2.

IV.3 The Singular Value Decomposition of the Radon Transform

In Section IV.1 we have seen that the singular value decomposition is a valuabletool in the study of ill-posed problems. After some preparations we shall derive

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the singular value decomposition of the Radon transform as an operator fromL2(

n) into L2(Z,w1-n) where w(s) = (1 _s2)1/2, see Theorem II.1.6.To begin with we show that for , S n - 1

where [0, ] is the angle between , i.e. cos = . For the proof wemay assume that = (0,. . . , 0, l)Tand = ( ,cos )T with | | = sin . Then,with x = (x', xn)

T,

and

Since the measure of for n > 2 is

for we obtain from Fubini's theorem

This holds also for n = 2 since | | = 1. Since |0'| = sin , (3.1) is proved.Next we apply (3.1) to the Gegenbauer polynomials C m

n/2, i.e. the orthogonalpolynomials in [ — 1, +1] with weight wn-l which we have normalized such that

(t sin + s cos ) is a linear combination of terms of the form

Substituting (1 — s2)1/2u for t we obtain for each of these terms

For j odd, this is zero. For j even, this is of the form

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with some polynomial Pk of degree k. Hence,

with Pm a polynomial of degree, m. From the orthogonality properties of theGegenbauer polynomials it follows that

for / > m. From (3.1) we see that R R is a symmetric operator, hence (3.3) holdsalso for l < m. Thus Pm is orthogonal, in the sense of L2 ([ -1, + 1], wn - l) , to C1

n/2

for l m, i.e. Pm coincides, up to a constant factor, with Cmn/2. Now (3.2) assumes

the form

with some constant am( ) which we now determine by letting s -> 1. Weobtain from (3.1)

For later use we record the following consequence of (3.3): putting

in we have

In the evaluation of the constant we have made use of (VII.3.1) and of well-knownformulas for the -function.

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After these preparations we derive the singular value decomposition of R byconsidering the operator RR* where the adjoint is formed with respect to theinner product in the space L2(Z, w 1 - n ) . We obtain as in II.1

hence

Putting

(3.4) assumes the form

i.e. um is an eigenfunction of the self-adjoint operator R R with eigenvalueam( ). For h a function on Sn-l we therefore obtain

where Am is the integral operator

For h = y1 a spherical harmonic of degree l we obtain with the Funk-Hecketheorem (VII.3.12)

This means that the spherical harmonics of degree / are in the eigenspace of Am

with eigenvalue . Thus the operator RR* which is self-adjoint on L2(Z, w1 -n)

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has eigenfunctions Ytum with eigenvalues , and since the functions Ytum

constitute a basis of L2(Z, w l - n ) we have found all eigenfunctions andeigenvalues of RR*.

Since am is a polynomial of degree m, we have, in view of the orthogonalityproperties of the Gegenbauer polynomials, = 0 for / > m. Also, axml = 0 ifl + m odd since C1

(n-2)/2 is even for / even and odd for / odd. Inserting am from (3.5)into (3.8) we find for the positive eigenvalues

Now let

be a normalized eigenfunction of RR*. In agreement with the general procedureof Section IV.1 we compute the functions

Again we use the Funk-Hecke theorem (VII.3.12) to obtain

Let

Then, because of the orthogonality properties of the Gegenbauer polynomials,

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where qml is a polynomial of degree (m —/)/2. Since the functions gml areorthogonal in L2(Z, w1 - n) , the functions fml are orthogonal in L2( ). From

we see that

where Pk,l, k = 0,1,. . . are the polynomials of degree k with

Up to normalization P k , l ( t ) coincides with the Jacobi polynomialGk(l +(n- 2)/2, / + (n - 2)/2, t) see formula 22.2.2 of Abramowitz and Stegun(1970). For fml we finally obtain

Combining (3.10), (3.12), (3.8) yields the singular value decomposition of R. Notethat each singular value has multiplicity N(m,l), see (VII.3.11). With Yik,k = 1, . . ., N(m, /) a normalized orthogonal basis for the spherical harmonics ofdegree /, with

compare (3.10), the singular value decomposition of the Radon transform reads

The dash indicates that the l-sum extends only over those / with / + m even.Incidentally, Theorem 1.2 leads to the following inversion formula: if g = Rf,

then

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Note that fmlk is a polynomial of order m. Hence truncated versions of (3.14)provide us with polynomial approximations to /.

We conclude this section by computing explicitly the singular values for n = 2.From (3.9), (VII.3.2-3),

With and we get

Using

we finally obtain for 0 < / < m, m + / even

and the same result holds for / = 0.We see that 0 as m , but the decay is rather gentle, indicating that the

ill-posedness of Radon's integral equation is not pronounced very much. Thisagrees with our findings in Section IV.2.

IV.4 Bibliographical Notes

For a thorough treatment of ill-posed problems see Tikhonov and Arsenin (1977),Groetsch (1984). For an error analysis for the Tikhonov-Phillips method in aHilbert scale framework see Natterer (1984b).

The results of Section IV.2 have been obtained in Natterer (1980a) for n = 2,see also Bertero et al (1980). A result related to Theorem 2.2—with a differentmeasure for the variation of f—can be found in Madych (1980).

For the singular value decomposition of the Radon transform see Davison(1981) and Louis (1985). These papers consider also more general weightfunctions. The singular value decomposition for the exterior problem (compareVI.3) has been given by Quinto (1985). Louis (1985) gives a singular value analysisof the Tikhonov-Phillips method with a Sobolev norm for the Radontransform.

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VReconstruction AlgorithmsIn this chapter we give a detailed description of some well-known reconstructionalgorithms. We start with the widely used filtered backprojection algorithm andstudy the possible resolution. In Section V.2 we give an error analysis of theFourier algorithm which leads to an improved algorithm comparable in accuracywith filtered backprojection. In Section V.3 we analyse the convergenceproperties of the Kaczmarz method for the iterative solution of under- andoverdetermined linear systems. The Kaczmarz method is used in Section V.4 toderive several versions of the algebraic reconstruction technique (ART). Thedirect algebraic method in Section V.5 exploits invariance properties of thesampling scheme by using FFT techniques for the solution of the large linearsystems arising from natural discretizations of the integral transforms. In SectionV.6 we survey, some further algorithms which are either interesting from atheoretical point of view or which are of possible interest in specific applications.

V.1 Filtered Backprojection

The filtered backprojection reconstruction algorithm is presently the mostimportant algorithm, at least in the medical field. It can be viewed as a computerimplementation of the inversion formula

see (II.2.4). However, a different approach based on the formula

of Theorem II. 1.3 gives much more insight. The basic idea is to choose wb suchthat Wb approximates the -distribution. More precisely we assume Wb to be alow-pass filter with cut-off frequency b which we write as

where and for An example is the ideal low-pass filterfor for which

102

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see (VII.1.3) and Wb -> for b .Theorem II. 1.4 gives the relation between Wb and wb in terms of Fourier

transforms:

Since Wb is a radial function we drop the first argument in wb. For wb even weobtain

From our assumptions on it follows that

The evaluation of (1.2) calls for performing the one-dimensional convolution orfiltering operation wb * Rf for each direction in S n - l , followed by the applicationof the backprojection operator R*. This explains the name of the algorithm.

The filtered backprojection algorithm is a discrete version of (1.2). Letf C ( ), and let g = Rf be sampled at ( , s1), j = 1, . . . , p, / = - q, . . . , q,where Sn-1 and s1 = hl, h = l/q. Then, the convolution wb * g can be replacedby the discrete convolution

For computing the backprojection in (1.2) we need a quadrature rule on Sn- l

based on the nodes , . . . , and having positive weights apj. We shall assumethat this rule is exact in H'2m, the even spherical harmonics of degree 2m (see III.2)for some m, i.e.

for v H'2m. The backprojection in (1.2) can now be replaced by the discretebackprojection

The complete filtered backprojection algorithm now reads

Before giving more explicit versions of (1.10) we study the effect of the variousdiscretizations.

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THEOREM 1.1 Let f C ( ). Assume that (1.8) holds on H'2m and that, forsome with 0 < < 1,

Then, with defined as in (III.2.4), we have

Proof We have

From (III. 1.7) we obtain

hence, by taking the inverse Fourier transform,

Since b /h we obtain from (1.6)

where we have used Theorem II. 1.1. Applying R*pand observing that, because ofapj > 0 and (1.8) with v = 1, i.e.

we obtain the estimate on e1.Next we consider the error e2 which comes from discretizing the backprojec-

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tion. We expand wb * g in terms of spherical harmonics, see VII.3:

Here, the k-sum runs over all N (n, /) spherical harmonics, and the y2l,k arenormalized such that

with c(n, /) from (VII.3.13).

We then have from (VII.3.13) that |y2l,k| 1.Note that the spherical harmonics of odd degree dropped out since

wb + g( ,x. ) is an even function of .Using the Fourier integral for wb * g, the rule R4 of VII. 1, Theorem II. 1.1 and

the Fourier integral for / we obtain

The inner integral can be evaluated by (VII.3.19) to yield

Combining this with (1.6) and remembering that |y2l,k| 1 we obtain

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Now let / > m. Then, because of b m < / and |x — y \ 2, we obtain for theinner integral the upper bound

see (III.2.5). It follows that for / > m with some constant c(n)

Hence, with (wb * g)m the series for wb * g truncated after the mth term, we have

where we have used (VII.3.11), (VII.3.13) and (III.2.7).Since (1.8) is exact on H'2m we obtain for e2 from (1.12)

and this is the estimate of the theorem.

A few remarks are in order.

(1) The crucial parameter in the filtered backprojection algorithm is b. Itobviously controls the resolution. Iffis essentially b — band-limited in the sensethat (f,b) is negligible, thenf fB is a reliable reconstruction offprovided bsatisfies (1.11) and is sufficiently large.

(2) The cut-off frequency b is subject to two restrictions: b < /h guarantees thatthe convolutions are properly discretized, while b m excludes those highfrequencies in wb * g which cannot be integrated accurately by the quadrature rule(1.8).

(3) There is obviously a connection between the existence of quadrature rulesexact on H'2m and resolution.

Since

and since the quadrature rule (1.8) contains n p free parameters a1, . . . , ap,, . . . , one might guess that one can find a quadrature rule exact on H'2m if

Below we shall see that this is in fact the case for n = 2, but for n = 3 we shall needp = (m +1)2 nodes what is roughly times the optimal number p =

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(m + 1) (2m +1). What is known about optimal quadrature on Sn-l can be seenfrom Neutsch (1983) and McLaren (1963), for a discussion of the subject in thepresent context see Griinbaum (1981).

(4) The evaluation of fFB(x) requires the computation of w g( , x. ) forj = 1, . . . , p. Thus we need 0(pq) operations for each x at which we want toreconstruct. In order to reduce this number to 0 (p) we evaluate (1.7) for s = sl,I = —q,...,q only and insert an interpolation step Ih in (1.10), i.e. we compute

rather than (1.10). For Ih we take the B-spline approximation

of order k (see III. 1.9).In Theorem 1.2 below we study the effect of interpolation independently of the

other discretization errors, i.e. we investigate the expression R*Ih(w*g).

THEOREM 1.2 We have for b /h

where the filter Gh is given by

otherwise

and the error e3 satisfies for and

Proof From Theorem III. 1.4 we get for b /h and a 0

where we have used R4 of VII. 1 and (1.6). From Theorem II. 1.4 we obtain for anyeven function a on Z

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Using this for a = ah and integrating over Sn-l we get from (1.15) and Theorem11.1.1

Applying R* to (1.14) we get

Since ( ) = 0 for \ \ /h it follows from Theorem II. 1.4 that e3( ) = 0for | | /h, and the estimate on e3 of the theorem follows from (1.16)with (n —1)/2— a replaced by a. With Fh = R Gh we have from TheoremII.1.1.

and from Theorem II. 1.2

Finally, from (1.2) applied to Gh*f, we see that

and the proof is finished.

From Theorem 1.2 we see that, apart from a highly oscillating additive error e3,interpolation has the same effect as applying the filter Gh. The error e3 has thesame order of magnitude, namely h ||f||H( ), as the discretization error in thedata which we found in Theorem IV.2.2 as long as a (n — l)/2. This result isquite satisfactory. Note that the order k of the splines does not enter decisively. Inpractical tests it has been found (Rowland, 1979) that k = 2 (broken lineinterpolation) is satisfactory but k = 1 (nearest neighbour interpolation) isnot.

Now we give a detailed description of the algorithm for special scanninggeometries.

V.I.I. The Parallel Geometry in the Plane

Here, is available for

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As quadrature rule on Sl we take the trapezoidal rule (VII.2.9) with p nodeswhich is exact on H'2p_2 and we use linear (broken line) interpolation. Then thefiltered backprojection algorithm (1.13) with interpolation goes as follows.

Step 1: For j = 1,. . . , p carry out the convolutions

For the function wb see (1.20)-(1.23) below.

Step 2: For each reconstruction point x, compute the discrete backprojection

where, for each x and j, k and u are determined by

The function wb depends on the choice of the filter in (1.3) or (1.5). For thefilter

containing the parameter [0,1] we obtain from (1.5)

An integration by parts yields

See Fig. V.1 for a graph of wb.If b is tied to h by b = /h, which is the maximal value of b in Theorem 1.1, and

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FIG. V. 1 Graph of the filter wb in the filtered backprojection algorithm for= 0 (top) and = 1 (bottom).

if wb is evaluated for s = sl only, considerable simplification occurs:

For = 0, this is the function suggested by Ramachandran andLakshminarayanan (1971). The filter Wb which belongs to this choice has beengiven in (1.4). Shepp and Logan (1974) suggested the filter

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This leads to

with the usual modification for s = ± /2. Again, for b = /h and s = s1 wesimply obtain

From our error analysis in Theorem 1.1 and Theorem 1.2 we can expect thealgorithm (1.17), with either of the choices (1.21), (1.23) for wb, to reconstructreliably an essentially b-band-limited function f, provided b < p and h < /b, bsufficiently large. This coincides with what we found out about the resolution ofthe parallel scanning geometry in III.3.

The number of the operations of the filtered backprojection algorithm (1.17) isas follows. For the convolutions we need 0 (pq2) operations what can be reducedto 0 ( p q l o g q ) if FFT is used, see VII.5. The backprojection requires 0(p)operations for each x, totalling up to 0 (pq2) operations if fFBI is computed on a(2q + l)x (2q + l)-grid which corresponds to the sampling of R f. For theoptimal relation p = q of III.3 we thus arrive at 0 (q3) operations, regardless ofwhether we use FFT or not.

The corresponding work estimate in n dimensions is 0 (pq log q) for theconvolutions and 0 ( p q n ) for the backprojection. If p = c qn-1 as suggested in(III.2.15) the total work is O(q2n-1).

V.1.2. The Fan-beam Geometry in the Plane

The simplest way to deal with fan-beam data is to compute the parallel data fromthe fan-beam data by suitable interpolations ('rebinning' the data, see Herman(1980)). In the following we adapt the filtered backprojection algorithm to fan-beam data. Again we start out from (1.2) which we write explicitly as

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We use the same notation as in III.3, i.e. the parallel coordinates s,

are related to the fan-beam coordinates a, B by

and The Jacobian of the transformation is

We have to express |x • — s\ in terms of a, B. This is the distance between x and thestraight line L (a, B) which is the dashed line in Fig. V.2. Let y be the orthogonalprojection of x onto L (a, B) and let y be the angle between x — a and — a, i.e.

FIG. V.2 Transformation of (1.24) to fan-beam coordinates.Compare Fig. 111.4.

Note that y depends only on x and B, but not on a. Considering the triangle xya wefind that

Now we can carry out the transformation in (1.24), obtaining

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At a first glance the inner integral looks as if it could be done by convolutions.Unfortunately, due to the presence of the factor |x — a|, the convolution kernelvaries with B and x. This means that the inner integral has to be computed for eachpair B, x, making the procedure much less efficient than in the parallel case. Inorder to circumvent this nuisance we remark that, in view of (1.5), we have

as is easily verified. Using this with n = 2 and p — \x — a \ we obtain

At this point make an approximation: the number |x — a\b in (1.25) plays the roleof a cut-off frequency and the inner integral depends only slightly on this cut-offfrequency if it is only big enough. Therefore, we can replace |x — a\b by asufficiently large cut-off frequency c which does not depend on x, B. We obtain

and we expect this to be a good approximation for essentially b-band-limitedfunctions f if c \x — a\b for all x, B, e.g. for c = (1 + r)b.

The implementation of (1.26) is now analogous to the parallel case: assume g tobe sampled for Bj = 2 (j- l)/p, j = 1,. . . , p and a1 = hl, I = -q,. . . , q,h = /2q. Then, the filtered backprojection algorithm reads:

Step 1: For j = 1,. . ., p carry out the convolutions

For the function wc see (1.20), (1.22).

Step 2: For each reconstruction point x compute the discrete backprojection

where, for each x and j, k and u are determined by

The sign is ' +' if a • x 0 and ' —' otherwise.In accordance with our findings in III.3 we expect this algorithm to reconstruct

reliably essentially b-band-limited functions / provided that c < p/2, c < 2q/rand c = (1 + r)b sufficiently large.

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The algorithm as described in (1.27) needs data from sources distributed overthe whole circle. Looking at Fig. V.2 we see that it should suffice to have sourceson an angular range of + 2 (r) since all lines hitting the reconstruction regioncan be obtained from sources on such an arc. a(r) is the maximal angle for whichL(B, a) meets , i.e. a(r) = arcsin(l/r).

It is very easy to derive such an algorithm. We go back to (1.24), replacing it bythe equivalent formula

The image in the a-B plane of the rectangle in the, plane is(see Fig. V.3)

FIG. V.3 Image in the a-B plane of therectangle [-1, + 1] x [0, ] in the

s- plane.

Therefore (1.25) assumes the form

and this formula is implemented exactly in the same way as (1.25X leading to analgorithm with the desired property.

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V.1.3. The Three-dimensional Case

Let the (m+1)2 direction S2 have spherical coordinates , , where

From (III.2.13) we know that the directions in (1.29) form a m-resolvingset, allowing the reconstruction of a function / with essential bandwidth m,0 < < 1, if the functions R fare given. We adapt the step-size h to thisresolution by requiring that h /m.

The problem is to reconstruct f from the data g( , s1) = Rf( , s1). There aretwo possibilities.

The two-stage algorithmThis algorithm uses the decomposition of the three-dimensional Radon trans-form R3 into the product of two-dimensional Radon transforms R2. Let S2

have the spherical coordinates and let For each

define the function fz on R2 by

Also, for each define the function on R2 by

where t Then we have for

This can be seen from Fig. V. 4 which shows the plane spanned by the x3-axes and0. For each (t, z) R2, gw(t, z) is the integral offalong the line through (t, z)perpendicular to that plane. Hence, integrating g along the orthogonal

FIG. V.4 Orthogonal projection ofthe plane x • = s onto the plane

spanned by the x3-axes and 6.

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projection of x • = s onto that plane, i.e. along the line (the dashed

line in Fig. V.4) yields precisely the integral of / over x • = s.The two-stage algorithm works as follows. In the first stage we solve (1.31) for

each of the directions 6. In the second stage we solve (1.30) for z fixed, obtainingfin the plane x3 = z.

Of course the algorithm has to be carried out in a discrete setting. This is veryeasy if the are equally spaced. Then we can run the algorithm (1.17) on (1.31) aswell as on (1.30) without any changes. In view of the resolution property of (1.17),the resolution of the two-stage algorithm is precisely what we expected above.

The direct algorithmThis is an implementation of the formula (1.2) analogous to the two-dimensionalcase. For the evaluation of fFBI in (1.13) we take the quadrature rule (VII.2.13)which is exact on H'2m for m odd. Note that the m + 1 angles in (VII.2.13) arearranged in an order different from the one in (1.29), requiring a slight change inthe notation. Again we chose broken line interpolation (k = 2) for Ih. FollowingMarr (1974a), we take as filter

Then we get from (1.5) for n = 3

For and s = s1 we simply obtain

We see that with this choice of , discrete convolution with wb amounts to taking

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second order differences. This makes sense since for n = 3, the operator-H2 is theidentity and (1.1) simply backprojects the second order derivative.

The complete direct algorithm now reads, with b = /h:

where, for each x, j and i, u and l are determined by

As in the two-dimensional case we can expect this algorithm to reconstructreliably essentially b-band-limited functions if b < m and b sufficiently large. Ouranalysis depends on the angles and the Aj being the nodes and weights of theGauss formula (VII.2.11), but from a practical point of view we can as well takethe equally distributed and Aj = /m +1 sin , compare the discussion inVII.2.

We conclude this section by doing some reconstructions in R2 with the filteredbackprojection algorithm. The function f we want to recover is the approximate

function (VII. 1.3), i.e.

This function has bandwidth c, but it is clearly not supported in . However itdecays fast enough to consider it approximately as a function which has a peak atthe origin and vanishes elsewhere, see Fig. V.5(a). The Radon transform g = Rfiseasily computed from VII. 1.2, Theorem 1.1.1 and III.l. We obtain

In Fig. V.5(b)-(d) we display reconstructions of f by the filtered backprojectionalgorithm (1.17) for a parallel scanning geometry, using the filter (1.20) with = 0.The number p of directions and the stepsize h have been chosen as follows:

In all three cases, the cut-off frequency b in the algorithm has been adjusted to thestepsize h, i.e. we have put b = /h.

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In reconstruction (b), p, h satisfy the conditions p b, h /b of Theorem 1.1(we have p = m, see the discussion following (1.23), and we have put 9 — 1). So weexpect the reconstruction (b) in Fig. V.3(b) to be accurate, and this is in fact thecase.

In reconstruction (c), p does not satisfy the condition p b. This angularundersampling produces the artefacts at the boundary of the reconstructionregion in Fig. V.3(c). The reason why these artefacts show up only for p = 32but not for values of p between 32 and 64 lies in the fact that our object as given byfdoes not fill out the whole reconstruction region but only a little part around themidpoint. A quick perusal of the proof of Theorem 1.1 shows that in that case thecondition b m can be relaxed to b 2 m.

In reconstruction (d), h does not satisfy the condition h /b. This results in aserious distortion of the original, see Fig. V.3(d). Note that the distortions are notmade up solely of high frequency artefacts but concern also the overall features.This is an example of aliasing as discussed in III.l.

The effect of violating the optimality relation p = q of Table III.l isdemonstrated in Fig. V.6. A simple object consisting of three circles has beenreconstructed by the filtered backprojection algorithm (1.17) with wb as in (1.21)from p = 120 projections consisting of 2q + 1 line integrals each, q varying from20 to 120. The reconstruction error as measured by the L2-norm is plotted in Fig.V.6. The optimal value for q is p/ ~ 38. We see that in fact the L2 error decreasesas q increases from 20 to 38 and is almost constant for q > 38. Thus there is nopoint in increasing q beyond its optimal value as given by p/ .

FIG. V.6 Dependence of the reconstruction error in aparallel geometry with p = 120 on q. The saturation if q

exceeds the optimal value of 38 is apparent.

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V.2 Fourier Reconstruction

By Fourier reconstruction we mean a direct numerical implementation of theprojection theorem

see Theorem II. 1.1. Other methods such as those in Section V.1 and Section V.5also use Fourier techniques but this is not what is meant by Fourierreconstruction.

Using the Fourier inversion formula

in (2.1) we immediately obtain an inversion formula for the Radon transform interms of Fourier transforms. The problem with this obvious procedure comeswith the discretization.

Let and let a = Rf be sampled atwhere and A straightforward discretiz-

ation of (2.1), (2.2) leads to the standard Fourier reconstruction algorithm inwhich the polar coordinate grid

plays an important role.The standard Fourier algorithm consists of three steps:

Step 1: For j = 1, . . . ,p, compute approximations by

From (2.1) we see that the first step provides an approximation to f onG p , q : f ( r j) = (2 ){1 - n ) / 2 g r j up to discretization errors.

An essential feature of Fourier reconstruction is the use of the fast Fouriertransform (FFT), see VII.5. Without FFT, Fourier reconstruction could notcompete in efficiency with other algorithms. Since we cannot do the FFT on G p , q

we have to switch to an appropriate cartesian grid by an interpolation procedure.This is done inStep 2: For each k Zn, \k \ < q, find a point closest to k and put

fk is an approximation to f( k) obtained by nearest neighbour interpolation in thepolar coordinate grid. Up to discretization errors made in step 1 it coincides with

Step 3: Compute an approximation by

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This is a discrete form of the n-dimensional inverse Fourier transform.

Incidentally, the standard Fourier algorithm provides a heuristic derivation ofthe optimality relation p = q between the number p of directions and thenumber 2q + 1 of readings per direction in the standard parallel geometry in theplane: the largest cells of the polar coordinate grid Gpq are rectangles withsidelengths ,( /p) • q respectively. These rectangles become squares for p = q.

Employing the FFT algorithm for the discrete Fourier transforms in steps 1and 3 and neglecting step 2 we come to the following work estimate for thestandard Fourier algorithm: the p discrete Fourier transforms of length 2q in step1 require 0(pq log q) operations, the n-dimensional discrete Fourier transform instep 3 requires 0(qn log q) operations. If p, q are tied to each other by the relationp = cq n - l from (III.2.15), the total work is 0(qn log q). This is much better thanthe 0 ( q 2 n - l ) work estimate we found in Section V.1 for the filtered backprojec-tion algorithm. This efficiency is the reason for the interest in Fourierreconstruction.

Unfortunately, Fourier reconstruction in its standard form as presented aboveproduces severe artefacts and cannot compete with other reconstructiontechniques as far as accuracy is concerned. In order to find out the source of thetrouble we make a rigorous error analysis. The algorithm as it stands is designedto reconstruct functions / with essential bandwidth q. By (III.1.6) the error instep 1 can be estimated by

which is negligible iff, hence g, is essentially q-band-limited in an appropriatesense. Likewise, the sampling of f in step 3 with step-size is correct in the senseof the sampling Theorem III.1.1 since/has bandwidth 1, and since all frequenciesup to q are included the truncation error in step 3 is negligible forfessentiallyq-band-limited in an appropriate sense. Hence step 1 and step 3 are all right, sothe trouble must come from the interpolation step 2. In fact we shall see that thisis the case.

Before we analyse the interpolation error what leads to a better choice of thepoints in step 2 and eventually to a competitive algorithm we give a heuristicargument showing that the interpolation error is unduly large and explaining theartefacts observed in practical calculations.

From Theorem III.1.4 we know that we can describe the effect of interpolationin terms of convolutions. More specifically, if the function/on Rn is interpolatedwith B-splines of order k and step-size the result is

where for and

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It follows that in

Hence, in , the inverse Fourier transform of the interpolated Fourier transformoffis (sinc ( /2) x)k f. This is a good approximation toffor \x\ small, but since(sinc ( /2) x)k decays at the boundary of , the overall accuracy is poor. This isnot helped by increasing the order k of the interpolation. Conversely, looking at(sinc( /2)x)k suggests that the distortion gets even bigger if k is increased. Ofcourse the interpolation in Fourier reconstruction is different from the simplecase of tensor product B-spline interpolation considered here but the artefactsobtained by Fourier reconstruction are similar to those predicted by our simplemodel.

We shall carry out the error analysis in a Sobolev space framework. FromLemma VII.4.4 we know that the norms

are equivalent on H ( ). This is partially true even iff is sampled at pointsclose to k in the following sense:There is a constant h such that for k Zn

LEMMA 2.1 Let the be distributed such that (2.4) holds. Let a 0 and a > 0.Then, there is a constant c(a, a, n) such that

for

Proof Define a function w by putting i.e.

by rule R4 of VII. 1. Since f=0 outside for Choosesuch that Then.for hence

or

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This relation is applied to the left-hand side of (2.5), yielding

Since x C (Rn) there is a number c(t) > 0 for each t > 0 such that

Now we make use of (2.4). For and we have

Combining the last two estimates with the inequality

we obtain for

Using this and Peetre's inequality

we get for the sum in (2.6) the estimate

Now we choose t big enough to make this series converge, e.g. t = 2a + n + 1 willdo. Then the series is a continuous periodic function of which is bounded, andwe conclude that

with some constant c1 (a, a, n). From (2.6) it now follows that

and this is (2.5).

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Now we do Fourier reconstruction with the points satisfying (2.4), ignoringthe discretizations of step 1, i.e. we compute the (hypothetical) reconstruction

The following theorem gives our error estimate.

THEOREM 2.2 Let be distributed such that (2.4) is satisfied, and let 0 a 1.Then there is a constant c(a, a, n) such that for

Proof The Fourier series for / in the cube reads

Hence, in C,

Parseval's relation yields

From the mean value theorem of calculus we find a on the line segment joiningk with such that

were V denotes the gradient. Since satisfies (2.4) we obtain

The satisfy (2.4) since the do, i.e.

Using (2.9) in (2.8) we obtain

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Since 0 a 1 the first sup is bounded by

and the second one by

Using this in (2.11) yields

Now choose x C (Rn) such that x = 1 on . Then, from rule R3 of VII.l,

Because of (2.10) we can apply Lemma 2.1 to the first sum in (2.12), obtaining

where the norm is to be understood as

Since, by Lemma VII.4.5, multiplication with the C (Rn) function xiX is acontinuous operation in H ( ) we have

with some constant c2(a,n). Using this in (2.13) we get the estimate

for the first sum in (2.10). The second sum in (2.10) can be bounded in the sameway since the norms (2.2), (2.3) are equivalent. Hence, with a new constant c(a, a, n)we finally arrive at

This proves the theorem.

We apply Theorem 2.2 to the reconstruction of a functionf C ( ) from theprojections R f, j = 1, . . . ,p where the directions satisfy

Choosing = \k\ with; such that is closest to k/\k\, (2.4) is obviouslysatisfied. It follows that the reconstruction /* for / has an L2 error of order

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ha||f||H ( ). This is precisely what we found out in IV.2 to be the best possibleaccuracy in the reconstruction of f from R f, j = 1,. . . ,p with directionssatisfying (2.14). Thus we see that the (hypothetical) Fourier algorithm withpoints satisfying (2.4) is in a sense of optimal accuracy.

Returning to the standard Fourier algorithm we find that the points of thatalgorithm do not satisfy (2.4) with h small, not even if the directions fulfil (2.14):the radial grid distance is , independently of h. Thus Theorem 2.2 does not apply,and we do not get an error estimate for the standard Fourier algorithm.

We consider the failure of the standard Fourier algorithm to satisfy (2.4) as anexplanation for its poor performance. This seems to be justified since the twoimproved Fourier algorithms which we describe below and which satisfy (2.4)perform much better.

The improved Fourier algorithms differ from the standard algorithm by thechoice of and by the way g ( / \ |, | \) is computed. For ease of exposition werestrict ourselves to the case n = 2 and we assume that = (cos , sin )T,

= (j— 1) /p,j = 1, . . . ,p. We do not need directions withj > p since g iseven.

The first algorithm simply uses

where is one of the directions closest to k / \ k \ . Then, (2.4) is satisfied withh = 2/(2p). With this choice the computation of g( /\ l ! !) calls for theevaluation of g ( , ) for arbitrary real a which is in no way restricted to a uniformgrid. This cannot be done by the ordinary FFT algorithm which gives only thevalues g( , r) for r — — q,. . . ,q — 1. The other values have to be computed byinterpolation. The failure of the standard Fourier algorithm taught us that thisinterpolation is critical. So we choose a rather complicated but accurate andefficient interpolation procedure which combines oversampling of g (not of g!)with the generalized sinc series of Theorem III. 1.2. The algorithm is as follows.

Step 1: For j = 1,. . . ,p, compute approximations gjr to g( ,( /2)r) by

Since has bandwidth 1, g is oversampled by a factor of 2, see III.l.Step 2: For each k Zn, \k\ < q, choose 0, closest to k/\k\ and put

Here, ' +' stands if k is in the upper half-plane and ' —' otherwise.This step deserves some explanations. First, y C (R l ) is the function in

Theorem III. 1.2. For L = , the series would represent the true value off( ). Aspointed out in the discussion following Theorem III. 1.2 the convergence of thisseries is very fast. So it suffices to retain only a few terms without significant loss inaccuracy, i.e. we can choose L relatively small. fk is an approximation to f( k)

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obtained by nearest neighbour interpolation in the angular direction and bygeneralized sinc interpolation in the radial variable.Step 3: Compute an approximation fm to f(hm), me Zn by

This step is identical with step 3 of the standard algorithm.

The work estimate is as follows. In step 1 we have to perform p FFTs of length 4qeach (put the missing 0s to zero) or equivalently, 2p FFTs of length 2q each, see(VII.5.12-13). This does not change the operation count of 0 (pq logq) of thestandard algorithm. Step 2 now requires 0(q2 L) operations. Since L can be keptalmost constant as q-> , this does practically not exceed the 0(q2logq)operations of the standard algorithm. Thus we see that the work estimate for thefirst improved algorithm is practically the same as for the standard algorithm.

In the second improved algorithm (Fig. V.7) which is simpler than the first onewe obtain the point for k = ( k 1 , k2)T 0 in the following way. For k1 > k2 wemove k vertically, for k1 < k2 horizontally onto the closest ray {t : t 0}. Thisprocedure extends in an obvious way to negative kt,k2. The satisfy (2.4) withh = 2/(P 2)- In order to compute f( ) for all such it suffices to computeg ( , a) for all a such that , lies on the vertical lines of the cartesian grid Z2 if\cos \ |sin | and on the horizontal lines otherwise. The algorithm goes asfollows.

Step 1: For j = 1,. . . , p, compute approximations= I/max {|sin |, |cos |} by

FIG. V.7 Choice of in the secondimproved Fourier algorithm.

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This amounts to computing the discrete Fourier transform for arbitrary stepsizein the frequency domain. This can be done by the chirp -z algorithm (VII.5.14).Step 2: For K Z2, | k \ q choose r, j such that | k — r c ( j ) \ is as small as possibleand put

fk is an approximation to f( k) obtained by nearest neighbour interpolationalong vertical or horizontal lines. It coincides withdiscretization errors made in step 1.Step 3: Compute approximations fm to f(hm), m Z2 by

up to

Again the work estimate is 0(q2 log q).For further improvement it is advisable to use a filter function F in the

computation of f from f i.e. to multiply by F (\k\). We found that the cos2

filter

gave satisfactory results.We finish this section by showing that the condition (2.4) which looks strange

at a first glance since it requires extremely non-uniform sampling off is in factquite plausible. Suppose we want to reconstruct the approximate -function

with

see (VII. 1.3), where x0 . Then, for \ k\, \ \ < b

We see that the effect of interpolation can be described by multiplying the Fouriertransform by an exponential factor. If the reconstruction is to be reliable thisexponential factor has to be close to 1 for k small and it should not be too faraway from 1 as k approaches b. It is clear that (2.4) guarantees this to be the casefor h sufficiently small, h < 2/(2b), say.

The superiority of the improved Fourier algorithms over the standardalgorithm is demonstrated by reconstructing the simple test object in Fig. V.8(a)from analytically computed data for p = 200 directions and 2q + 1, q = 64 lineintegrals each. Note that this choice of p, q satisfies approximately the optimalityrelation p = q of Table III.l. The result of the standard Fourier algorithm isdisplayed in Fig. V.8(b). We see that the reconstruction shows serious distortionswhich make it virtually useless. In Fig. V.8(d) we see the result of the secondimproved Fourier algorithm with the cos2 filter (2.16). The improvement isobvious. For comparison we displayed in Fig. V.8(c) the reconstruction produced

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by the filtered back projection algorithm (1.17) using the filter (1.21) with £ = 0. Itdiffers only slightly from Fig. V.8(d), demonstrating that the second improvedFourier algorithm and the filtered backprojection algorithm are of comparablequality.

V.3 Kaczmarz's Method

Kaczmarz's method is an iterative method for solving linear systems of equations.In this section we give an analysis of this method quite independently ofcomputerized tomography. First we study Kaczmarz's method in a geometricalsetting, then we look at it as a variant of the SOR method of numerical analysis.Both approaches provide valuable insights into the application to ART in thenext section.

Let H, H j , j = 1, . . . ,p be (real) Hilbert spaces, and let

be linear continuous maps from H onto Hj. Let gj Hj be given. We want tocompute f H such that

We also write Rf=g for (3.1) with

Let PJ be the orthogonal projection in H onto the affine subspace Rj f = gj, andlet

where is a relaxation parameter. Then, Kaczmarz's method (with relaxation) forthe solution of (3.1) reads

with f° H arbitrary. We shall show that, under certain assumptions, fk

converges to a solution of (3.1) if (3.1) is consistent and to a generalized solution ifnot.

For = 1, (3.3) is the classical Kaczmarz method (without relaxation). Itsgeometrical interpretation is obvious, see Fig. V.9: Starting out fromf0 we obtainf1 by successive orthogonal projections onto the affine subspaces Rjf =gj,j = 1 , . . . ,p.

For the actual computation we need a more explicit form of (3.3). To begin withwe compute PJ. Since Pjf—f ker (Rj) there is uj Hj, such that

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FIG. V.9 Kaczmarz's method forthree equations in R2.

From RjPjf= g, we get

Since Rj is surjective, R and hence RjRj* is injective, and we can write

Thus we obtain for Pj

This leads to the following alternative form of (3.3):

For the convergence proof we need some lemmas. As a general reference werecommend Yosida (1968).

LEMMA 3.1 Let T be a linear map in H with \\T\\ 1. Then,

Proof For an arbitrary linear bounded map A in H we have

The lemma follows by putting A = I — T and showing that

Let i.e. Then

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It follows that

Hence geker(I — T*). This shows that

and the opposite inclusion follows by symmetry. This establishes (3.6).

Rather than dealing with the affine linear orthogonal projections Pj we startwith linear orthogonal projections Qj in H. As in (3.2) we put

Since Qj = Q2j = QJ one verifies easily that

LEMMA 3.2 Let (fk)k=0, 1, be a sequence in H such that

Then we have for

Proof The proof is by induction with respect to the number p of factors inFor p = 1 we have

where we have used (3.8). If ||Fk|| 1, ||Q Fk | |->l it follows that, forhence the lemma for p = 1. Now assume the lemma

to be correct for p — 1 factors. We put

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We then have

Because of (3.7) we haveNow let

hence ||s FK|| 1 and, of course, ||s FK|| 1. Since the lemma is correct for p — 1factors, (/ — S )fk ->0. Applying the lemma for the single factor Q to thesequence F'K = S Fk we also obtain (I - Q )S fk -> 0. Hence, from (3.9), it followsthat (/ — Q ) fk 0. This is the lemma for p factors, and the proof is finished.

LEMMA 3.3 For 0 < > < 2, (Q )k converges, as k , strongly to the ortho-gonal projection onto ker(J — Q ).

Proof Let T be the orthogonal projection onto ker(I — Q ). From Lemma 3.1and (3.7) we know that

hence / — T is the orthogonal projection onto range Thus,

From the first equation we get T= TQ , from the second one T = Q T. Inparticular, T and Q commute.

Now let f H. The sequence (11 (Q )kf \ \ )k = 0, 1 , . . . is decreasing, hence its limitc exists. If c = 0 we get from

as hence

i.e. If we put

Then we have

hence we obtain from Lemma 3.2

or

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It follows that (Q )k converges strongly to 0 on range (i — Q ), and this extends tothe closure since the (Q )k are uniformly bounded. On kerconverges trivially to /. The lemma follows from (3.10).

LEMMA 3.4 For 0 < < 2 we have

Proof A fixed point of Qj is also a fixed point of Q . This settles the inclusion

On the other hand, If f= Q f, then, from (3.7),

Hence, and from (3.8), Since Ql is a projection,Since now

we can show in the same way that Q2f = f and so forth,This settles the inclusion

LEMMA 3.5 For 0 < > < 2, (Q )k converges strongly, as k , to the ortho-gonal projection onto

Proof This follows immediately from the two preceding lemmas.

Now we can state the convergence theorem of the geometric theory.

THEOREM 3.6 Assume that (3.1) has a solution. Then, if 0 < < 2 andf° range(R*) (e.g.f0 = 0),fkconverges,as k , to the solution of (3.1) withminimal norm.

Proof Let Qj be the orthogonal projection onto ker(Rj), and let / be anysolution of (3.1). Then, for h H,

According to Lemma 3.5,

as k where T is the orthogonal projection onto

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If f° range(R*) = range(R*1)+ ... +range(R*p), then Tf° = 0 and (/- T)fis the solution of (3.1) with minimal norm, see I V.1.

So far we have considered the geometrical theory of Kaczmarz's method. Weobtain an entirely different view if we resolve the recursion (3.5). Putting

in (3.5) we get

Using the injectivity of Rj* we solve for uj, obtaining

Now let

If we decompose

then we can rewrite (3.11) as

Solving for u we obtain

hence

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The Kaczmarz method for solving

now simply reads

with f° H arbitrary.If (3.13) has a solution, the solution with minimal norm i s f= R*u where

see IV.l. We want to relate (3.14) to the SOR method (successive over-relaxation,Young (1971)) for (3.15) which reads

After some algebra this can be written as

Note that R*C = BWR* and R*c = b . Hence, carrying out the SOR method(3.16) with f* = R*uk leads precisely to the Kaczmarz method (3.14). This makesit possible to analyse Kaczmarz's method in the framework of SOR. In order toavoid purely technical difficulties we assume in the following that H and Hj, arefinite dimensional.

LEMMA 3.7 Let 0 < CD < 2. Then, range (R*) is an invariant subspace andker(K) the eigenspace for the eigenvalue 1 of B . The eigenvalues of therestriction B' of B to range (R*) are precisely the eigenvalues 1 of C .

Proof From the explicit expressions for B , Cw it is obvious that range (R*),ker(R) are invariant subspaces of Bw and range ((D + ) L ) - l R), ker(R*) areinvariant subspaces of C . For the latter ones we have

For if

with some z H, then

Hence

Since 2D + L + L* is positive definite for 0 < ) < 2 it follows that y = 0.Now let A be an eigenvalue of B' , i.e.

and f 0. Because of (3.17) we must have 1. It follows that

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with some g 0, and

Because of B R* = R*C this can be rewritten as

Since range (D + L) 1 R is an invariant subspace of C satisfying (3.17) itfollows that

hence is eigenvalue of C .Conversely, if C u = u with 1, u 0, then

hence u range (D + L) - 1 R and therefore, because of (3.17),f= R*u 0. Itfollows that

i.e. is eigenvalue of B` .It remains to show that ker (R) is the eigenspace for the eigenvalue 1 of Bw. Let

u 0 and B u = u. Then,

and (3.17) implies u ker (R). Since B u = u for u ker (R) is obvious the proof iscomplete.

The spectral radius p(B) of a matrix B is the maximum of the absolute values ofthe eigenvalues of B. The following lemma uses standard arguments of numericalanalysis.

LEMMA 3.8 Let 0 < CD < 2. Then, the restriction B' of B to range (R*) satisfies

Proof According to Lemma 3.7 it suffices to show that the eigenvalues i ofC , are < 1 in absolute value. Let C u = u with 1 and u 0, Then,

We normalize u such that (u, Du) = 1 and we put (Lu, u) — a = a + ib with a,real. Forming the inner product of (3.18) with u we obtain

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hence

Since RR* = D + L + L*is positive semi-definite we have 1+a + d = 1 + 2 0,hence a —1/2. We can exclude a = -1/2 since in that case | | = 1. Fora > —1/2 we have |1 — — \ < \1 + \ as long as 0 < < 2, and (3.19)implies that | | < 1.

Now we come to the main result of the SOR theory of Kaczmarz's method.

THEOREM 3.9 Let 0 < < 2. Then, for f° range (R*), e.g. f0 = 0, theKaczmarz method (3.14) converges to the unique solution f , range (R*) of

If (3.13) has a solution, then f is the solution of (3.13) with minimal norm.Otherwise,

where fM minimizes

infl .

Proof Since f°,b range (R*) the iteration takes place in range (R*) where Bis a contraction, see Lemma 3.8. Hence the convergence of fk to the uniquesolution f range (R*) of

follows from the elementary theory of iterative methods, see e.g. Young (1971),equation (3.23) is equivalent to (3.20). For the proof of (3.21) it suffices to remarkthat fM is determined uniquely by

see Theorem IV. 1.1 with A = D - l / 2 R and that (3.23) differs from (3.24) onlyby 0( ).

We want to make a few comments.

(1) The convergence rate in Theorem 3.9 is geometrical.(2) Even though the treatment of Kaczmarz's method in Lemma 3.8 is quiteanalogous to the usual proofs for SOR, there are some striking differencesbetween Theorem 3.9 and the classical SOR convergence results. In the latterones, the parameter co is used only to speed up convergence. In Theorem 3.9,

not only has an effect on the convergence behaviour (see the next section) but italso determines, at least in the inconsistent case, what the method converges to.All we know from (3.21) is that for small, fm is close to the Moore-Penrose

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generalized solution of Rf=g with respect to the inner product (f, g)D

= ( D - 1 f , g). Furthermore, the limit f not only depends on > but also on thearrangement of the equations Rjf = gj in the system Rf = g.

V.4 Algebraic Reconstruction Technique (ART)

By ART we mean the application of Kaczmarz's method to Radon's integralequation. Depending on how the discretization is carried out we come to differentversions of ART.

V.4.1. The Fully Discrete Case

Here, Radon's integral equation is turned into a linear system of equations bywhat is called a collocation method with piecewise constant trial functions innumerical analysis. We consider only the two-dimensional case, the extension tohigher dimensions being obvious.

Suppose we want to solve the equations

with straight lines L; for the function / which is concentrated in 2. First wediscretize the function / by decomposing it into pixels (= picture elements), i.e.we cover 2 by little squares Sm, m = 1, . . ., M and assume / to be constant ineach square. This amounts to replacing the function / by a vector F R M whosemth component is the value of / in Sm. Now let

Since Lj meets only a small fraction ( M, roughly) of the pixels, most of the ajm

are zero. Putting

(4.1) can be rewritten as

It is to this linear system that we apply Kaczmarz's method with

Since for R1

with the euclidean norm, the iteration (3.5) reads

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This describes one step of the ART iteration, transforming the feth iterateFk = F0 into the (k + l)st iterate

The implementation of (4.4) requires a subroutine returning for each; a list ofpixels which are hit by L, and the non-zero components Oji of a,. Since there areonly Q(^/M) such pixels, each step of (4.4) changes only 0(^7M) pixel values andneeds only 0(>/M) operations. Thus, one step of the ART iteration requires0(NN/M) operations. For the parallel scanning geometry with p directions and2q + 1 readings each we have M = (2q + I)2, N = (2q + l)p. If p = nq we obtain0(p3) operations per step. This is the number of operations of the completefiltered backprojection algorithm (1.17). We see that ART is competitive—for theparallel geometry—only if the number of steps is small.

A definite advantage of ART lies in its versatility. It can be carried out for anyscanning geometry and even for incomplete data problems. This does not say thatthe results of ART are satisfactory for such problems, see Chapter VI.

The convergence of ART follows from Theorem 3.9: if F° is in <«!,. . ., aN)(e.g. F° = 0), then Fk converges to a generalized solution (in the sense of (3.20)) F^of (4.3) if 0 < to < 2. If (4.3) is consistent (which is unlikely in view of thediscretization errors) then F^ is the solution of least norm, otherwise F^converges to the minimizer of

as fo -> 0.

V.4.2. The Semidiscrete Case

Here we use a moment type discretization of the Radon integral equation. Againwe restrict ourselves to the plane. Let LJt,j = 1,. . . , p, I = 1,. . . , q be sub-domains of Q2 (e.g. strips, cones,...) and let Xji be a positive weight function onLJI. We assume that Lj{ n LJk = 0 if / ̂ k. As an example, Lfl, I = 1,. . ., q maybe parallel non-overlapping thin strips orthogonal to the direction 0,. This is amodel for parallel scanning in CT, allowing for finite width of the rays anddetector inhomogeneities. It is clear that other scanning modes can also bemodelled in this way, see Section V.5.

The system of equations we want to solve is

Putting, with (,) the inner product in

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we can apply Kaczmarz's method to (4.5) with H = L2 ( ), H, = Rq. We have forh Rq

Because of our assumption on Ljl, this reduces to

Thus the iteration (3.5) reads

or

where l = l (j, x) is the unique index for which x L j l. With j running from 1 to p,(4.6) describes a complete step of the semi-continuous ART algorithm withfk = f0 ,fk + 1 = f p . An application of Theorem 3.6 shows that the methodconverges for 0 < > < 1 to the solution of least norm of (4.5), providedf° < X 1 1 i » • • • • XPq) X e.g.f° = 0- Since in that case all the iterates are in the finitedimensional space <X11, • • • , Xpq> (4.6) can in principle be executed withoutfurther discretizations. However, the computational cost is clearly prohibitive.Therefore we use a pixel decomposition of fas above, resulting in an algorithmvery similar to (4.4).

V.4.3. Complete Projections

Here we solve the system

where j Sn-1 are such that j ± i for i j and f is a function on n. FromTheorem II.1.6 we know that R : L2 (

n)-> L2 ([- 1, + 1], w 1 - n ) , w (s) =(1 — s2)1/2 is bounded. It is also surjective since

is a solution of R f= g. Therefore we can apply Kaczmarz's method (3.3) to (4.1)with

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From 11.1 we obtain for the adjoint R * of R as an operator from H onto Hj:

According to (3.4) the orthogonal projection Pj is

and the iteration in (3.5) assumes the form

From Theorem (3.6) we know that, if (4.1) is consistent and if f° range (R*),i.e. if

(e.g.f0 = 0),then Kaczmarz's method (3.3) or (3.5) converges to a solution of (4.1)with minimal norm.

Thus, in principle, the question of convergence (at least in the consistent case) issettled. However, this result is not very satisfactory from a practical point of view.It does not say anything about the speed of convergence, nor does it give any hintas to how to choose the relaxation parameter to speed up convergence. In orderto deal with such questions we shall give a deeper convergence analysis by makingfull use of the specific properties of the operators R .

As in IV.3 we introduce the functions

in n. We show that Cm , l , . . . , Cm,p are linearly independent for m p - 1. For,assume that

in n, Consider the differential operator

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of order p -1. Since 01 j, for; 1 we have 0f + 0} = R" for) = 2,. . . , p.Thus, each derivative in D can be written as

where D}, D'j are derivatives in directions perpendicular to 0i, 0j, respectively.Now we write D = D' + D" with

and D' is a sum of differential operators each of which contains a derivative D'}.Hence

and (4.11) implies that

in Q". Without loss of generality we may assume that the components /1 of 1 donot vanish. If this should be violated we rotate the coordinate system a little. Since

it follows that Cm2 is a polynomial of degree < p — 1, unless at = 0. Hence,for m > p — 1, we must have ai =0. In the same way, we show that a, = 0,j = 2,. . . , p. This establishes our claim that C m t l , . . . ,Cm>p are linearly in-dependent for m > p — 1.

Now we define

Note that because of (IV.3.7), <#m 1 <#k for m * k in L2 ( "). Because of (4.8)and since the Gegenbauer polynomials form a complete orthogonal set inL2 ([-1, +1], vv"'1) we have

hence

Incidentally, (4.12) implies that R* has closed range.Let fM be the minimal norm solution of (4.1), i.e. the solution of (4.7) in

range (R*), see I V.I. The errors

satisfy

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where Qj is the orthogonal projection on ker (R j), compare the proof ofTheorem 3.6. Qj can be obtained from (4.10) bv putting gj = 0, i.e.

We can rewrite (IV.3.4) in the form

hence

This means that m is an invariant subspace of Qj, hence of Qwj and of Q . In

order to compute the norm pm ( ) of Qw in m we use a matrix representation ofQ in m. Let m < p. Then, the Cm , 1 , . . . , Cm,m+ 1 form a basis of m, and wehave for each f m

for suitable numbers ai, ai'. Equation (4.13) translates into

Introducing the m + 1 x m + 1 matrices

which coincide, except for thejth row, with the unit matrix, we obtain the matrixrepresentation of Qw in m in the product form

From (IV.3.7) we obtain

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where a = ( a 1 . . . , am +1 )T, c (m, n) is some constant, and

It follows that

i.e. p2m ( ) is the largest eigenvalue of the m + 1 x m +1 eigenvalue problem

Note that m, being the Gram matrix of the linearly independent functionsCm,.1,. . .,Cm,m + 1 in L2 ( n), is non-singular.

Now let f° range (R*). Since fM range (R*) and since range (R*) is aninvariant subspace of Qw, all errors ek are in range (R*). Because of (4.12) we canwrite

and

It follows that

We see that pkm ( ) determines the rate of decay of that part of ek which is in m.

This holds for m < p. There is no point in studying the contribution of m withm p. The reason is simply that the part of f in p, p+1,. . . is not determinedby the data. From our results on resolution in III.2 we know that / can bedetermined reliably from an m-resolving set of directions if/is essentially fe-band-limited with bandwidth 9m, 0 < 3 < 1. According to (III.2.2) the number p ofdirections of an m-resolving set is at least

with equality for n = 2. Hence we cannot expect p directions to resolve functionsof essential bandwidth p or larger. The Fourier transform of Cm,i can be computedfrom Theorem II. 1.1, (IV.3.4) and (VII.3.18): we obtain with some constant c (n)

From Debye's formula in the form of (VII.3.20) we see that this assumes valuessignificantly different from zero for \ \ (m + n/2) only. It follows that the

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functions in m cannot be recovered from p projections if m > p. We also notethat, in the language of II. 1, the subspaces m do not represent details of sizeapproximately 2n/m or larger. Hence the #m with m small are responsible for theoverall features of/while the #m with m large contribute to the small details.

The numbers pm(u)) can easily be computed numerically by solving theeigenvalue problem (4.14) on a computer. We did some calculations in the two-dimensional case. Figure V.10 shows a plot of pm(co) for p = 32 equally spacedangles cpj and for co = 1,0.5 and 0.1. The angles are arranged in natural order, i.e.q>j = (j — l)n/p,j = 1, . . . , p. We see that the behaviour of pm(co) depends in adecisive way on the choice of co. For co large (e.g. co = 1), pm decreases fromPo = 0.86 to Pp-i virtually zero, while for to small (e.g. a> = 0.1), pm increasesfrom the small value p0 = 0.26 to pp~ t = 0.90. This has the following practicalconsequence for the iterates fk: for co large, the small details in / are picked upquickly, while the smooth parts of / (e.g. the mean value) converge only slowly.For co small, the opposite is the case. Thus, choosing co small has an effect similarto digital filtering (see I V.I) on the early iterates. This may account for thesurprisingly small values of co which are being used in practice. The situationchanges drastically if the angles cpj are permuted in a random way, see Fig. V.I 1.Now the pm(co) are much more favourable for co large, with values virtually 0 onthe first few #m, and with maximum significantly smaller than for the naturalorder in Fig. V.10. In fact it is observed in practice that ART performs muchbetter for orderings other than the natural one.

From the point of view of numerical analysis one might want to choose co suchthat the largest pm is as small as possible. In Fig. V.10 this is the case for co close to0.5. One could also work with different values of co at each step, mimicking thenon-stationary methods of numerical analysis, see Young (1971).

As has been pointed out in I V.I, an iterative method such as ART can only besemiconvergent when applied to an ill-posed problem. An example for thissemiconvergence is given in Fig. V.I2. The original in Fig. V.12(a), a brainphantom created by Shepp and Logan (1974), has been reconstructed fromp = 256 projections with 2<? + 1, q = 75 line integrals each using ART withrelaxation parameter co = 0.05. The data have been contaminated by addingrandom errors. Figure V.I2(b)-(d) show the iterates 2, 4, 40. Obviously, thereconstruction deteriorates after step 4.

V.5 Direct Algebraic Methods

These methods assume a certain rotational invariance of the scanning geometry.They reduce Radon's integral equation to a huge linear system which, due to therotational invariance, is block Toeplitz or even block cyclic what makes* itamenable to the methods of VII.5.

Again let / be supported in the unit ball Q" of R". We assume the data to comein p groups numbered 0 to p — 1, the jth group containing q measurements gjt,

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FIG. V.10 The norms pm (w) of Q" in <gm, m = 0,. . . , 31 for p = 32equally distributed angles in natural order.

FIG. V.I 1 Same as Fig. V.10, but with random ordering of the angles.

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/ = 1, . . . , q. The measurements are assumed to be

As in Section V.4 the functions Xji model the scanning geometry. The supports Ljl

of Xji are the rays of the scanning geometry and may be thought of as narrowstrips or cones. Such a scanning geometry is called rotational invariant if there is arotation U such that, for j = 0, . . . , p — 1,

where ° means composition. If, in addition, V = / then the geometry is calledcyclic.

We give a few examples of rotational invariant scanning geometries.

V.5.1. Parallel Geometry in the Plane

Let j,J = 0, . . . , p — 1 be p equally spaced directions, i.e. j = (cos j, sin j)T,j =j with the angular increment . Let w1, l = 1, . . . , q be some weight

functions in C[— 1, + 1]. Put

From the explicit form of R * in II. 1 we get

i.e. we have

The scanning geometry is obviously rotational invariant, U being the rotation byan angle of . This is true even in the limited angle case, see VI.2. Typically, thefunctions w, have small support, modelling thin beams. Since the supports ofthe functions w, are not required to cover all of [ — 1, + 1], we can deal with theexterior and the interior problem, see VI. 3 and VI.4. If = 2 /p, then thescanning geometry is cyclic.

y.5.2. Fan Beam and Cone Beam Geometries

Let j,J = 0,. . . , p — 1 be equally spaced sources on a circle surrounding n, e.g.in R2 aj = r (cos ,, sin j)

T, b j,- = j b, b the angular increment. Let w, be some

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weight function in C(S"~l). Put

where U is the rotation in the plane of the circle of sources around the origin by anangle of A/?. On introducing x = a, +16 we get

i.e. we have

The scanning geometry is obviously rotational invariant and even cyclic ifA/? = 2n/p. Note that this geometry comprises the majority of the incompletedata problems in Chapter VI.

Rather than solving Radon's integral equation we now solve the linear system

Arranging into groups yields the system

and this in turn is written as

Here, R: L2( n) -»RM is a discrete version of the Radon transform.

For solving the underdetermined system (5.4) we can use the

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Tikhonov-Phillips method of I V.I; for other methods see the end of this section.This amounts to compute the minimizer fy of

with the euclidean norm in Rpq and the L2-norm in L2(n"). From (IV.1.6) we get

The adjoint R* of R is computed as in Section V.4 to yield

where h = (h0,. . . , /ip-i)T, hj = (hn,. . . , hjq}^. Recalling that the £,, aresupported by narrow strips or cones defining the rays we see that (5.6) is basicallya discrete backprojection.

The computation of the vector h to be backprojected calls for the solution ofthe qp x qp linear system

In general, this system is far too large for being solved directly on a computer.However, we shall see that, due to the rotational invariance, (5.7) can be solvedquite efficiently by FFT techniques.

The operator RR* in (5.7) is a p x p matrix of q x q blocks R,R* whose k, Ielement is

From (5.1) we see that

depends on j — i only, i.e. we can write

Note that S-j•, = Sj. With this notation, (5.7) assumes the form

We see that we end up with a system whose matrix is block Toeplitz, and, if thegeometry is cyclic, even with a block cyclic convolution, since in that caseSp - jf = S- j,. In either case, (5.9) can be solved with 0(pq2 + pq log p) operations bythe methods in VII.5.

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We work out the details of the direct algebraic algorithm for a simple parallelscanning geometry in R2 in which the functions w, in (5.2) are simply

where s,, / = 1,. . ., q, are arbitrary numbers in [— 1, +1]. This means that ourdata are

where Lfl is the strip of width d perpendicular to j whose axis goes through thepoint sl j, i.e.

In practice the strips Ljl for fixed j will be non-overlapping but this is notnecessary for our approach.

The (k, I) element of the matrix Sj is now simply

and this is d2/sin2 ( j • 0) if L0k Ljl is inside 2,0 if Lok Ljl, is outside 2 andsomething in between if Lok LJt meets the circumference on 2. Numericalexperiments show that the method is quite sensitive with respect to changes inthese latter entries of Sj. So these entries have to be carefully computed.

In the cyclic case, i.e. if = 2 /p, the direct algebraic algorithm goes asfollows:

Step 1: Compute the matrices Sj from (5.11) and carry out the Fourier transform

Compute and store the matricesStep 2: Carry out the Fourier transform

compute the vectors

and do the inverse Fourier transform

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Step 3: Perform the 'backprojection'

Note that step 1 is preparatory and can be done once and for all as soon as thespecifications of a certain CT machine (including a good value for theregularization parameter y) are known. For each set of data, only steps 2 and 3must be carried out.

The number of operations is as follows. In step 1 we have to compute a Fouriertransform of length p on q x q matrices. This can be done with 0(^2plogp)operations if FFT is used. The Fourier transforms of length p on q—dimensionalvectors in step 2 require 0(<?plogp) operations while (5.12) needs 0(p<j2)operations. The backprojection in step 3 on a q x q grid can be done with Q(pq2)operations. Thus the total number of operations for step 2 and 3 isQ(pq2 + pq\ogp) which is essentially what we found for the filtered backprojec-tion algorithm (1.17).

We conclude this section by pointing out that direct methods are in no wayrestricted to the computation of the Tikhonov-Phillips regularized solution fy.For instance, we may compute the estimate (IV.1.16) for/very much in the sameway. In this case we have to replace (5.5) by

see (IV.1.16). Here, F is to be interpreted as integral operator with kernel F(x, x'),i.e.

If F is a radial function such as (IV.I. 17), then

depends only on j — i provided (5.1) is satisfied. Thus RFR* is Toeplitz as well asRR*, and (5.13) can be dealt with, up to an additional smoothing by the integraloperator F, exactly in the same way as (5.5). A further possibility is to compute theminimum norm solution of the system (5.4).

Actual reconstructions done with the direct algebraic algorithm can be foundin Chapter VI.

V.6 Other Reconstruction Methods

Many other types of algorithms have been suggested. We mention only a few.

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V.6.1. The Davison-Griinbaum Method (Davison and Grunbaum, 1981)

Again we take Q" to be the reconstruction region. Assume that the projections0j = R0 /are available for p directions 6l,. . . , 6P. Then, the form of Radon'sinversion formula suggests to try a reconstruction f^ to / in the form

This is basically what we did in the filtered backprojection algorithm. But now wedisregard everything we know about Radon's inversion formula, determining thefunctions w, so as to make f^ a good approximation to /.

To begin with we compute

where we have put y = s0j +1. Thus,

Now we try to determine W. i.e. the functions w,, such that W approximates the-function. The basic difference to the approach in the filtered backprojection

algorithm in Section V.I is that the function w, in (6.1) depends on the directionOj, allowing for arbitrary spacing of the angles.

Now we have to determine the functions w,. Davison and Grunbaum suggestto choose a 'point spread function' O freely and to find functions w7 such that

is as small as possible. For xeiO" we then have the error estimate

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provided that/is supported in yQ". Thus, making || W—O \\Ll(fr) small guaranteesthat/xj is close to O*/which is a smoothed version of/.

In order to solve the minimization problem (6.2) we use the machinerydeveloped in IV.3. There we have seen that Re is a bounded operator from L2 (O

n)into L2([- 1, -I-1], w1-"), w(s) = (1 -52)1/2 with adjoint

Formula (IV.3.4) reads

Putting

we obtain W = R*U and we have to minimize

The normal equations for this minimization problem are

see Theorem IV. 1.1. From (6.4) we get for each ceR p

Thus RR* is a bounded operator in (L2([—1, +1], w1 ")p with invariantsubspaces <%„ = M m R p whose matrix representation on ^lm is Am. Writing

see (VII.3.1), (6.5) reduces to

These p x p systems have to be solved numerically to obtain the functions vv,.Once these functions have been determined the Davison-Grunbaum algorithmcan be set up.

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V.6.2. The Algorithm of Madych and Nelson(Madych and Nelson, 1983, 1984)

This algorithm differs from the Davison-Grunbaum algorithm only in the waythe functions Wj are chosen. Let n = 2. We assume $ to be a polynomial of degreep — 1. Such a polynomial can be written as

with (one-dimensional) polynomials w,-, provided the directions ±6\, . . . , ±9P

are different. This follows from the fact that the polynomials Cm> _,- ,y = 1,. . . , p,in V.4 are linearly independent for n > p — 1. Again the idea is to choose apolynomial $ which is a reasonable point spread function, i.e. which resemblesthe ^-function, and to solve (6.7) for the functions Wj.

V.6.3. The p-filtered Layergram Method (Bates and Peters, 1971)

This is an implementation of the inversion formula (II.2.10). In terms of Fouriertransforms it reads

Thus we have to do a backprojection, followed by n-dimensional Fouriertransforms.

The discrete implementation of this formula is by no means obvious. There aretwo difficulties. First, in orter to compute (R * g}" one needs R * g on all of R", notonly in the reconstruction region. Second, it is clear from Theorem II. 1.4 that(R* gY(£) has a singularity at ^ = 0. See Rowland (1979) for details. Because ofthese difficulties, the p-filtered layergram method has not been used much.

V.6.4. Man's Algorithm (Man, 1974a)

This is an implementation of the inversion formula (IV.3.14) for the PETgeometry, see III.3. This means that we work in two dimensions with Q2 thereconstruction region, and g = R/is measured for the p(p — l)/2 lines joining psources which are uniformly distributed on the circumference of fi2. Equivalentlywe may say that g (0, s) is sampled at the points (9j, st) where, for p even,

with j + k even, compare Fig. III.5. Note that each of the p ( p — l)/2 lines isrepresented twice by these p ( p — 1) points.

We use the inversion formula (IV.3.14) for n = 2 in a slightly different notation.Putting

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where w(s) = (1 — s2)1/2, Um is the Chebyshev polynomial of the second kind, andPk,i, k = 0, 1,. . . are the normalized orthogonal polynomials in [0, 1] withweight function f'. Then, (IV.3.14) reads (the dash stands for l + m even)

We remark that a^ = a^ = 4n/(m + 1) in the case n = 2, see (IV.3.15).For the approximate evaluation of the inner product we use a discrete version

of this inner product which is adapted to the sampling of g, namely

where the dash indicates that j assumes only values for which j + k even. Putting

and truncating (6.8) at m = M we obtain Marr's reconstruction formula

As it happens the functions gmi are not only orthogonal with respect to the innerproduct in L2 (Z, 1/w) but also with respect to the inner product (•, -)p:

whenever m, m' < p — 1. Equation (6.11) is established as follows. We have

In the ;-sum we put for k even and for k odd, obtaining

By the usual orthogonality relation for the exponential function this vanishes for|/ — /' | < 2p except / = /' and |/ —1'\ — p. In the latter case,

and

If / = /' then we have the same result without the factor (— 1 )*. In either case, / and

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I hence m and m' have the same parity. Thus (6.11) follows from (6.12) and

for m + m' even and m, m' < p — 1 where ek = 1 or ek = (— l)k.The discrete orthogonality relation (6.11) permits an interesting interpretation

of Marr's formula (6.10): assume we want to compute among all functions of theform

that one which minimizes

where || • ||p is the norm derived from the inner product ( • , • ) ?> then /M =/M aslong as M < p— 1. This follows immediately from (6.11) and

since this is minimal for <rm/ym, = (g, gml)p, hence ym/ = y*/- Thus Marr'sreconstruction formula provides a polynomial minimizing the discrete residual(6.13).

V.6.5. Methods based on Cormack 's Inversion Formula

In its original form, Cormack's formula (II.2.18) is virtually useless due to severenumerical instabilities, compare the discussion in VI.3. However, the modifiedform (II.2.20) does not suffer from instabilities. In fact, successful reconstructionshave been done with (II.2.20), see Cormack (1964), Hansen (1981) and Perry(1975).

V.7 Bibliographical Notes

The filtered backprojection algorithm has been introduced in the medical field byShepp and Logan (1974) and Ramachandran and Lakshminarayanan (1971), butit has been applied in radio astronomy by Bracewell and Riddle (1967) as early as1967. They gave the following alternative form for the filter wb of (1.20) for e = 0:

with db the approximate <5-function (VII. 1.2). For the reconstruction of essentiallyfe-band-limited functions db can be replaced by d. This leads to a slightly different

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algorithm where the filter involves only the sine function. The investigation of theresolution achieved by the filtered backprojection algorithm goes back toBracewell and Riddle (1967) and Tretiak (1975); our proof of Theorem 1.1 isbased on Lewitt et al. (1978a). Filtered backprojection algorithms for fan-beamdata have been given by Lakshminarayanan (1975), Herman and Naparstek(1977) and others. Our treatment is based on Horn (1973). The idea of using (1.2)as starting point for reconstruction algorithms is due to K. T. Smith and otherswho systematically investigated the interplay between the filters wb, \vb in a seriesof papers, see Leahy et al. (1979) (which paper contains the n-dimensionalanalogue of (1.25)), Smith (1983, 1984), Smith and Keinert (1985), see also Changand Herman (1980). The material on three-dimensional filtered backprojectionalgorithms has been taken from Marr, et al. (1981). An error estimate for thethree-dimensional problem has been given by Louis (1983).

Fourier reconstruction in R2 has been suggested by Bracewell (1979) in radioastronomy and by Crowther et al. (1970) in electron microscopy. Both paperscontain a heuristic derivation of the relation p = q by a loose application of thesampling theorem to the polar coordinate grid. Bracewell (1979) observed thatthe failure of the standard Fourier algorithm comes from the interpolation. Healso showed that interpolation which is correct in the sense of the samplingtheorem requires a huge number of operations. Condition (2.4) for thedistribution of the interpolation points k has been introduced in Low andNatterer (1981), see also Natterer (1985). Stark et al. (1981) replaced nearestneighbour interpolation by a more sophisticated interpolation procedure andfound experimentally that radial interpolation is far more critical than angularinterpolation. Since (2.4) puts a severe restriction on radial interpolation, theseexperiments confirm (2.4). Of course, one can avoid interpolation altogetherby introducing polar coordinates. This leads to the filtered backprojectionalgorithm.

ART has been introduced by Gordon et al. (1970) independently ofG. Hounsfield (1973) who built the first commercially available CT scanner forwhich he received the 1979 Nobel prize for medicine, jointly with A. Cormack.The connection of ART with the Kaczmarz method has been discovered byGuenther et al. (1974). This paper gives also a convergence estimate forKaczmarz's method in the case = 1 in terms of the angles between ker (Rj) andi- > • ker (^i); f°r tne derivation see Smith et al. (1977). Our proof of Theorem 3.6is based on Amemiya and Ando (1965).

The connection between Kaczmarz's method and the successive over-relaxation method (SOR) has first been seen by Bjbrck and Elving (1979); ourproof of Theorem 3.9 is based on this fact. The observation that /u -»/M as co -* 0has been made by Censor et al. (1983). The rediscovery of Kaczmarz's method andits obvious usefulness in CT has led to a vast number of publications on this andrelated methods. We mention only Herman and Lent (1976), ch. 11 of Herman(1980) and the survey articles of Censor (1981, 1983). The convergence analysisfor Kaczmarz's method applied to complete projections in Section V.4 is based onHamaker and Solmon (1978) and on unpublished lecture notes of D. Solmon.

The basic idea of the direct algebraic algorithm is due to Lent (1975). First

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numerical experiments have been reported in Natterer (1980b). The algorithmsuggested by Buonocore et al (1981) is essentially a direct algebraic algorithm for(5.12), the 'natural pixels' of that paper being the supports of our functions Xjiwhich in fact come into play quite naturally in our framework without anydiscretization of the image. A different kind of direct algebraic methods isobtained by representing the image by trial functions with rotational symmetrywhich also results in a block-cyclic structure of the relevant matrix. Suchalgorithms have been suggested by Altschuler and Perry (1972) (see alsoAltschuler (1979) and chs 13-14 of Herman (1980), Pasedach (1977), and Ham(1975)). An explicit representation of the minimum norm solution for a finitenumber of equally spaced directions in R2 has been given by Logan and Shepp(1975); this corresponds to a direct algebraic algorithm in the limit case of aninfinite number of strip integrals in each direction. We remark that theprobabilistic approach based on (5.13) has been suggested by Hurwitz (1975) andTasto (1977). For an iterative implementation see Herman et al (1979).

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VIIncomplete DataThe relevance of incomplete data problems has already been pointed out inChapter!. In Section VI. 1 we discuss some features which are shared by allincomplete data problems, such as the nature of the artefacts, the degree of ill-posedness, and data completion. In Sections VI.2-5 we work out the details forspecific problems. In Section VI.5 an inversion formula which is adapted to aspecial three-dimensional scanning mode is given. In Section VI.6 we de-monstrate that homogeneous objects can be recovered from very few data.

VI. 1 General Remarks

Let / be a function with support in Q". If g = Rf is given (possibly in discretizedform) on all of Sn-l x [ — 1, + 1] ve call the data complete, otherwise incomplete.Analogous definitions apply to the other integral transforms introduced in II. 1.Typical examples of incomplete daU problems are:

(i) The limited angle problem. Here, R f is given only for 9 in a subset of somehalf-sphere,

(ii) The exterior problem. Here, Rf( , s) is given only for |s| a where0 < a < 1. Of course, f ( x ) is to be determined for |x| a only,

(iii) The interior problem. Here, Rf( , s) is given only for |s| a, where0 < a < 1. Of course, f(x) is to be determined for |x| a only,

(iv) The restricted source problem. Here, Da f is given only for a on a curvesurrounding n.

In 1.1 we have seen that these problems arise quite naturally in practicalapplications.

For incomplete data problems only a few of the results from Chapter II remainvalid. For (i), (ii) the stability estimates of II.5 do not hold, (iii) is uniquelysolvable for n odd only, and in (iv) we have stability only under seriousrestrictions on the curve on which the sources run. In contrast to the mild ill-posedness of the problems with complete data (see IV.2), incomplete dataproblems tend to be severely ill-posed. This ill-posedness is the most seriousdifficulty in dealing with incomplete data problems.

The structure of Radon's inversion formula (II.2.5) gives considerable insight

158

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into the nature of the ill-posedness. For n = 3 it reads

It averages over the planes through x. The biggest contributions come from theplanes in whose neighbourhood g varies strongly. This is the case for those planeswhich are tangent to surfaces of discontinuity of f If the integrals over suchplanes are missing, then (1.1) cannot even approximately be evaluated on suchplanes. In two dimensions the same argument applies except that the effectundergoes some blurring due to the presence of the non-local Hilbert transform.Thus we come to the following rule of thumb.

In incomplete data problems, artefacts show up mainly in the vicinity of lines(planes) which are tangent to curves (surfaces) of discontinuity of / and for whichthe value of Rf is missing.

In order to illustrate this rule we reconstructed the phantom of Fig. VI. 1 (a)from limited angle data. We used a parallel scanning geometry, leaving out lineswhich make an angle 15° with the vertical axes. We did the reconstructionusing the filtered backprojection algorithm (V.I. 17), (V.1.21) with = 0, puttingthe data in the missing angular range to zero. The artefacts in the reconstructionin Fig. VI. 1 (b) are precisely as predicted by our rule. Note that the artefacts extendup to the boundary, indicating that the reconstructed function does not even havecompact support. This is not surprising since the filtered backprojectionalgorithm computes, apart from discretization and filtering, a reconstruction f E

satisfying

where gE is the function g extended by zero in the missing angular range. gE is notlikely to be in the range of R, hence fE does not decay fast at infinity. Rememberthat the crucial point in the proof of Theorem II.4.2 was to show that theconsistency conditions for g imply that fe .

Figure VI.l(c) shows the reconstruction from lines outside the black hole. Theartefacts are much less pronounced since all tangents to curves of discontinuityare available.

In order to avoid the severe artefacts caused by inconsistent data we do theextension in a consistent way. Let / be the subset of Sn- l x [ - 1, + 1] on whichg1 = Rf is given. We want to find a function gc in range (R) such that gc = g1 on /.In case there is more than one solution to this extrapolation problem we choosethe solution of minimal norm. This leads to the definition of gc as the solution ofthe minimization problem

Here, R is considered as operator on C ( n) and the closure is in L2(Z, w). Wechoose the solution of least norm to make the extension as stable as possible, w is asuitable weight function. We shall compute gc in the following sections, and it willturn out that consistent completion of data significantly reduces the artefacts.

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Methods such as ART or the direct algebraic methods (if applicable) do notneed any preprocessing of the incomplete data. This does not mean that thesemethods produce always good reconstructions, but in favourable cases they maybe useful.

In the following sections we discuss the peculiarities of the incomplete dataproblems mentioned above.

VI.2 The Limited Angle Problem

We consider only the plane problem. We assume g = R/ to be given on/ = S*, x [- 1, + 1] where S* = {0: 0 = (cos<p, sin<p)T, \(p\ < <D}. The data areincomplete if O < n/2.

From Theorem II.3.4 we know that / is uniquely determined as long as 0 > 0.However, the stability estimate from Theorem II.5.1 does not hold for O < n/2.More precisely we can show that an estimate of the form

cannot hold for arbitrary s e R1, not even if t is arbitrarily large. For the proof weproduce a function /+ with support in Q2 which is not continuous butR/+ eC°°(Six R1). With /eC«(Q2) a radial function such that/(0) * 0 wemay define

Then R/+ is a smooth function of the straight lines as long as the straight lines arenot parallel to the Xt-axes for which q> = n/2. Thus R/+ eC°°(S£x [— 1, +1])but/is not continuous. This shows that (2.1) cannot hold for s > 1 because ofLemma VII.4.6.

In the terminology of I V.I the failure of (2.1) to hold means that the limitedangle problem is severely ill-posed. To confirm this we compute the singularvalues of the operator

For read

with Um the Chebyshev polynomials of the second kind (VII.3.3). Puttingum = w Um this leads precisely as in IV.3 to

where, for each

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Hence RR* has the invariant subspace L2 (SiK, and coincides with the operatorAm(Q>) on this subspace. Putting co = (cosi//, sini^)T, B = (cos9, sin<p)T we get

with the usual modification for Since

we find that

Hence Am (O) has the eigenvalue 0 with multiplicity oo, plus the eigenvalues of thematrix (2n/m + l)A'm(Q) where A'm(Q>) is the Toeplitz matrix

The eigenvalues of the matrix A'm(Q>) have been studied by Slepian (1978). Theyare denoted (p. 1376) by Aj (m 4-1,0/n), / = 0, . . . , m. This means that the non-zero eigenvalues of the operator RR* are (2n/m+ l)A,(m + l,<b/n), and thesought-for singular values of R are

For <D = n/2, Xi(m + 1, <D/7t) = 1, i.e. <r£, = 2n/(m + 1). This agrees with (IV.3.15)since we worked in IV.3 with L2 (S

1 x R *) rather than with L2 (5^/2 x Ri) as we didin the derivation of (2.3) for <b = n/2.

We are interested in the asymptotic behaviour of the singular values. Makinguse of the //-notation in (III.2.4) we can express the basic result of Slepian (1978) asfollows:

This means that for m large, a fraction slightly smaller than 2<b/n of theeigenvalues is close to 1, a fraction slightly smaller than 1 - (20>/n) is close to zero,with a very small transition region in between. For a plot of the eigenvalues see

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FIG. VI.2. Eigenvalues A,(m+ 1,0.4), / = 0,. . . ,m of the matrix A'm(0.4n)from (2.2) for ro = 40, m = 60 and m = 80. The dichotomy expressed by (2.4) is

apparent.

Fig. VI.2. We conclude that the singular values a^ with l> 0~l (2O/7t)m decayexponentially as m -> oo, and this confirms that the limited angle problem isseverely ill-posed.

Now we compute the complete data gc defined in (1.3). The expansion (II.4.2)for gc in the case X = 1 reads with w = (1 — y2)112

is a trigonometric polynomial of degree m of the form

where the dash indicates that the sum extends only over / with / + m even. Sincegc = g1 on / we must have

Of course the polynomials gcm are uniquely determined by (2.8) provided that

<t> > 0. Thus its coefficients gcm / can be computed from the incomplete data via

(2.8).We outline the procedure for discrete data. We assume g' = R/ to be given for

the 2r+ 1 directions 0, = (cos</>j, sin<p,)T, </>, = nj/(2p), j = — r, . . . , r wherer < (2/7r)Op at the equidistant points

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Step 1: For m = 0,. . . , Af compute approximations g ' m i j to g'm((pj) by

This is a discrete version of (2.9).Step 2: From m = 0, . . . , M, solve the over-determined (for m < 2r) linearsystem

Step 3: For w = 0, . . . , M, compute approximations by

Step 4: Approximate the data in the missing range by

This is obtained by truncating (2.5) at m = M.

Step 2 is the crucial step of the algorithm. The reason is that (2.10) is highly ill-conditioned. This is to be expected since the limited angle problem is severely ill-posed. It is also plausible because we perform an extrapolation for trigonometricpolynomials, and extrapolation is notoriously unstable. We want to investigatethe nature of this ill-conditioning. Let us write

for (2.10) where gcm contains the m+ 1 unknowns gc

mj and g'm the 2r+ 1 right-hand sides of (2.10). Bm is the (2r + 1) x (m +1) matrix with;, / element e'*'^2^.The components of gc

m are numbered m, m — 2,. . ., — m and so are the columnsof Bm. The (m+ 1) x (m+1) matrix Cm = B*Bm, with rows and columnsnumbered m, m — 2, . . . , — m has k, I element Cm ,_k with

Cm is the discrete analog of A'm (<I>). In fact, if r, p tend to infinity in such a way thatr/p -> (2/n) <D, then

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as is easily verified by comparing (2.12) with (2.2). This means that for r, p large,the singular values of Bm, i.e. the square roots of the eigenvalues of Cm areapproximately (2p A, (m + 1, <b/n))112,1 = 0,1,. . . , m. In view of the behaviourof the Aj (m + 1, O/TT) for m large, see (2.4), Bm is highly ill-conditioned for m large.

In order to solve (2.10) or (2.11) we have to resort to the methods of IV.l 1 forthe solution of ill-posed problems. In principle we could work with the truncatedsingular value decomposition. However it is simpler and computationally muchmore efficient to use Tikhonov-Phillips regularization. This amounts to solving

or

Since Cm is a Toeplitz matrix, (2.13) can be solved efficiently even for m large usingFFT techniques, see VII.5. FFT techniques can be employed in the other stepstoo.

This means that the computing time for the completion procedure is smallcompared with the time needed for doing the reconstruction from the completeddata.

In the application of the algorithm we have to make two decisions: we have tochoose M and y. We found that M = q is a good choice, y has to be chosen by trialand error. If y is too large, then the reconstruction is too smooth and lacks smalldetails. For y too small the reconstruction contains rapid oscillations.

In order to see what can be achieved by the completion procedure wereconstructed the object in Fig. VI.3(a) from line integrals making an angle lessthan 75° with the horizontal axes. In this range we used 215 equally spaceddirections and 161 line integrals for each direction. This amounts to running ourcompletion algorithm with p = 128, r = 107, q = 80 and to rotate the object byan angle ofn/2. In Fig. VI.3(c) we display the reconstruction done by applying thefiltered backprojection algorithm to the incomplete data. Except for a differentscaling, this picture is identical to Fig. V.l(b). In Fig. VI.3(b), the result of thefiltered backprojection algorithm for the completed data set is shown. We see thatthe strong artefacts outside the object are reduced by the completion procedure,but new artefacts are added in the interior. The cross-sections in Fig. VI.3(c)reveal that the completion procedure also corrects the severe damping of thefunction values within the annulus. In total, the completion procedure correctsthe most apparent distortions but it tends to introduce new artefacts which arecomparatively small but nevertheless quite disturbing.

Now we turn to the reconstruction techniques for the limited angle problemwhich do not need a complete data set. From a computational point of view, ARTcan be used without any changes, and even the convergence analysis of V.4 can becarried out as in the full angle case. However, the norms pm (co) ofQ01 behave quitedifferently in the limited angle case. We computed pm ((o), m = 0, ... ,31 for

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FIG. VI.4 The norms pm (co) of Q° in tfm, m = 0,. . ., 31 for p = 32equally distributed angles in an angular range of 150° in natural order.

p = 32 equally distributed angles in an angular range of 150°,(pji = (j— 1) 150/180 n,j = 1,.. ., 32. In Fig. VI.4 we used the natural order, inFig. VI.5 a random arrangement of the (pj. In both cases, and irrespective of thevalue of co, most of the pm (co) are virtually 1. However, those first few pm (co)which are significantly less than 1 do depend on co and on the ordering,qualitatively in the same way as in the full angle case. That means that choosing asmall co and arranging the angles in a clever way improves convergence only on

FIG. VI.5 Same as Fig. VI.3, but with random ordering of the angles.

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the first few subspaces <tfm, while nothing can be achieved on the remaining <#ms.This again reflects the inherent ill-posedness of the limited angle problem.

VI.3 The Exterior Problem

The exterior problem is uniquely solvable. This follows from Theorem II.3.1.However, uniqueness hinges on the fast decay of/at infinity, as is shown by theexample following that theorem. Again we restrict ourselves to the plane problem.We assume that g' = R/ is given on

As in the limited angle case we can show that the exterior problem is severely ill-posed. It suffices to produce a function/.,, in Q2 which is not continuous in \x\ > abut for which R/+ is smooth outside Sl x [-a, a]. With/eC,? (Q2) a radialfunction such that /^ 0 in a neighbourhood of {x: \x\ < a}, the function /+which agrees with /+ in the upper half-plane and is zero elsewhere is obviouslysuch a function.

In principle, the exterior problem can be solved by Cormack's inversionformula (H.2.18) which expresses the Fourier coefficients/ of/in terms of theFourier coefficients gl of g:

It permits to compute / (r), r > a from the values of g{ (s) for |s| > a. Since,for u > 1,

see (VII.3.5), the kernel 7^ (s/r) in (3.1) increases exponentially as |/| -> oo. On theother hand, we derive from Theorem II.4.1

which means that g{ is strongly oscillating for |/| large. Hence in (3.1) a functionassuming large positive values is integrated against a rapidly oscillating function.This causes intolerable cancellations and reflects the severe ill-posedness of theexterior problem. The formula (II.2.20) does not suffer from this instability but itis clearly not a solution to the exterior problem.

In order to carry out the data completion procedure of Section VI. 1 we needrange (R) in the sense of (1.3). By Theorem II.4.2, a function g ^ S f (Z) whichvanishes for |s| > 1 is in range (R) if and only if

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Figure V. 5: Filtered backprojection. a(top left): Original, given by a function withmaximal function value 900. b(top right): Reconstruction from correctly sampled data.c(bottom left): Reconstruction from too few directions. d(bottom right): Recon-

struction from undersampled projections.

Figure V. 8: Fourier reconstruction. a(top left): Original, consisting of two circles.Function value within the large circle is 900. b(top right): Standard Fourier algorithm.c(bottom left): Filtered backprojection. d(bottom right): Second improved Fourier

algorithm.

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Figure V. 12: Semiconvergence phenomenon of ART. a(top left): Original. b(topright): Step 2. c(bottom left): Step 4. d(bottom right): Step 40.

Figure VI. 1: Artifacts in the reconstruction from incomplete data, a(left): Original,b(middle): Reconstruction from limited angle, c(right): Reconstruction from exteriordata. The lower row shows horizontal cross-sections through the middle of the object.The reconstructions have been done with the filtered backprojection algorithm, putting

the data in the missing range 0.

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Figure VI. 3: Reconstruction from limited angle data. a(top left): Original. Functionvalue in the ellipsoidal annulus is 900. b(top right): Reconstruction from completeddata. c(bottom left): Reconstruction from incomplete data. d(bottom right): Horizontal

cross section through the upper green spot of b(yellow) and c(blue).

Figure VI. 6: Reconstruction from exterior data. a(top left): Original, consisting oftwo circles. Function value in the bigger circle is 900. b(top middle): Filteredbackprojection algorithm, putting the missing data 0. c(top right): Direct algebraicmethod. d(bottom left): Filtered backprojection with the full data set. e(bottom

middle): Filtered backprojection with completed data, f(bottom right): ART.

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Figure VI. 7: Reconstruction of the original in fig. l.la from exterior data, a(left):Filtered backprojection with completed data, b(right): Direct algebraic method.

Figure VI. 8: Reconstruction from interior data by the direct algebraic algorithm.a(top left): Original. Function value in the ellipsoidal annulus is 900. b(top right):Reconstruction by the direct algebraic algorithm within |x| 0.7 c(bottom): Cross

section through original (yellow) and reconstruction (red).

Figure VI. 9: Reconstruction of I-1f. a(top left): Original. b(top right):Reconstruction with the filtered backprojection algorithm, c(bottom): Reconstruction

of I-1f.

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where — (cos < , sin < )T and Pm are the Legendre polynomials (VII.3.6). If wechoose w = 1, then range (R) consists of those functions g in L2 (Z) which satisfy(3.3). In terms of the Fourier coefficients g\ of g this means that

In view of the orthogonality properties of Pm we finally get

where

Now we want to compute the completed data gc from (1.3). Let cl, §\ be the

Fourier coefficients of gc, g1, respectively. In view of (3.4), gcl is the solution of the

minimization problem

|| • || being the norm in L2 ([— 1, + 1])• (3.5) has the solution

where the dash indicates that m + \l\ even, and the y,m are given by

for m < |/| and m+| / | even. To verify this it suffices to remark that (3.7)implies #/ € <% and u/ -L <#, in L2 ( — !, + !), hence, for any z e % with z = 0for a < \s\ < 1,

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In (3.7), the Legendre polynomials with argument > 1 show up. Since theLegendre polynomials—as the Chebyshev polynomials—increase exponentiallyoutside their interval of orthogonality the determination of vlm is highly unstablefor / large. This means that v{ can be computed reliably for moderate values of/ only, |/| < L, say. In view of the ill-posedness of the exterior problem this is notsurprising. In order to find a good value for L and to decide what to do for |/| > Lwe remind ourselves that our main concern is consistency of the data. Thereforewe introduce a measure of consistency

and choose L such that this measure of consistency is reduced by the completionprocess. This means that we use (3.7) only as long as, for the actually computed vh

where gf is obtained by extending g{ by zero in |s| < a. L is taken to be themaximal index for which (3.8) holds for |/| < L. For |/| > L, t?/ is replaced by 0.

The implementation of the completion procedure for a parallel scanninggeometry is obvious. Since a linear map takes straight lines into straight lines wecan use any algorithm for circular holes for ellipsoidal holes after a suitable lineartransformation.

Apart from the completion procedure we have two further methods for dealingwith the exterior problem: ART and the direct algebraic method. In Fig. VI.6 wecompare these methods. The original in Fig. VI.6(a) is simple enough: it consistsof two circles with midpoints at a distance 0.7 from the origin and with radius 0.1,0.2, respectively. In Fig. VI.6(d) we show the reconstruction with the filteredbackprojection algorithm from p = 256 projections with 2q +1, q = 80 lineintegrals each. Figure VI.6(b) gives the reconstruction by the same algorithm,with the integrals along lines hitting the circular hole of radius a = 0.4 put equalto zero. The strong artefacts are precisely as we expect from our rule of thumb inSection VI. 1: artefacts show up mainly along those tangents to the circles whichhit the black hole |x| < 0.4. In Fig. VI.6(e) the reconstruction from completeddata (L = 6) with filtered backprojection is displayed. The typical artefacts arestill there, but they are much weaker. They same applies to Fig. VI.6(c) (directalgebraic method, the line integrals being replaced by strip integrals) and toFig. VI.6( f) (ART, 15 iterations, o> = 0.1). We conclude that for an object such asFig. VI.6(a), all three methods improve the 'brute force approach' of simplyignoring the missing data, but they do not give results comparable to the case ofcomplete data.

The object displayed in Fig. VI. 1 (a) is much easier to reconstruct from exteriordata. This is demonstrated in Fig. VI.7. Figure VI.7(a) gives the result of thecompletion procedure (L = 5), followed by the ordinary filtered backprojectionalgorithm. In Fig. VI.7(b) we display the reconstruction using the direct algebraicalgorithm. In both cases we used p = 256 (hollow) projections consisting of 129line or strip integrals outside the black hole. We see that the results are much

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better than the reconstruction of Fig. VI.l(c) obtained by ignoring the missingdata, and they are quite close to the original in Fig. VI. 1 (a).

We remark that failure and success in the exterior problem are in fullagreement with our rule of thumb in Section VI. 1.

VI.4 The Interior Problem

The interior problem in two dimensions, in which g' — R/is given on / = S1

x [ — a, a], is not uniquely solvable, as can be seen from the following radiallysymmetric two-dimensional example: let JieC00^1) be an even function whichvanishes outside [a, 1]. Then, g(0,s) = h(\s\) satisfies the hypothesis of TheoremII.4.2, hence there is u eCg5 (Q2) such that Ru = g. The inversion formula (II.2.15)for radial functions yields for / = 0, n = 2 the solution

of Ru = g. Obviously, Ru(0, s) = 0 for \s\ < a, and u does not vanish in | x | < afor h suitably chosen. On the other hand, the interior problem is uniquely solvablein odd dimensions. This follows from the local character of Radon's inversionformula (II.2.5).

However, we shall see that functions u with Ru vanishing on / do not vary muchfor x well in the interior of | x | < a. By Radon's inversion formula in the form of(II.2.7) we obtain for such a function u by an integration by parts

It follows that for

The values of C\ (a, b) are surprisingly low, as can be seen from Table 4.1 whichhas been obtained by numerical integration.

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TABLE 4.1 Values of Cj

b 0.4

0.2 0.130.40.6

a

0.6

0.0390.11

i (a, b).

0.8

0.0160.0390.078

Equation (4.1) is applied as follows. Suppose a complete data set gc has beenfound such that gc range (R) and gc = g' on /, e.g. the function gc from (1.3) forw = 1, and Rfc = gc has been solved. Then, u =f-fc has the property thatRu = 0 on /, and we obtain from (4.1) with M an unknown constant(Af = (/-/<) (0))

for | x | < b. In view of the smallness of C\ (a, b), a crude approximation to themissing part of g suffices to determine/accurately for | x | < b up to an additiveconstant. In spite of the non-uniqueness we thus can compute a solution to theinterior problem which is satisfactory if only changes in the values of fare sought.

The computation of the completed data gc is quite analogous to Section VI.3.Now we turn to algorithms which do not need completed data. ART and the

direct algebraic methods of V.4 and V.5 can be used without any changes on thesedata. In the (hypothetical) consistent case, ART converges to the solution withminimal norm of a discretized version of g' = Rfon I (compare Theorems V.3.9and V.3.6) if properly initialized. The same is true for the direct algebraic methodin the limit ->0 (compare I V.I). Thus, both methods provide us with anapproximation to the minimal norm solution fM of the equation Rf = g on / and itis interesting to know something about the variation of u = f—fM. From (4.1) weget for

On the other hand, we have because of

Since we also have in

Combining the last three estimates we obtain for

If C2(a, b) is sufficiently small, then (4.4) states that fM differs only little fromf,except for an additive constant.

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As an example we reconstructed the phantom in Fig. VI.8(a) from p = 256projections consisting of 2q + 1, q = 56, strip integrals all of which meet the circleof radius a = 0.7. In particular we did not use strips lying entirely outside the blueellipse which has half-axes 0.74 and 0.8. We used strip integrals rather than lineintegrals because we did the reconstruction with the direct algebraic algorithm ofV.5. The reconstruction is displayed in Fig. VI.8(b). We see that the small circlesand the annulus in the 'region of interest' | x \ < 0.7 are clearly visible, but thecolours are shifted towards white. This becomes even more obvious in the cross-section through the original and the reconstruction shown in Fig. VI.8(c).

An alternative method which reconstructs density differences but not theabsolute values is as follows. The inversion formula (II.2.1) for n = 2, a = — 1reads

By R3 from VII.l, — l - 2 g is simply the second derivative of g. Hence,

What is important here is that the operator on the right-hand side is local. Thismeans that I-1 f(x) can be computed from integrals along lines which meet anarbitrarily small neighbourhood of x. In particular we can reconstruct I - 1f in| x | < a from the data of the interior problem.

Of course what one wants to reconstruct is f, not I- 1f. But we shall show thatthere is an interesting relation between f and I -1 f: I -1f is smooth where f is, andvice versa. Thus,for its derivatives have discontinuities precisely where I -1 for itsderivatives have. Practically, this means that I-1 f gives the same information as fif only rapid changes in /, such as edges, are sought for.

The fact about the smoothness of I-1f we referred to above is essentially thepseudo-local property of pseudo-differential operators, see Treves (1980),Theorem 2.2. We give a version of that theorem which suits our purpose.

THEOREM 4.1 Let f Lv (OT), and let/e C°°(17) for some open set U s R". Then,for k an integer with \k\<n, /~k/eC°°([;).

Proof In terms of Fourier transforms we have in the sense of distributions

Let ak(0 = | £ \k for | £ | > 1, say, and ak e C* (Rn). It suffices to prove the theoremfor the operator

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sinceLet V be an open bounded set in U such that F c U, and let ¥ e C% (U) be 1 in

K Then

Since We show that

is in C°°( V}. With A? the Laplacian acting on the variable £, we have for anyinteger p

hence

Note that the integral makes sense for x € V since (1 — ¥)/= 0 in a neighbour-hood of x. By an integration by parts we get

where

as | £ | -+ oo. For 2p — k>n we can interchange the order of integration, obtaining

Since This holds for any open boundedwith This proves the theorem

In order to demonstrate the method of reconstructing I 1f we created thethorax phantom in Fig. VI.9(a). The reconstruction of this phantom from parallel

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data with p = 500, q — 160 using our standard filtered backprojection algorithm(V.I. 17) with wb from (V. 1.21) and e = 0 is displayed in Fig. VI.9(b). In Fig. VI.9(c)we show the reconstruction of P1/by an algorithm very similar to (V.I. 17),replacing the convolution in step 1 by taking central second differences. We seethat the colours are different from the original, but the geometry of the phantomcomes out clearly. Remember that the latter reconstruction is purely local.

Summing up we come to the conclusion that in the interior problem it ispossible to reconstruct features which are characterized by rapid changes in thedensity function. If only such features are sought for, the reconstruction suffersonly little from the non-uniqueness of the interior problem.

VI.5 The Restricted Source Problem

In a typical three-dimensional reconstruction problem, the sources are restrictedto a curve A surrounding Q3, i.e. g = Df is given on A x S2. From Theorem II.3.6we know that / is uniquely determined by these data.

However, a stability estimate corresponding to Theorem II.5.1 does not hold ingeneral. To see this we assume that there is a plane E which meets Q3 but missesthe curve A which we assume to be smooth and of finite length. Then we shallshow that there is a function /+ which is not continuous in Q3 but for whichD/+ 6 C00 (A x S2). For this purpose we choose a function /e CQ (Q3) which doesnot vanish on all of Q3 n E. Let E+ be one of the half-spaces generated by E, andlet

Since the distance between A and E is positive, straight lines through A can meet Eonly transversally, hence D/+eC°°(>l x S2). It follows that a stability estimatecannot hold whenever there exists such a plane E. Unfortunately, this is the casefor the practically used source curves which consist of a circle surrounding fi3 asin Wood et al. (1979) or two parallel circles as in Kowalski (1979) and Hamaker(1980).

In order to obtain a stability estimate we have to exclude planes E meeting Q3

but missing A. Let x = a (A), A e A c Rl be a parametric representation of A witha e C1 (A). A is said to satisfy Tuy's condition if for each x e Q3 and 0eS2 there isA = A(x,0)eA such that

The first condition requires that for each x in Q3 and for each plane containing xthis plane contains also some point of A, and the second condition says that thecurve intersects the plane transversally in that point. We remark that Tuy'scondition is satisfied, for example, by two orthogonal circles surrounding Q3 butnot for one circle or two parallel circles.

Below we shall give an inversion formula for D using only data on / provided A

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fulfils Tuy's condition. In this formula we extend Da/ to a function on R3 byputting

This is equivalent to extending Dfl/ as a function homogeneous of degree — 1.Such a function is locally integrable and it makes sense to compute its Fouriertransform (Da/)

A, see (VII.l).

THEOREM 5.1 If A satisfies Tuy's condition (5.1), then, for /eCo (ft3),

j

where A = A(x, 0) as in (5.1).

Proof From VII.l we have with convergence in y

In the inner integral we introduce x = a + ry, obtaining

For we get on putting

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We compare this with the Fourier inversion formula in polar coordinates whichreads

We see that the inner integral is similar to the inner integral in (5.2). To enhancethis similarity we put in (5.2) a = a (A) and differentiate with respect to A. Thisyields

Now we make use of Tuy's condition. For each x e Q3 and 6 € S2 we can findA = A(x, 6) such that (5.1) holds. Thus, in (5.3), we can replace 0-a(A) by 6-x,yielding

The integral on the right-hand side is now precisely the inner integral in theFourier inversion formula. Since, also by (5.1), a'(X)'0 ^ 0, we finally obtain

where

The formula of Theorem 5.1 is called Tuy's inversion formula. It is not clear if itcan be turned into an efficient reconstruction algorithm, and even if this werepossible it would not help much because Tuy's formula requires Da/ on all of S2,while in practice Daf can be measured only in a cone.

What is more important is that Tuy's formula makes one expect a certaindegree of stability if the curve of sources meets Tuy's condition. The reason is thatno highly unstable operations show up in Tuy's formula. In fact Finch (1985)proved a stability estimate in a Sobolev space framework.

Not much can be said about algorithms at present. Direct algebraic methodsare applicable for source curves with circular symmetry such as a circle or twoparallel circles, but we know that such source curves lead to severely ill-posedreconstruction problems. We do not know of possibilities to use direct algebraicmethods for curves satisfying Tuy's condition.

VI.6 Reconstruction of Homogeneous Objects

If only very few projections of an object are available, then even approximate

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reconstruction is impossible. However, if the class of objects is suitably restrictedit may be possible to restore uniqueness. As an example we consider convexhomogeneous objects in the plane, i.e. we assume that the function / we arelooking for is the characteristic function of a convex set fi in R2. Two typicaluniqueness theorems read as follows:

THEOREM 6.1 There are four directions 0lt . . . , 04 such that for each convex Q,/ is determined uniquely by

THEOREM 6.2 Let alt a2, a3 be not collinear and let be a convex setwhich does not contain either of the aj. Then / is uniquely determined by Daj.f,j = 1, 2, 3.

For the proofs—which are entirely different from what we did in II.3—seeGardner and McMullen (1980) and Falconer (1983). For more results in thisdirection see VolCiC (1984).

These positive results encourage to think about numerical methods forreconstructing characteristic functions for very few, say three to four, projections.An approach entirely within discrete mathematics has been proposed by Kuba(1984). We shall describe a method based on Theorem II. 1.1 which works well inpractice but which lacks theoretical justification.

We consider a domain which is starlike with respect to the origin, i.e. eachhalf-line starting out from 0 meets the boundary of precisely once. Theboundary of Q can therefore be represented in the polar coordinate form

where F( ) > 0. The Fourier transform of the characteristic function f of canbe computed as follows:

Using the formula

f,j = 1, . . . ,4.

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which is easily verified by an integration by parts, we obtain

Now let g = R/. Theorem II. 1.1 holds for/, hence

Combining the last two relations we obtain

For each direction 6, Ae is a nonlinear integral operator mapping functions on S1

into functions on R1. If g is available for p directions 01 ? . . ., Op, we have thesystem

for the unknown function F. Equation (6.3) is a system of nonlinear first kindintegral equations.

In general, we cannot expect (6.3) to be uniquely solvable. However, we knowfrom Theorem 6.1 that there are four directions Oit. .. ,0P such that (6.3) forthese directions is uniquely solvable within the class of functions F representingconvex objects. In this case we even have a result on continuous dependence, seeVolciC (1983), but nevertheless we expect (6.3) to be severely ill-posed in theterminology of I V.I. Thus, solving (6.3) requires some kind of regularization. Thefollowing variant of the Tikhonov-Phillips method has proven to be successful:determine an approximation Fy to F by minimizing

Here, y is the regularization parameter which has to be chosen carefully.In our numerical experiments we used a straightforward discretization of (6.4)

by the trapezoidal rule and a standard minimization subroutine. The results areshown in Fig. VI. 10. The original (a) has been reconstructed from the directionsq>j = (cosq>j, sin<jDj)T with the angles q>j

0° 45° 90° 135° for reconstruction (b),0° 30° 60° for reconstruction (c),0° 10° 20° for reconstruction (d).

We see that even in the case of only three directions in an angular range as small as20° there is some similarity between the reconstructed object and the original.

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FIG. VI. 10 Reconstruction of a homogeneous object in the plane, (a, topleft): Original, (b, top right): Reconstruction from the directions 0°, 45°,90°, 135°. (c, bottom left): Reconstruction from the directions 0°, 30°, 60°.

(d, bottom right): Reconstruction from the directions 0°, 10°, 20°.

VI.7 Bibliographical Notes

Consistent completion of data as a means to reduce artefacts in incomplete dataproblems was first suggested by Lewitt et al. (1978b).

Our treatment of the limited angle problem is based on Louis (1980), see alsoPeres (1979). For a different approach to the limited angle problem based onextrapolation of band-limited functions see Tuy (1981), Lent and Tuy (1981), andTarn et al. (1980). For a numerical method based on the Davison-Griinbaumalgorithm in V.6 see Davison (1983).

The singular value decomposition for the exterior problem has been given byQuinto (1985). This paper also contains a characterization of the null space of theexterior problem considered in the whole space. Quinto (1986) gives an exterior

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inversion method. A reconstruction formula for the exterior problem along thelines of Marr (1974) has been given by Perry (1977).

The idea of reconstructing 1- lf rather than / for the interior problem is due toSmith (1984), also the observations based on (4.1), see Hamaker et al. (1980).

Tuy's inversion formula has been published in Tuy (1983). The method ofproving that the restricted source problem is severely ill-posed in the absence ofTuy's condition is due to Finch (1985). We used this method throughout thechapter.

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V I IMathematical ToolsIn this chapter we collect some well-known results for easy reference. We giveproofs only for less accessible results and in cases where the proof serves as amotivation. Most of the time we simply refer to the literature.

VII.1 Fourier Analysis

The Fourier transform plays an important role throughout the book, particularlyin the theory of the integral transforms in Chapter II and in the sampling theoremin Chapter HI. For our purpose it suffices to study the Fourier transform in theSchwartz space (Rn) and in its dual (Rn). This theory is easily accessible intextbooks (Yosida, 1968), so we restrict ourselves to the basic facts andconcentrate on some less well known examples which we have used in thepreceding chapters.

(Rn), or , is the linear space of those C functions f on Rn for which

is finite for all multi-indices k, l . For f L1 (Rn), the Fourier transform andthe inverse Fourier transform are defined by

The Fourier inversion formula reads

i.e. the transforms are inverses to each other and map in a one-to-one manneronto itself. If has compact support, the extends to an analytic function in Cn.

180

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If not stated otherwise the following rules hold in L1 (Rn). Later on in thissection they will be generalized.

Rl: For r > 0 let fr (x) = f (rx). Then

R2: For y Rn let fy, (x) = f(x + y). Then

R3: For k Zn+ we have for f

R4: We have for g

where the convolution f * g is defined by

R5: We have Parseval's relation

For the inverse Fourier transform we obtain the corresponding rules from =is the space of linear functionals T over which are continuous in the

following sense: there are k, l Zn and c < such that

for f . The elements of are called (tempered) distributions. Distributionsare differentiated according to

If g C°° and all of its derivatives increase only polynomially, then the productgT = Tg is the distribution

Finally, if g , then we define the convolution

where gx (y) = g (x - y).If g is a measurable function on Rn such that

for a suitable q, then we can define the distribution Tg by

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We say that Tg is represented by g. Integration by parts yields

DkTg = TDk

g

if Dk g satisfies the same growth condition as g. This relation suggests to identifyTg with g and to write simply g for Tg.

Now we extend the Fourier transform to by defining

and correspondingly for T. Because of R3, is in if T is. If g LI (Rn) we haveby R5

hence Tg = Tg, i.e. the distributional Fourier transform and the Fouriertransform in the sense L1 (Rn) coincide. If f is locally integrable and fr =f in|x| < r but 0 otherwise, then fr -> f pointwise in as r-» . Similarly, ifg L2 (Rn), then Plancherel's theorem states that Tg = Tg with some L2 function gwhich we call the Fourier transform of g. One can show that

the limit being understood in the sense of convergence in L2 (Rn). The extensionof rule R3 to is obvious. Rule R4 holds if f and g . R5 holds in L2 (Rn)as well, and it readily follows that

with the inner product in L2 (Rn), i.e. the Fourier transform is an isometry inL2 (R

n).Now let f L2 ([ — a, a]n). The functions

constitute a complete orthogonal system in L2 ([ — a, a]n), hence with the innerproduct in L2 ([ — a, a]n,

converges in L2([ —a, a]n). Written out this gives the Fourier series

fk are the Fourier coefficients of f If f= 0 outside [ — a, a]n, then we can write

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and we have Parseval's relation

if g = 0 outside [ — a, a]n.In the following we compute the Fourier transforms of some special

distributions. To begin with we consider Dirac's -function

We have

i.e. is the function

Applying formally the Fourier inversion formula we obtain for 6 therepresentation

which of course does not make sense since the exponential function is notintegrable. However, if we put for some b > 0

then we have for f

i.e. -» pointwise in . Hence is an approximate -function. From (3.19) weobtain for / = 0

where we have used (VII.3.25).

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The distribution

is called the shah-distribution in communication theory. To compute its Fouriertransform we use the Fourier expansion of the periodic function

in [ — ( ), which reads

Comparing the two expansions for g in terms of f we see that

which holds for f . For = 0 this is Poisson's formula

which we can also write as

This can be interpreted as a representation of the quadrature error of thetrapezoidal rule. For a periodic function with period a the corresponding formulareads

where the restriction on k holds componentwise and c1 are the Fourier coefficientsof f in [0,a].

Now let g be an arbitrary locally integrable function with period 2a. Then,

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g , and we claim that

with gk the Fourier coefficients of g. For the proof we apply the distribution Tg tof substituting for g its Fourier expansion. We obtain

and this is (1.7).Another example is the Cauchy principal value integral

which can be defined for f by either of the ordinary integrals

Using the second definition we obtain for the Fourier transform of T

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The inner integral is bounded as a function of , uniformly in a and convergespointwise to sgn ( ) as a -» , except for = 0. Letting a we thus obtain

i.e. T is the function

The Hilbert transform H is defined by Hf= i.e.

Applying R4 yields

VII.2 Integration Over Spheres

In this section we collect some formulae involving integrals over Sn - 1 . Somefamiliarity with integration over spheres is required throughout the book.

To begin with we introduce spherical coordinates in Rn. Letx = (xl, . . . , xn)T Rn and x + ... + x = r2. Assume that x + . . . + x 0,m = 2,. . . , n. Then there is a unique [0, ] such thatxn = rcos , and

Likewise, there is a unique angle n_2 [0, ] such that xn-1

= rsin cos , and

We continue in this fashion until we have found angles ,. . . , [0, ] suchthat, for i = n,. . . , 3, xi- = rsin . . . sin cos , and

Now there is a unique angle [0, In] such that

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The numbers . . . , for which we now have

are called the spherical coordinates of x.We want to compute the integral of a function / over Sn - 1 . Putting

x'= ( x 1 , . . . , x n _ 1 ) T we can represent the upper and lower halvesSn-1 = {w S n - 1 : ±(wn>0} of S n - l by

Then, with the usual definition of surface integrals,

Let 01. . ., 0n_1 be the spherical coordinates of x. Then,

hence

In particular we obtain for the surface area \Sn- l \ of Sn- 1

where denotes the -function. For example, |5°| = 2, \S1\ = 2 and |S2| = 4 .The integral

for w Sn-1 does not depend on w, so we may assume that w = (0,. .., 0, 1)T.Introducing spherical coordinates 01 , . . ., 0n_1 for x we have w.x = cos0n_1and

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The integrals over . . . , n-2 add up to \ S n - 2 |, hence

where we have made the substitution t = cos0n-1.Integrals over Rn can be expressed by integrals over spheres by introducing

polar coordinates x = rw, w Sn-1:

Similarly we write the integral over the hyperplane xn = s as

where s > 0. For the proof we project the hyperplane onto Sn-l by puttingw = x/ x |. The Jacobian of this projection is easily evaluated. For l < n we have

hence, for k, I < n

where is Kronecker's symbol. Using the formula

where / is the unit matrix and aa T the dyadic product, it follows that

Thus, carrying out the substitution w = x/\x\ we obtain

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In view of (2.1) this is identical with (2.5).Occasionally we will have to integrate over the special orthogonal group SO(n),

i.e. the set of real orthogonal (n, n) matrices with determinant +1. For thispurpose we introduce the (normalized) Haar measure on SO(n), i.e. the uniquelydetermined invariant normalized Radon measure on SO(n).

Let m-1 Sm-1 have the spherical coordinates ,. . ., . Then,can be obtained from the mth unit vector em Rm by rotations in the x1 — x1+1

plane with angle , l = m - 1,. . ., 1. Let um-1 ( m - l ) be the product of theserotations. We show that each u SO(n) admits the representation

with S m - l uniquely determined. Let 0n-1 = uen. Then, (un-1 ))"^leaves en fixed, hence

with some v SO(n — 1). Because of

we have reduced the problem to n — 1 dimensions. Since the case n = 2 isobvious, the proof of (2.6) is finished. The spherical coordinates of ,. . . , 0l

are the Euler angles of u.The (normalized) Haar measure on SO(n) can now be defined simply as the

normalized Lebesgue measure on Sn-1 x ... x S1, i.e.

As an immediate consequence we obtain for 00 Sn- 1 and f a function on Sn- 1

Since the left-hand side is independent of 00 we may assume 00 = en. With u as in(2.6) we get u00 = 0n-1, hence

and this is (2.7).

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We use (2.7) to prove the important formula

For the proof we apply (2.7) to the function

on S n - l . With 00 Sn-1 we obtain

where we have used (2.7) with 00 replaced by y/ \ y |. Putting y = rw,w Sn-1

yields

which proves (2.8).We conclude this section by giving quadrature rules on Sn-1 which are exact

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for even polynomials of degree 2m. For n = 2 we simply use the trapezoidal ruleon [0, ]. We then have from (1.6)

for f an even polynomial of degree 2m. For n = 3 we base our quadrature rule onthe nodes Qij whose spherical coordinates are ( , ,) where, for m odd,

i.e. we assume the to be uniformly distributed in [0, ] and the to besymmetric with respect to /2. If f is an even polynomial of degree 2m, then f is alinear combination of terms

where a1 + a2 + a3 is even and 2m. Upon introducing spherical coordinates weget from (2.2)

In order to derive a quadrature formula for the integral we start out from theGauss-Legendre formula of degree m + 1. This formula reads

where t±j, tj= —t-j are the zeros of the Legendre polynomial Pm + 1, and theweights Aj = A_j are given by

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Abramowitz and Stegun (1970), 25.4.29. (2.10) is exact for polynomials of degree2m +1, hence

This is a special case of (2.10) if al + a2 and a3 are even. It is trivially satisfied, dueto the symmetry of the Gaussian weights, if a1 +a2 and a3 are odd. Puttingt = cos we finally obtain

This is our quadrature rule for the integral. It is exact whenever al + a2 + a3 iseven and 2m. It is interesting to study the distribution of the nodes for mlarge. From Table VII. 1 we see that are almost equally distributed in (0, ), andthe weights are very close to the weights ( /m + 1) sin in the straightforwardrule

We conclude that for practical purposes we can replace (2.11) by the simpler rule.

TABLE VII.1 Nodes and weights of the quadrature rule (2.11) for m+ 1 = 16.

j tjA

j

12345678

0.095010.281600.458020.617880.755400.865630.944580.98940

0.189450.182600.169160.149600.124630.095160.062250.02715

1.475641.285331.095030.904760.714530.524390.334500.14572

0.19030.19030.19030.19020.19010.18990.1888

1.03171.03181.03181.03201.03231.03311.03551.0502

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with the equally distributed in (0, ).The -integral needs to be evaluated for a3 even only since for a3 odd our

quadrature rule for the -integral gives the correct value 0. Hence a1 + a2 is evenin the -integral, and (sin )x1 (cos )x2 is an even trigonometric polynomial ofdegree 2m. Such a function is integrated exactly by the trapezoidal rule with thenodes ,., i = 0, . . . , m, hence

for a1 + a2 even and 2m. Combining this with (2.11) we finally arrive at the rule

which is exact for even polynomials of degree 2m if the , Aj are chosen as in(2.11). However, from a practical point of view we can as well choose equallydistributed angles with weights Aj = ( /m+ l )sin without much loss inaccuracy.

For what is known about quadrature on Sn-1 see Stroud (1971), Neutsch(1983), and McLaren (1963).

VII.3 Special Functions

In this section we collect some facts about well known special functions, such asGegenbauer polynomials, spherical harmonics and Bessel functions. We giveproofs only for a few results which are not easily accessible in the literature.

VII.3.1. Gegenbauer Polynomials

The Gegenbauer polynomials C > - , of degree / are defined as theorthogonal polynomials on [-1, + 1] with weight function (1 -x2) . Wenormalize C by requiring C (l) = 1. We then have

with the Gamma function. For = 0 we have to take the limit -»0. Note thatdue to our normalization, (3.1) and other formulas differ from the usual one asgiven in Abramowitz and Stegun (1970). We remark that C is even for / even andodd for / odd.

For = 0 we obtain the Chebyshev polynomials of the first kind

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and for — 1 the Chebyshev polynomials of the second kind

For x 1 we have

A simple proof of the first of these relations uses the recursion

which follows immediately from (3.2) and which holds also for x 1. Observingthat

we obtain for x 1

i.e. the T1 grow exponentially with / for x > 1.For = 1/2 we obtain the Legendre polynomials

for which

The Mellin transform M is an integral transform on (0, ) which is defined by

It is easily verified that

where the convolution is now

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From Sneddon (1972) we take the following Mellin transfom pairs:

For = 0 we again take the limit -»0. Note that the formulas of Sneddon (1972)have to be modified according to our normalization of the C .

VII.3.2. Spherical Harmonics

A spherical harmonic Y1 of degree / is the restriction to Sn-l of a harmonicpolynomial homogeneous of degree l on Rn (Seeley, 1966). There are

linearly independent spherical harmonics of degree /, and spherical harmonicsof different degree are orthogonal on S n - l . An important result on sphericalharmonics is the Funk-Hecke theorem: for a function h on [— 1, + 1], we have

For n = 2, (3.12) reads

see (3.14) below, and follows by simple substitutions from (3.2).

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As a simple application we derive a pointwise estimate for the normalizedspherical harmonics. Putting h = C\n - 2 ) / 2 we obtain from (3.12)

From the Cauchy-Schwarz inequality we get

where we have used (2.3). Inserting the value of c(n, l) and using (3.1) yields

This does not make sense for n = 2,l = 0, in which case we simply havec(2,0) = 27 .

For dimensions n = 2,3 we shall give explicit representations of sphericalharmonics. For n = 2 we have N(2, l) = 2 linearly independent sphericalharmonics of degree / if / > 0, namely

and Y0 = 1. For n = 3 we have N(3, l) = 2l+ 1 linearly independent sphericalharmonics of degree / if l > 0. They can be expressed in terms of sphericalcoordinates, see Section VII.2. We put for Sn-1

Then,

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are 2l+ 1 linearly independent spherical harmonics of degree / > 0, and Y0 = 1.The Legendre functions P are given by

with Pl from (3.6). For k even, P is a polynomial of degree l with the parity of l,while for k odd

where Q is a polynomial of degree / - 1 with the parity of / — 1.

VII.3.3. Bessel Functions

By ,jk we denote the Bessel function of the first kind of integer order k, seeAbramowitz and Stegun (1970), ch. 9. It can be defined by the generating function

i.e. Jk (x) is the coefficient of zk in the Laurent expansion of the left-hand side.Hence,

where C is the circle of radius r centered at the origin. On putting z = rei ,0 2 this becomes

For r = 1 we obtain the integral representation

If a, b R and a2 < b2 we put

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obtaining for k = 0

The Bessel functions of real order v can be obtained as Fourier transforms ofGegenbauer polynomials. More precisely we have for a > 0 with w(s) =

Gradshteyn and Ryzhik (1965), formula 7.321. For = 0, we take the limit 0.Combining (3.18) with the Funk-Hecke theorem (3.12) we obtain for Y1 aspherical harmonic of degree /

For n = 2 this is identical with (3.16). In the evaluation of the constant we havemade use of well-known formulas for the -function.

We need a few results on the asymptotic behaviour of Bessel functions. Theasymptotic behaviour of Jv(x) as both v and x tend to infinity is crucial in theinvestigation of resolution. Debye's formula (Abramowitz and Stegun(1970), 9.3.7) states that Jv(x) is negligible if, in a sense, v > x. More precisely, wehave from Watson (1952), p. 255 for 0 < < 1

Using the expansion (Abramowitz and Stegun (1970), 4.1.28)

for 0 t < 1 we obtain

hence

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We see that for < 1, Jv( v) decays exponentially as v -» .An asymptotic estimate for the Weber-Schafheitlin integral

for r > 1 and 0 v u is obtained from its representation (see Abramowitz andStegun (1970), 11.4.34)

in terms of the hypergeometric function (Abramowitz and Stegun (1970), 15.1.1)

Putting

we obtain

Using the reflection formula (see Abramowitz and Stegun (1970), 6.1.17)

of the -function we obtain

In order to estimate the sum we remark that 0 < b a < c. Choosing m such thata + n c + m<a + n + l we have

Similarly, choosing m such that n b + m < l + n we have

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Using the last two estimates in (3.22) we obtain

since c = a + b. It follows that

with some C(r) which is a decreasing function of r.Finally, the asymptotics of Jv (t) as t and v fixed is

see formula 9.2.1 of Abramowitz and Stegun (1970).We finish this section by collecting a few integral formulas.

for v > 0, z > 0, see Abramowitz and Stegun (1970), formula 11.3.20

for > 0, v 0, see Gradshteyn and Ryzhik (1965), formula 6.623.

where / is an integer, see Abramowitz and Stegun (1970), 11.4.17.

VII.4 Sobolev Spaces

Sobolev spaces are used extensively in II.5, IV.2, V.2 and also in Chapter VI. Asgeneral reference we recommend Triebel (1978), Adams (1975). We give only a fewdefinitions and facts which are of direct significance in our context.

The Sobolev space H (Rn) or H of real order a is defined by

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H is a Hilbert space with norm and inner product

For a = 1 we obtain from the rules R3 and R5 in VII. 1

with the Laplacian:

An integration by parts shows that

Thus we see that H1 consists of the functions whose (distributional) derivatives oforder 1 are in L2(Rn). Likewise, H for a 0 an integer consists of thosefunctions whose derivatives of order a are in L2(Rn).

If is an open subset of Rn we put

where the support of f is the complement of the points having aneighbourhood in which / vanishes (i.e. / vanishes for all C -functions withsupport in that neighbourhood). Note that the definition of the spaces H ( )varies in the literature. Our spaces are the Ha( ) spaces of Triebel (1978). Theycoincide with the usual H ( ) spaces only for a k + 1/2, k an integer. The innerproduct and the norm in H ( ) are those of H(Rn).

An important tool in the theory of Sobolev spaces is interpolation, see Lionsand Magenes (1968), ch. 1.2. Two Hilbert spaces H0, H1 are said to be aninterpolation couple if H1 is dense in H0 with continuous embedding. Let S be aself-adjoint strictly positive operator in H0 with domain H1 such that the normsf H1 , Sf | H0 are equivalent on H1. It can be shown that there is such anoperator. Then we define for 0 1

H is a Hilbert space with norm

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For 9 = 0, 1 we regain the spaces H0, HI. H is called an intermediate orinterpolation space. It can be shown that H is determined uniquely by H0, HI

even though S is not, and the norm in H is determined by (4.2) up to equivalenceonly. We quote a few basic facts on interpolation spaces.

LEMMA 4.1 For 0 < 1 there is a constant C(9) such that for f H1 theinterpolation inequality

holds.

LEMMA 4.2 Let H0, HI and K0,K1 be two interpolation couples and let H , Kbe the corresponding interpolation spaces. Let A: H0 -» K0 be a linear operatorsuch that there are constants A0, AI with

for f Hi, i = 0,1. Then there is a constant C(6) for each 0 [0, 1] such that

in H . C( ) is independent of A0, AI..

LEMMA 4.3 For 0 a l we have the reiteration property

for 0 0 1, with equivalence of norms.

As an example we have, for a and 0 < 1

This follows immediately from the choice

A less obvious example is

which holds for sufficiently regular, e.g. = , see Triebel (1978), Theorem4.3.2/2.

A third example is provided by the spaces H( ). For a an integer 0, thesespaces consist of those functions for which

is finite. This definition extends to real non-negative orders in the following way.

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For a = s + , s an integer, 0 < a < 1, we put

If is sufficiently regular, then

These spaces are used in Adams (1975); they agree with the Ha( ) spaces ofTriebel (1978) for sufficiently regular, e.g. = . The definition (4.6) isconsistent because of the reiteration property of Lemma 4.3.

The interpolation result (4.4) can be used to find equivalent norms in HO ( ).For a 0 an integer we can expand f H ( ) in its Fourier series in [- 1, + 1]n,

and we obtain from R3 of Section VII. 1 and Parseval's relation

For a 0 real we define

LEMMA 4.4 The norms (4.1), (4.8) are equivalent on H( ) for a 0 real.

Proof Let 0 a be integers. We compute a norm for

by considering the operator

S is a self-adjoint operator in H ( ) with domain H ( ), as can be seen from(4.7). Hence

defines a norm on Hg. Since, by (4.4), H = H + ( ), the lemmafollows.

The interpolation inequality in Sobolev spaces on Rn reads simply

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for a y . This follows from a simple application of the Cauchy-Schwarzinequality to the integral defining ||f||H,. For the spaces H ( ), H ( ) thisinequality, possibly with a constant C(a, , y), follows from Lemma 4.1.

LEMMA 4.5 Let x C (Rn). Then, the map f x f is bounded in H .

Proof For a 0 an integer, the lemma is obvious since in that case the norm inH can be expressed in terms of derivatives. For a 0 real, the lemma follows byapplying Lemma 4.2 to the operator Af= xf and to Hi, = Ki = H , i = 0, 1where is an integer such that a < + 1. For a 0 the lemma follows byduality.

In the sequel we need two embedding theorems for Sobolev spaces. The firstone is the Sobolev embedding theorem, the second one the embedding theorem ofRellich - Kondrachov.

LEMMA 4.6 Let be sufficiently regular, and let a > n/2. Then, Ha( ) c C( ),and

with some constant c independent of f.

For a proof see Adams (1975), Theorem 7.57. By Ha( ) c C( ) we mean thatfor f Ha( ) there is a continuous function f* such that f = f* a.e.

LEMMA 4.7 Let be bounded and sufficiently regular, and let a > . Then,Ha( ) H ( ), the embedding being compact.

For the proof see Lions and Magenes (1968), Theorem 16.1.We finish this section with an estimate of functions in Ha( ) which vanish at

many points in .

LEMMA 4.8 Let be bounded and sufficiently regular. For h > 0 let h be a finitesubset of such that the distance between any point of and h is at most h. Leta > n/2. Then there is a constant c such that

for each f Ha( ) vanishing on h.

Proof We carry out the proof for a not an integer and indicate the minorchanges for a an integer (which in fact are simplifications) at the end of the proof.

Let a = s+ with s an integer and 0 < a < 1. We introduce the seminorm

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Let x1 , . . . , xm be such that no non-trivial polynomial of degree s vanisheson x1 , . . . , xm. We show that

with some constant c independent of f. If this is not the case, then there is asequence (//) in Ha( ) such that

as l . (f1) is bounded in Ha( ). Because of Lemma 4.7 we can assume thatf1 f in Hs( ). From the second part of (4.11) we see that f1 converges even inHa( ), and | f H = 0. Hence the derivatives of order s of f are constant, i.e. f is apolynomial of degree s. Again from the second half of (4.11) we conclude with thehelp of Lemma 4.6 that

By our choice of X 1 , . . ., xm it follows that f= 0. This being a contradiction tothe first half of (4.11), (4.10) is established.

From (4.10) it follows that for f Ha( ) with f (xi) = 0, i = 1,. . ., m, we have

Applying this to the domain h we obtain by a change of coordinates

if f vanishes on hx1,.. ., hxm. The constant c in (4.13) is independent of h; in factit is the constant in (4.12). Now we subdivide into subdomains , l = 1,. . ., q,with the following properties: no non-trivial polynomial of degree s vanisheson , n , and the diameter of is at most c h with c independent of h. Then wecan apply (4.13) to each of the subdomains , obtaining for f Ha( ) vanishingon h

with c independent of h and f. Summing up yields

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This finishes the proof for a not an integer. If a is an integer, then (4.10) is replacedby

with the seminorm

This is proved very much in the same way as (4.10).

VII.5 The Discrete Fourier Transform

In this section we study the discrete analogue of the Fourier transform from acomputational point of view. We use the discrete Fourier transform to computediscrete convolutions, to solve linear systems with Toeplitz structure, and toevaluate Fourier integrals.

The (one-dimensional) discrete Fourier transform of length p is given by

In view of the orthogonality relation

we obtain immediately the discrete inverse Fourier transform

The evaluation of the discrete Fourier transform of length p, as it stands, requires0(p2) operations. This number can be reduced to 0(p log p) by fast Fouriertransform (FFT) techniques (Nussbaumer, 1982). We describe the well-knownFFT algorithm of Cooley and Tukey for p a power of 2, p = 2m, say.

Let q = p/2 = 2 m - l and w = e -2x i /p . Then wp = 1, wq = - 1, and (5.1) reads

The basic idea in the Cooley-Tukey algorithm is to break the sum into one partwith / even and the rest with / odd. This yields

Since w2 = e 2 i/q, gk and uk have period q. Hence

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where k runs only from 0 to q — 1. Since gk, uk can be computed by a discreteFourier transform of length q — p/2 we see that the discrete Fourier transform oflength p can be computed by two discrete Fourier transforms of length p/2,followed by p/2 complex multiplications and p complex additions. If this is donein a recursive way we arrive at the following number of operations: if Mm, Am arethe numbers of complex multiplications and additions, respectively, for thediscrete Fourier transform of length p = 2m, then

with M0 = A0 = 0. It follows that

where Id is the dual logarithm. Thus the total number of operations is 0(p Id p) asclaimed above.

There is a close connection between the discrete Fourier transform andconvolutions

of length p. If the sequence (xk)k Z1 has period p the convolution is called cyclic. In

that case, the discrete Fourier transform takes (5.4) into

as is easily verified. Hence cyclic convolutions can be carried out by doing twodiscrete Fourier transforms, followed by p complex multiplications and aninverse Fourier transform. Using FFT we need 0(p log p) operations. The sameapplies to linear systems of the form

z = Xy

where X is a cyclic convolution, i.e.

Equivalently, the (k + l)st row of X is obtained by making a ring shift in the kthrow. In V.5 we apply this method to block cyclic convolutions for which theentries x1 in (5.5) are q x q matrices. In that case FFT leads to an algorithm with0(pq2 + pq logp) operations, provided the inverses of the q x q matrices

are precomputed and stored.

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We also can deal with arbitrary convolutions in which the sequence (xk)k z1 isnot necessarily periodic. Here we extend yk outside [0, p— 1] by zero and xk

outside [ — p+ l ,p — 1] by periodicity with period 2p, the value of xp beingirrelevant. Then, the equations

together with the (irrelevant) equations for k = p,. . . , 2p — 1, form a cyclicconvolution of length 2p which can be carried out by FFTs of length 2p, resultingin a 0(p log p) algorithm for arbitrary convolutions of length p.

The matrix X of a linear system defined by an arbitrary convolution has theform

Such a matrix, whose rows are obtained by successive right shifts, is called aToeplitz matrix. In order to handle such systems by FFT we need some prep-arations, see Bitmead and Anderson (1984), Brent et al. (1980) and Morf (1980).

Let M be the p x p matrix

whose only non-zero elements are in positions (/ + 1, /), / = 1, . . . , p — 1. For anarbitrary p x p matrix A we define

+ (A), a_ (A) are called the ( + ) or (-) displacement rank of A. The relations

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show that for the Toeplitz matrix X from (5.6)

hence a+(X), _(*)< 2.

LEMMA 5.1 If A is invertible, then

Proof First we show that for arbitrary p x p matrices A, B,

In fact, if rk(I — AB) — m, then there are p — m linearly independent vectors xi

such that (/ — AB)Xi = 0. The vectors Bxi must be linearly independent as well,and

Hence

Interchanging A, B we get (5.7).Now we apply (5.7) to the matrices MA, M T A - l in

For x the vector (x0, . . . , xp _ 1)T we introduce the triangular Toeplitz matrices

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LEMMA 5.2 Let xj, y j Rp,j = 1, . . . , a. Then, the following statements areequivalent:

Proof Putting x{ = y = 0 for k < 0 we have

Let

be the matrix (5.9). Then,

and, with ak,1 = 0 if k = 0 or l = 0,

Hence (5.8) follows from (5.9). Vice versa, let A satisfy (5.8). A is uniquelydetermined by (5.8) since A — MAM T = 0 implies A = 0. On the other hand, wejust showed that the matrix A from (5.9) is a solution of (5.8). Hence (5.8) implies(5.9) and the proof is complete.

LEMMA 5.3 For each pxp matrix A there are vectors xj, yj Rp,j = l , . . . , a + ( A ) such that

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Proof Let B a p x p matrix with rank a, let x1, . . . , xa be a linearly independentcolumns of B, and let y be the coefficient of xj in the representation of the kthcolumn bk of B in terms of the vectors x1, . . . , xa, i.e.

This means that B admits a representation

in terms of a dyadic products where yj — (y , . . . ,j )T. Applying this to thematrix B = A — M A M T we get

with suitable vectors xj yj, and the lemma follows from Lemma 5.2.The following theorem is our main result on inverting Toeplitz matrices.

THEOREM 5.4 Let X be an invertible Toeplitz matrix (5.6). Then there areToeplitz matrices LI, L2, U1, U2 such that

Proof According to Lemma 5.1 we have a+ (X -1 ) = a_ (X) 2. The theoremfollows from Lemma 5.3.

The significance of Theorem 5.4 lies in the fact that the linear system Xy = Zcan be solved by convolutions, provided the matrices L1,U1, L2, U2 areprecomputed and stored:

If FFT is used, this can be done with 0(p logp) operations. Thus p x p systemswith Toeplitz matrices can be solved with 0(p logp) operations.

In V.5 we apply the algorithm (5.10) to matrices X which are block-Toeplitz, i.e.for which the entries x, in (5.6) are q x q matrices. In this case Theorem 5.4 stillholds with L1, U1, L2, U2 being block-Toeplitz as well, and the algorithm (5.10)needs 0(pq2 + pq logp) operations.

One of the main applications of the discrete Fourier transform is theapproximate calculation of the Fourier transform

Assume / = 0 outside [ — a, a] and let f1 =f(hl), l = — q , . . . , q — 1 , h = a/q. Wewant to compute approximations fk to f(uk), k = —q,...,q— 1, where u > 0 is

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the step-size in the frequency variable . Discretizing the integral by thetrapezoidal rule we obtain

For u = /a this is a discrete Fourier transform of length 2q. Since f hasbandwidth a, u = /a is precisely the Nyquist sampling rate for f, see II.1. We seethat (5.11) can be done by FFT in 0(p logp) operations if the sampling in Fourierspace is done at the Nyquist rate. For the error in (5.11) see Theorem III.1.3.

Sometimes (see V.2) we want to compute f( ) with a step-size u other than theNyquist rate. If u is one-half of the Nyquist rate, u = /(2a), then we can write

where f1 has been put zero for / < — q and l q ('padding by zeros'). Thisamounts to doing a discrete Fourier transform of length 4q. A more economicaland more elegant procedure consists in separating odd and even k by writing

Now we perform two discrete Fourier transforms of length 2q each, one with theoriginal data f1, the other one with the modified data e~ f1. We see that the caseu = /(2a) (and more general, the case u = /(ma) with m an integer) can behandled with FFT methods as well, using 0(q logq) operations.

For arbitrary step-size u > 0 we have to resort to a less direct method, the so-called chirp z-algorithm (Nussbaumer (1982), ch. 5.1), We write

in (5.11), obtaining

The second formula of this algorithm is a convolution of length 2q which can bereplaced by two discrete Fourier transforms of length 4q each. Again we come to a0(q log q) algorithm by the help of FFT.

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IndexA priori information, 90, 94, 95Abel type integral equation, 23, 24, 26Adjoint, 18, 140Algebraic reconstruction technique (art),

102, 137, 138, 140, 144, 160, 164, 170Aliasing, 60, 113Approximate delta-function, 183Artefacts, 118, 120, 121, 159, 164, 168Attenuated radon transform, 46, 52, 53

B-spline, 60, 107, 120, 121Backprojection, 103, 113, 148, 151, 153Band-limited, 54Bandwidth, 54Beam hardening, 3Bessel functions, 197

Cauchy principal value, 185Chebyshev polynomials, 154Chirp z-algorithm, 127, 212Collocation method, 137Communication theory, 54, 184Completion of data, 159, 162, 166, 168,

170, 178Conditional expectation, 91Cone beam scanning, 3, 147, 174Consistency conditions, 37, 49, 159, 168Cormack's inversion formula, 28, 155,

166Covariance, 91Cut-off frequency, 60

Debye's formula, 198Diffraction tomography, 5Digital filtering, 89Dirac's delta-function, 183Direct Algebraic method, 102, 146, 149,

160, 168Discrete Fourier transform, 206Displacement rank, 208Distributions, 181

Divergent beam transform, 10, 33Dual operators, 13, 47

Electron microscopy, 3Emission CT, 4, 47Essentially band-limited, 55Exponential radon transform, 47Exterior problem, 28, 30, 101, 146, 158,

166, 168

Fan-beam geometry, 82Fan-beam scanning, 2, 10, 75, 83, 111Fast Fourier transform (FFT), 11, 119,

120, 125, 126, 148, 164, 206, 207Filtered backprojection, 49, 102, 103,

106,113,117,128,151,159,164,168Filtered layergram, 22, 48, 153Filtering, 60Fourier coefficients, 182Fourier expansion, 184Fourier inversion formula, 180Fourier reconstruction, 102, 119, 120,

156Fourier transform, 180Funk-Hecke theorem, 195

Gamma function, 193Gauss-Legendre formula, 191Gegenbauer polynomials, 193Generalized inverse, 85

Haar measure, 189Hilbert transform, 186Hole theorem, 30

Ideal low-pass, 60Ill-posed, 33, 42, 85, 95, 158, 164, 166,

168Incomplete data, 2, 85, 147, 158Indeterminacy, 36, 90Integral geometry, 5, 14

221

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Interior problem, 146, 158, 169Interlaced parallel geometry, 71, 74, 83Interpolation of functions, 107, 108, 120,

125, 126, 127Interpolation of spaces, 46, 60, 201, 202,

203Inversion formulas, 18, 48, 49, 100, 102,

119, 153, 171Isotropic exponential model, 92

Jacobi polynomials, 100

k -plane transform, 52Kaczmarz method, 102, 128, 134, 136,

137, 139

Legendre functions, 197Legendre polynomials, 194Limited angle problem, 3, 158, 160, 163,

164, 166Locality, 21, 172

m -resolving, 64Mellin transform, 194Mildly ill-posed, 91Minimal norm solution, 170Modestly ill-posed, 91

NMR imaging, 8Normal equations, 86Nyquist condition, 56

Optimality relation, 83, 118, 120, 127Oversampling, 56

Parallel scanning, 2, 71, 111, 117Parseval's relation, 183Pet (positron emission tomography), 4,

76, 82, 83, 153Picture densities, 94Pixels, 137Plancherel's theorem, 182Poisson's formula, 184Projection theorem, 47, 119Pseudo-differential operators, 172

Quadrature rule, 103, 106, 190

Radar, 8Radon transform, 9

Radon's inversion formula, 22, 151, 158,169

Random variables, 91Rebinning, 111Regularization, 86, 178Relaxation parameter, 128Resolution, 64, 68, 71, 106, 111, 115,

116Restricted source problem, 158Riesz potential, 18Rotational invariance, 146, 147, 148Rytov approximation, 6

Sampling, 54, 71, 75, 180Scanning geometries, 2, 71, 83, 108Schwartz space, 9Semi-convergence, 89, 144Severely ill-posed, 91, 160, 163, 166Shah-distribution, 184Sinc function, 55Sinc series, 57Singular value decomposition, 85, 86, 88,

95, 100, 101, 160, 161Sobolev spaces, 42, 92, 94, 95, 121, 200SOR, 136Spectral radius, 135Spherical coordinates, 186Spherical harmonic, 195Stability estimate, 160, 174

Three-dimensional CT, 3, 32, 174Tikhonov-Phillips method, 89, 91, 148,

164Toeplitz matrix, 146, 148, 161, 164, 206,

208, 209Tomography, 8Transmission CT, 1Trapezoidal rule, 56, 109, 184, 191Tuy's condition, 174

Ultrasound CT, 4Undersampling, 57, 118

Weber-Schafheitlin integral, 199White noise, 91Worst case error, 90

X-ray transform, 9