exact image representation via a number-theoretic radon transform

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3&

Published in IET Computer VisionReceived on 1st May 2013Revised on 5th November 2013Accepted on 25th November 2013doi: 10.1049/iet-cvi.2013.0101

38The Institution of Engineering and Technology 2014

ISSN 1751-9632

Exact image representation via a number-theoreticRadon transformShekhar Chandra1, Imants Svalbe2

1Australian e-Health Research Centre, Division of Computational Informatics, CSIRO, Australia2School of Physics, Monash University, Melbourne, Australia

E-mail: [email protected]

Abstract: This study presents an integer-only algorithm to exactly recover an image from its discrete projected views that can becomputed with the same computational complexity as the fast Fourier transform (FFT). Most discrete transforms for imagereconstruction rely on the FFT, via the Fourier slice theorem (FST), in order to compute reconstructions with low-computational complexity. Consequently, complex arithmetic and floating point representations are needed, the latter of whichis susceptible to round-off errors. This study shows that the slice theorem is valid within integer fields, via modulo arithmetic,using a circulant theory of the Radon transform (RT). The resulting number-theoretic RT (NRT) provides a representation ofimages as discrete projections that is always exact and real-valued. The NRT is ideally suited as part of a discretetomographic algorithm, an encryption scheme or for when numerical overflow is likely, such as when computing a largenumber of convolutions on the projections. The low-computational complexity of the NRT algorithm also provides anefficient method to generate discrete projected views of image data.

1 Introduction

A common problem in medicine and the natural sciences is tonon-destructively determine the internal structure of anobject. For example, in medicine it is desirable to view theorgans of a patient without an invasive medical procedure.This problem is known as tomography and has a very longhistory [1]. The basis of tomography is the inverse of theRadon transform (RT) first constructed independently byFunk [2] and Radon [3].In this work, an inverse of a discrete RT (via an extension of

the discrete Fourier slice theorem (FST) [4]) is constructedusing integer fields, so that grey scale images can berepresented and reconstructed exactly via cyclic sets ofunique integer sequences. The result is a scheme with thesame computational complexity as the fast Fourier transform(FFT) [5], while utilising only integers throughout itscomputation. In the subsequent sections, the preliminaries oftomography will be introduced as developed by Radon [3],followed by the classical FST, its extension to the discretedomain in Section 1.3 and a discussion of the discrete Fouriertransform (DFT) over integer fields in Section 2 beforepresenting the proposed transform in Section 3.

1.1 Radon transform

Radon [3] showed that a two-dimensional (2D) object f can be‘reconstructed’ from its one-dimensional (1D) projectedviews or ‘projections’ along lines. An example of aprojection within the RT is given in part (a) of Fig. 1.Explicitly, a projection μθ at an angle θ from a 2D object f is

mu =∫Lu

f (x, y) dℓ (1)

where Lθ is a set of parallel lines (constrained by the imagingcircle D that has compact support) within the projection μθ onR2 and dℓ is the measure on these lines. The Radon transformR of the object f is then

Rf = m = mu|u [ [0, p){ }

(2)

where μ is the set of projections in the RT and μθ = μθ + π.Note that this set is uncountably infinite, which haspractical implications in acquiring projections and on theuniqueness of the reconstruction [6–8]. Radon [3] provedthat the inverse of the transform (2) is possible via the FST,also known as the central slice theorem.

1.2 Fourier slice theorem

The FST facilitates the inverse problem, one of recovering anobject from its projections, by reconstructing the Fouriertransform (FT) of the object rather than the object itself. TheFST states that the 1D FT of the projections are slices,perpendicular to their view angles, of the 2D FT of the object.An example of a slice in the FST is given in part (b) of Fig. 1.Theoretically, the Fourier slices provided through the FT of

projections at different angles can fully populate 2D Fourierspace, f, can then be recovered via the inverse FT.However, one requires an infinite number of slices, andhence an infinite number of projections, to completely fill

IET Comput. Vis., 2014, Vol. 8, Iss. 4, pp. 338–346doi: 10.1049/iet-cvi.2013.0101

Fig. 1 Radon transform (within the imaging circle D) and theFourier slice theorem

a Shows a projection μθ of an object f at an angle θ. The point ρ quantifies theattenuation of an x-ray because of absorption, which is proportional to thedensity of fb Shows the same projection whose 1D Fourier transform mu is a slice in the2D Fourier transform f of the object. The object f is fully recovered onlywhen f is completely filled.

Fig. 2 Finite equivalent of Fig. 1 for a line with slope 1 in a 5 × 5image

a Shows the wrapping behaviour of the lines in an image lattice. Each greyscale shows a different translate tb Shows the corresponding coefficients in discrete Fourier space of theprojections taken along the lines in Fig. 2a

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continuous Fourier space and provide an unambiguousreconstruction. In practice, this requirement is overcome bytaking a large number of projections, which hasconsequences on both the amount of radiation to which theobject or patient is exposed to and on the uniqueness of thereconstruction [7].The continuous polar geometry of the FST is also

problematic when attempting to utilise the low-computationalcomplexity of the FFT [5], which is Cartesian coordinatebased. Filtering and interpolation of the projection data arecommon practice to overcome these issues. Filtered back-projection (FBP) utilises filtering of the projection data thatare in a polar coordinate system to apply a weighting so thatthe data are appropriate for back-projection onto a Cartesianimage space. This filtering is necessary to account for thenon-uniform sampling of the polar grid proportional to theradius and to reduce the effects of ghosts or reconstructionartefacts caused by the non-uniqueness of the solution space[6–8]. The term back-projection refers to the smearing of theprojections across image space during reconstruction as eachslice is placed into Fourier space. In another popular methodcalled Fourier inversion, the polar grid of the Fouriertransform is directly interpolated onto the discrete Fouriergrid. The interpolation is the most significant source of errorsin reconstructions involving this method and to a lesserextent in FBP [9, 10].Grigoryan, Bolker, Gertner, Fill and others [11–19]

independently introduced the discrete FST (also known asthe discrete Radon transform (DRT) or the Finite RadonTransform (FRT),which overcomes these problems of the FSTby defining and mapping discrete projections using theCartesian geometric form of the DFT, the basis of the FFT [4].

1.3 Discrete Fourier slice theorem

The discrete Fourier slice theorem defines projections as sumsalong lines present on a finite torus, so that the image isperiodic in both rows and columns. These lines take theform of the congruences

y ; mx+ t(mod N ) (3)

x ; psy+ t(mod N ) (4)

with slopes m and s, intercepts t and x, y, m, s, t [ Z for a


square image of size N = pn, where p is prime. Rectangular(N1 ×N2) images have to be zero padded to the nearest squaresize. An example of the line in (3) is shown in part (a) ofFig. 2. Note that the lines (3) and (4) can also be representedas the vectors [1, m] and [ps, 1], respectively, colloquially as‘one down and m across’ and ‘ps down and one across’,respectively. The discrete projection bins R(m, t) and Q(s, t),for a fixed slope m and translate t, is then defined as

R(m, t) =∑N−1

x=0

I x, kmx+ tlN( )

(5)

Q(s, t) =∑N−1

y=0

I kpsy+ tlN , y

( )(6)

for an image I(x, y) where ⟨a⟩N represents a(mod N). Forexample, the pixels of the same grey scale in Fig. 2a end upin the same projection bin. A discrete projection is the set ofbins with all translates 0≤t , N having a fixed m. Forinstance, Fig. 2a shows the projection for m = 1 for a 5 × 5image.The congruences (3) and (4) partition an N ×N image into a

set of 1D slices. For N = pn, where n is a positive integer and pis prime, a total of N +N/p slices tile all of the images at leastonce when 0≤m , N and 0≤ s , N/p. An example of thesimplest case, that is, when n = 1 so that N is prime, isshown in Fig. 3. The N + 1 1D DFT values create one entryat each and every element of the 2D array (i.e. the tiling ofthe discrete Fourier plane is complete and uniform). Thetiling is made possible because the greatest common divisor(GCD) of N and m is always unity, represented as gcd(N,m) = 1, so the lines only intersect at the origin.For the cases n > 1, there is some degree of oversampling,

which is removed by a simple exact filter S. The N = p casedoes not require any filtering. The filter S is determined bygcd(u, v, N ) or, equivalently, gcd{gcd(u, N ), gcd(v, N )}.Furthermore, the gcd can be precomputed as the uniquecoordinate values are {1, 2, …, N−1} . All other pixels inthe space are sampled once, except the DC coefficient,which is oversampled by N +N/p [20]. The oversampling isremoved exactly by dividing all the frequencies by S.Kingston and Svalbe [21] showed that S can be reduced toa single exact 1D filter, equivalent to a digital version ofthe Ram–Lak filter used in FBP [22], which is the same forall projection angles.Thus, the image can be reconstructed using the inverse

DFT [4]. The reconstruction process is detailed in Section 3.

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Fig. 3 Discrete slices within the geometry of the DFT for a 5 × 5 image. Each colour represents a slice of a different slope with the DCcoefficient centred (black). Note that each vector shown is computed modulo N

a–e Shows the slices with slopes 0≤m , 4(mod 5) in DFT spacef Shows the row sum (perpendicular) slice in DFT space

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1.4 Previous work

Matús and Flusser [4] showed that reconstructions of an N ×N image can be computed using only integers with acomplexity of O(N3). However, this method is only suitedfor small image sizes. Do and Vetterli [23] showed that the1D wavelet transform can be applied to the slices of thediscrete FST for image compression and de-noising.However, their algorithm is not designed for reconstructingimages from projections. Other applications of the discreteFST include tomography [24–27], image processing andoperations [28–30], encryption and encoding [31, 32].The discrete FST was developed for arbitrary composite

sizes by Grigoryan [15] and Fill [18]. The extension of theDiscrete FST to non-prime sizes was rediscovered byHsung et al. [33] (for power of two sizes), Kingston [20](for prime power sizes) and Kingston and Svalbe [21] (forarbitrary composite sizes). Orthogonal forms of the discreteFST have been constructed by Lun et al. [34, 35] andKingston [20]. A recent detailed review of the DRT can befound in Kingston and Svalbe [36].


Related discrete RT includes the work of Beylkin [37] thatutilises projections along arbitrary curves and reconstructionvia block-circulant matrices. This discrete RT can only bean approximation to the RT [37]. The Mojette transform(MT) of Guédon et al. [38] is a direct discretisation of theRT. However, its reconstruction methods either havesignificant computational complexity or are sensitive tonoise [39, 40]. Other notable discrete RTs include theworks of [41–45].

1.5 Overview

This paper presents a digital Radon transform which is bothinteger-only and of the same computational complexity asthe FFT. It shows that the discrete FST is valid within the2D number-theoretic transform (NTT) and defines digitalprojections as sums along the congruences (3) and (4)modulo an integer M. The NTT is a transform analogous tothe DFT in terms of geometry, but replaces thecomplex-valued unit circle e2πiσ = 1 (where σ is a


Fig. 4 Example of the NRT for a 512 × 512 image

Digital projections of the image are taken (shown as NRT space) and placedinto 2D NTT space [46, 48] via the proposed NST. The inverse 2D NTT willthen reconstruct the image with the same computational complexity as theFFT

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non-negative integer and i2 =−1) with a number field, wherethe results of addition, subtraction, multiplication and divisionof integers are also integers [46]. The slices are then 1D NTTsof discrete projections computed modulo M and theirmapping to the 2D NTT will be referred as thenumber-theoretic slice theorem (NST). The number fieldsused are very similar to Rivest, Shamir and Adleman (RSA)encryption [47] and results in a discrete RT that isimpervious to round-off error and numerical overflow.An example of the number-theoretic Radon transform

(NRT) of an image is given in Fig. 4. The definition andproperties of the NRT are described in Section 3 afterpresenting the preliminaries of the NTT in Section 2. Notethat the initial work of this paper was presented at [49]. TheNRT, together with the discrete FST, are developed in fullusing a new discrete theory of the RT in Section 1.2.

2 Number-theoretic transform

The DFT is possible because the unit circle provides an Nthroot of unity α so that αN = 1 and where α = e2πi/N. This rootmakes the circular convolution property (CCP) result h(x)possible for two 1D finite (periodic) sequences f (x) and g(x) with 0≤ x , N , as

h(u) = f (u) g(u) (7)

where f represents the DFT of f. The CCP is especiallyadvantageous when using the FFT, so that thecomputational complexity becomes O(N log2 N ), asopposed to O(N2) for the direct computation.

2.1 Definition

In the NTT, which is also defined on the finite torus, α isredefined as an integer modulo a prime number M so thatone obtains Fermat’s (little) theorem

aN ; rM−1 ; 1(mod M ) (8)

where a, r, M [ Z, which results in an integer field whengcd(N, M ) = 1 so that N−1 (the multiplicative inverse) exists.


Definition 1: (NTT [50]). The NTT I(u, v) of an N ×N imageI(x, y) is defined as

I(u, v) =∑N−1

x=0

∑N−1

y=0

I(x, y)a−xua−yv(mod M ) (9)

where α is given by (8) and where the intensities andcoordinates of the pixels of image I(x, y) are integers. Theinverse NTT (iNTT) is then defined as

I(x, y) = 1

N 2

∑N−1

u=0

∑N−1

v=0

I (u, v)axuayv(mod M ) (10)

One needs to choose α or rM−1 as well asM for a given imagelength N.The NTT also preserves the CCP, but now the computation

of circular convolutions can be done using only integers,hence avoiding all floating point representation issues andround-off errors [46], as well as numerical overflow [51].The important aspect of the NTT is the selection of α andM which then defines the maximum transform length N.When N is a power of 2, one may use the Fermat numbertransform (FNT) [52, 50] or choose a prime of the formM = k × N + 1 [53]. The FNT uses the Fermat numbers F t,which are 2b + 1 where b = 2t, as the choice for M. Only thefirst five Fermat numbers (t = 0, ..., 4) are known to beprime. The dyadic transform lengths allowed for the FNTare N ≤F t − 1 when α = 3 or N ≤ b when α = 2. The latterhas the advantage of being multiplication free, becausebit-shifts can be used [50].Using a prime of the form M = k × N + 1 allows for large

transform lengths [53]. Such a selection is the prime M =63 × 225 + 1 and primitive root r = 5 for all dyadic lengthsup to N = 225. A primitive root is a number whose valuerw(mod M ) cycles through all possible {1, …, M−1} values(in some order) when w cycles through all {1, …, M−1}values. The primitive root(s) r in these cases have to befound by trial and error and can be computed by dividingM−1 by the prime factors pj of M−1 such thatr(M−1)/pj � 1 modM( ) and the trial value of r is prime.

2.2 Properties

The result is that the NTT is an integer-only transform thatalways represents all terms exactly within a finite precision.For example, Bailey [54] showed that the FFT losesprecision when calculating π to 19 million digits. The NTThas been applied to fast multiplication of very large integers[52], fast digital convolutions and filtering [48, 55] andencryption [56, 57]. A new C library implementation [58] isshown to have better performance than the FFT for largeimages, improving on the results of Agarwal and Burrus[48] and Bailey [54]. See Bhattacharya et al. [59] for ahistorical review of the NTT.Complex-valued definitions of the NTT are also possible

and were introduced by Nussbaumer [60]. The fastalgorithms of the DFT are applicable to the NTT becausethe NTT preserves all the properties of the DFT [46]. As aresult, one may use the Cooley and Tukey [61] algorithm toconstruct a fast NTT for dyadic lengths and Rader’s [62]algorithm for prime lengths.


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2.3 Performance
The performance of the NTT (via [58]) is compared with thatof the FFT, using the well-known FFTW library of [63], inFig. 5. The test images consisted of the image of Lena withdyadic sizes up to 213 and zero padded where appropriate.The performance advantage of the NTT is approximatelyexponential with image size. The benchmarks wereconducted using a 16 bit modulus and integers of the samesize. The integer size is increased to 32 bit integers only formultiplications as they require double width. The FFT wascomputed using double precision. The benchmarks wereconducted on two different central processing units (CPUs)to show the dependence of the FFT on the floating pointunit (FPU). The Turion™ is a single core CPU with 1 GBof random access memory (RAM) and the Core2Duo™ is anewer dual-core CPU possessing a performance-driven FPUwith 4 GB of RAM. See Appendix 1 for more technicaldetails. Despite the faster FPU, the NTT still out-performsthe FFT for large images. For the largest array size on theTurion™, the computation of the FFT did not completebecause of memory limitations. The NTT had no problemsfor this image size since it maintains full representation ofthe image using only 16 bit (half precision) integers.These results improve on the results of Agarwal and Burrus

[48], which showed better performance of the NTT for smallsequence sizes, and contradict the results of Bailey [54],which showed that the NTT was slower. The performanceadvantage applies despite the fact that most CPUs have aslower (relative to floating point) integer divisioninstruction, which is critical to the modulo instruction. Theperformance of the modulo operation can be increased onthese architectures by computing the integer divisionoperation on the FPU if one is present [64]. Architectureshave also been constructed specifically to implement NTTsby McClellan [65] (see also [66]). Computation of NTTusing streaming SIMD extensions (SSEs), graphicalprocessing units and multi-core CPUs should yield further

Fig. 5 Performance of the 2D NTT (via [58]) compared with that of th

Times (in ms) are shown on a logscale and the results were obtained using M = 5×


significant performance gains. In the following section, thedigital NRT is defined using the NTT.

3 Number-theoretic Radon transform

The NRT provides a mapping of discrete projections to the2D NTT. The discrete projection bins N (m, t) and M(s, t)are defined as

N (m, t) =∑N−1

x=0

I x, kmx+ tlN( )

(modM ) (11)

M(s, t) =∑N−1

y=0

I k psy+ tlN , y

( )(modM ) (12)

along the lines (3) and (4), where M is an integer definedwithin the NTT (see Definition 1). Let N0 denote the set ofnon-negative integers including zero, then the discreteprojections of the NRT are the set of bins with the slopes mand s as

m = m:m , N , m [ N0

{ }(13)

s = s:s , N/p, s [ N0

{ }(14)

respectively, for the set of translates t as

t = t:t , N , t [ N0

{ }(15)

The NST partitions 2D NTT space into slices along thevectors [−m, 1] and [1, −ps]. The vectors represent thelines perpendicular to the lines (3) and (4). The NRT iscomputed (in O(N2 log2 N ) using the fast NTT) via theNST, as follows:

(1) compute the 2D NTT of the image.

e 2D FFT for two machine types

213 + 1 and α = 3


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(2) extract each slice m and s from the 2D NTT space at thevectors [−m, 1] and [1, −ps], since the slices are rotated by90° in NTT space.(3) compute the 1D inverse NTT of all the N + N/p slices togive the NRT projections.
The NRT may also be extended to arbitrary composite sizesusing the work of Kingston and Svalbe [21].Although the NRT projections do not appear to be practical

because of their wrapping nature, projections within the MT,which are good approximations of parallel-beam x-rayintensity profiles, can be mapped/repacked as NRTprojections. Svalbe and others [24–27] have mappeddiscrete projections of the discrete FST in order toreconstruct objects from conventional tomography using theMT. These methods will also be applicable to the inverseNRT, which is computed (in O(N2 log2 N ) using the fastiNTT) as the following:

(1) compute the 1D NTT of all NRT projections. The resultswill form slices in NTT space. If the projections arecomplex-valued, one may use the transforms ofNussbaumer [60].(2) place each slice m and s in 2D NTT space at the vectors[−m, 1] and [1, −ps].(3) if N = pn with n > 1, filter the space by using the exactdigital filter S for oversampling.(4) invert the 2D NTT space using the 2D iNTT.

3.1 Properties

The main advantages of the NST, as opposed to the FST, arethe following:

(1) the NRT is a purely digital transform with the sameperformance as other fast methods, but with no round-offerrors.

Fig. 6 Performance of the NRT (via [58]) compared with that of the di

Times (in ms) are shown on a logscale and the results were obtained using M = 5×


(2) the NRT does not require an FPU for digital projectiondata and so the performance of the NRT on any given CPUis independent of the FPU. The NRT is also well suited forcomputation on multi-core CPUs for this reason.(3) for certain choices of N, M and α = 2 for the NTT, it ispossible to compute the NRT without any multiplications,that is, with only bit-shifts and additions [50]. See recentwork of Chandra [67].(4) there is no need for complex numbers, so the NRT is idealfor real data. Complex NTT can be used to extend thetransform lengths or for complex-valued projection data [60].(5) the NRT also uses less memory than FST schemes since itis real-valued and can utilise 16 bit integers to maintain fullrepresentation.(6) exact digital convolution can be evaluated using the NTT,even when intermediate results overflow modulo M, providedthe initial values are less than M [51].(7) pre-processing of projection/slice data via exact digitalfiltering or convolution is possible because arithmetic(including division) can be done exactly within an integerfield.

3.2 Performance

The performance of the NRT (using a new C library [58]) iscompared with that of the DRT, using the well-known FFTWlibrary by [63], in Fig. 6. The test images consisted of theimage of Lena with dyadic sizes up to 213 and zero paddedwhere appropriate. One can see that the performanceadvantage of the NTT to the FFT is preserved within theNRT when compared with the discrete FST as thecomputational complexities for acquiring the projectionsand their placement into the transform space areapproximately the same. It is important to note that theNRT always has zero reconstruction error, which has beenshown to be particularly important when computing a verylarge number of convolutions [68]. The performance of the

screte Fourier slice theorem for two machine types

213 + 1 and α = 3


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NRT could be boosted by utilising hardware implementationsof the NTT by McClellan [65] (and also [66]) or byimplementation on integer-only computer architectures.
3.3 Significance

The NRT is a purely digital RT. Up till now, all previousdiscrete RTs have relied on discrete versions of the FST forfast reconstructions, making them susceptible to round-offerrors. For instance, Chandra et al. [26] found that 20 2Dconvolutions for an N = 479 image overflowed and requiredusing infinite precision integers to complete convolutions tosolve discrete ghosts for this size. The NRT does notdepend on the DFT and does not have this weakness. TheNST is also the first purely digital slice theorem, with nocounterpart in integral (continuous) geometry. Appendix 2presents the NRT in a more general way using circulantmatrices [69]. The result is a circulant theory of the Radontransform [70] that could allow a number of Vandermondematrix and root of unity combinations to construct otherdiscretised Radon transforms. The NRT projections areideal for number-theoretic cryptology such as RSAencryption. Further work needs to be done to ascertain whatnumber-theoretic properties are present and useful withinNRT projections.

4 Conclusion

In this paper, a discrete RT, called the NRT, was constructedthat preserves all the properties of the RT in digital form. TheNRT has the same computational complexity andperformance as the DRT computed through the FFT (seeFig. 6) while achieving exact reconstructions using onlyintegers. NRT also has no round-off errors, can bemultiplication free (via bit-shifts) for all cases where anappropriate modulus can be chosen (see Section 3.1) and isimpervious to numerical overflow. The transform was madepossible by showing that the slice theorem is also validwithin the NTT via a circulant theory of the Radontransform. This new NST has no integral (continuous)transform counterpart because it is only valid on a pointlattice and within the realm of integers. Future work willexploit number-theoretic properties of discrete Radonprojections for encoding and encryption schemes based onthe NRT.

5 Acknowledgment

S. Chandra would like to thank the Monash University,Australia for a Ph.D. scholarship and a publications award.

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7 Appendix

7.1 Appendix 1: technical notes

The benchmarks were done using the Ubuntu 64 bitoperating system with the GNU compiler collection(GCC). All benchmarks were performed with the samecompiler flags, FFTW library version (3.1.2) withESTIMATE and optimisations. The average time from a100 runs were taken for each image size. The duration foreach run was taken as the average of the forward andinverse transforms. The error bars indicate the maximumand minimum values involved within the 100 runs. Theofficial (non-SSE) FFTW build (with ESTIMATE) wasused which is provided by the FFTW team. Code for thebenchmarks and the C library for NTT/NRT, as well asthe FFT code used for comparison, can be found in thenew open source finite transform library (FTL) [58].

7.2 Appendix 2: circulant theory of the Radontransform

In this section, it will be shown that the discrete FST ispossible using circulant matrices and the DFT. For a moredetailed description, see the thesis of Chandra [70]. Thistheory will be extended to include the NTT, resulting in theNST. The theory can be constructed by noting that thestructure of the congruences (3) and (4), when representedin matrix form, is a circulant matrix (or simply a circulant,see part (b) of Fig. 2).

Definition 2: (m-circulant (Also referred to as a generalisedcirculant or g-circulant [69].). An m-circulant is an N ×Nmatrix containing a unique row f(x) with x = 0, …, N−1replicated on each row, but where each row is cyclicallyshifted (mod N ) by an additional m elements to the right.Hence, the line of slope 1 given in part (a) of Fig. 2 is a1-circulant. A line of slope 2 will be a 2-circulant and soon. The important property of the circulant is that itrepresents a cyclic convolution.

7.2.1 Circulant diagonalisation: In Section 2, it wasmentioned that the CCP is an important property of boththe DFT and the NTT. In the circular convolution (7), eachcyclically shifted version of f can be represented as a rowof a circulant matrix (i.e. it is a 1-circulant as part (b) ofFig. 2). The matrix multiplies with the elements of g incolumn vector form to compute the circular convolution.Equation (7) implies that the circulant form of f is‘diagonalised’, that is, transformed into a monomial matrix,a matrix with one entry per row, in order to multiply it bythe DFT of g to obtain h.


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Definition 3: (monomial matrix [69]). A monomial matrix Mis an N × N matrix of the form
M = PL (16)

where Λ is an N ×N diagonal matrix and P is an N ×Npermutation matrix with N non-zero elements representing adiscrete line at slope m(mod N ). The matrices M then mapΛ matrices to discrete lines with slopes m(mod N ).

The transformation of a circulant into a monomial matrix isknown as m-circulant diagonalisation [69].

Definition 4: (circulant diagonalisation [69]). An m-circulantC is diagonalised by

C = V†MV (17)

where M represents eigenvalues along a discrete line withslope m (see Definition 3) and † represents the Hermitianconjugate. The matrix V is a Vandermonde matrix, whichalways has an inverse [71] and is a matrix of the form

V xy = ay−1x (18)

A special case of the Vandermonde matrix is the Fouriermatrix F, where α is the root of unity e2πiσ so that

Fxy = e2pixy (19)

Matrix (20) shows an example of F for N = 4, where thepowers of α may be expressed modulo N making the matrixsymmetric

F4 =1 1 1 11 a1 a2 a3

1 a2 a4 a6

1 a3 a6 a9

⎡⎢⎢⎣

⎤⎥⎥⎦ =

1 1 1 11 a1 a2 a3

1 a2 a0 a2

1 a3 a2 a1

⎡⎢⎢⎣

⎤⎥⎥⎦ (20)

Combining Definition 4 with the Fourier matrix, one obtains a


special case of the circulant diagonalisation known as Fourierdiagonalisation and the discrete FST.The discrete inverse problem requires that the set of

discrete projections tile all of DFT space evenly. This canbe shown using the well understood tiling of discrete lineson a finite torus [11] and diagonalisation of Definition 4.

Proposition 1: (finite tiling) the lines (3) and (4) tile all of anN ×N space for the slopes 0≤m < p and 0≤ s <N/p when t isfixed.

Proof: for the case N = p, Gertner [17] showed that the set oflines in (3), for 0≤m < p, span the quadratic grid exactlyonce. The tiling is shown in Fig. 3 for p = 5 and is madepossible because the gcd(m, p) = 1 always (i.e. m is alwayscoprime to p) since p is prime. Hence, all the lines in theset (3) for a fixed t may intersect only once.For the cases N = pn with n > 1, Kingston [20] showed thatline (3), for 0≤m <N, under-samples the N ×N space. Theunder-sampling occurs because N/p of the unique slopes mare not coprime to N. Therefore the lines having slopescoprime to N intersect only once for a fixed t, but theslopes not coprime to N will intersect p times leading tounder-sampling. The line (4), for 0≤ s <N/p, then providesfor the sampling of the remaining space because it is‘perpendicular’ to (3). However, the union of these twoprojection sets leads to some oversampling, which is easilyremoved by the exact filter S.To construct the NRT, it is necessary to show that the

circulant diagonalisation is possible using the NTT. Sincethe the number-theoretic matrix N, whose entries are

N xy = axy(mod M ) (21)

with α andM defined by Fermat’s (little) theorem (8), is also aVandermonde matrix and its entries consist of powers of aroot of unity, both Definition 4 and Proposition 1 areapplicable and make the NRT possible (see Chandra [70]for details). □


exact image representation via a number-theoretic radon transform

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