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Inverse ProblemsCT Inverse Problem

ReconstructionAdvanced Issues

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CT Scans and an Introduction to Inverse Problems(or what I did on my summer vacation)

Ryan Walker

November 15, 2010

Ryan Walker CT Scans and an Introduction to Inverse Problems



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Acknowledgements

I draw much of the theoretical background and intuition onthe CT scan problem from a series of brilliant lecturesdelivered by Gunther Uhlmann, Peter Kuchment, and LeonidKunyansky at the IPDE Summer School 2010 at theUniversity of Washington.

A comprehensive introduction to the mathematics of CTscans can be found in Charles Epstein’s book, Introduction tothe Mathematics of Medical Imaging [1].

A more advanced treatment can be found in the classic bookof Frank Natter, The Mathematics of Computed Tomography,[2].




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Inverse Problems: An Overview

An exciting and expanding area of mathematics

A rich source of theoretical and applied problems

A direct problem: Know the cause; find the effect. Example:Know how X-rays waves attenuate as they pass through abody in response to size, shape, and material composition ofthe body. Knowing the cause, we can determine theattenuation of the radiation.

An inverse problem: Know the effect; find the cause.Example: If the attenuated X-ray radiation is measured, what(if anything) can we say about the size, shape, and materialcomposition of the body?




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Inverse Problem Schematic

Have a system configuration f (x)

A process L acts on the system configuration.

Measure data d

Symbolically, the inverse problem is:

L(f (x)) = d What is f (x)?




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Inverse Problem Schematic




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The CT Problem

Application: The Computed Tomography problem: want toview the internal structure of something without cutting itopen.

1 Send radiation through the object and look at how itattenuates.

2 Determine the density by looking at the attenuation patterns.3 Recover detailed image of internal structure from the density

Example Problem: (Medical Computed Tomography (CT))Tumors and other abnormalities have specific densities,distinct from healthy tissues. Would like to look at a patient’sbrain without cutting him open. Recover detailed images ofthe brain by knowing the densities at different points.




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Patient lies between a rotating source and detector. The sourceemits x-ray radiation in straight beams and the attenuated signalbeam is collected by the detector.




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A model for X-ray Attenuation

Goal: Image a 2D slice of the patient’s head. First, need tosolve the direct problem.

Let I(x) be the flux of radiation at the point x ∈ R2. and letI (x) = ‖I(x)‖ be the intensity of the radiation at x.

Let µ(x) be the attenuation coefficient for X-ray radiation.This coefficient describes the energy loss for radiation passingthrough x and is determined by the density and materialsproperties of the body.




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A model for X-ray Attenuation

For X-ray radiation traveling along a straight line we haveBeer’s Law:

dI

dx= −µ(x)I

So if L is the line segment on the source and detector and thesource emits initial intensity I0, then the detector on theopposite side feels a radiation intensity

I = I0e−

∫L µ(x) dx




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Radon TransformSinograms

The Inverse Problem for CT

We want to find µ(x) which is essentially a proxy for thedensity of the patient’s head in the slice at the point x.

We know all the source intensities and we can compute all thedetector intensities by computing all the integrals over all thelines on the sources and detectors.

So solving the inverse problem is equivalent to resolving thefollowing question: If we know all the line integrals of afunction f (x) in a domain Ω, is this enough information torecover the function in Ω?




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The Inverse Problem for CT




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The Radon Transform

Let L be a line in R2. Let t denote the perpendicular distance tothe origin, ω a unit vector perpendicular to L, and ω⊥ a unit vectorperpendicular to ω. Then the points x ∈ L are those which satisfy

< x , ω >= t




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The Radon Transform

The Radon Transform of f ∈ C∞0 (R2) is the map given by:

Rf (t, ω) =

∫Lt,ω

f (x) dx =

∫ ∞−∞

f (tω + sω⊥) ds.

R maps functions in C∞0 (R2) into functions mapping lines inthe plane into R.

Even: Rf (−t,−ω) = Rf (t, ω)

Domain? Range?




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Sinograms

The raw output of a CT scan procedure is called a sinogram.

Typically, we visualize this data by a “heat” plot of the valuesof the Radon transform against t and θ.

Darker and lighter areas of the sinogram plot correspond todifferent values of the Radon transform.

We won’t usually have real X-ray data to use, but image filesare a great substitute. Pixel locations correspond to points inspace and gray scale colors correspond to densities.




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A Phantom:




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Sinogram




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Whose sinogram is this?




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Image Source http://images.hollywood.com/site/homer-simpson.jpg




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Sine Waves in Sinograms

The appearance of sine waves in sinograms is due to the factthat the Radon transform of a delta function has support on atrigonometric curve.

Why? Regard the point x as a vector with angle θ0. Bytrigonometry the only lines containing x will be those withangle θ and affine parameter t satisfying

t = |x | cos (θ − θ0).

Since Rδx(t, θ) = 0 whenever x is not in Lt,θ, the support ofδx is a cosine curve in the θ-t plane.

We can show that the Radon transform is continuous and thusan object made up of many small,sharp-edged features willhave a sinogram that is a combination of blurred sine curves.


http://images.hollywood.com/site/homer-simpson.jpg



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Back-ProjectionRadon InversionThe Reconstructions

Back-Projection

Idea: Know the integrals of f over every line; reconstruct f (x)by averaging all the integrals of the lines passing through x.

f (x) =1

2π

∫ 2π

0Rf (< x, ω(θ) >, θ) dθ

Theory: Regarding the Radon transform R as a map betweentwo appropriate function spaces, we can prove thatback-projection (R#) is the adjoint of R.




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Back-Projection

For functions f (x) from a “good domain” space, recalling thatt = x · ω, compute:

R#Rf (x) =

∫S1

Rf (x · ω, θ) dω

=

∫S1

∫ ∞−∞

f ((x · ω)ω + sω⊥) dsdω

Basically, a polar integral. Rewrite:

R#Rf (x) =

∫2f (y)

|y − x |dy =

2

|x |∗ f (x).

Thus, we should recover not f (x) but a blurred version of f (x)with back-projection.




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A Phantom:




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Back-projection: Blurring




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Back-projection: Fail




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Fourier Transform

Our task is to “invert” the Radon transform. To do this, we willmake use of the Fourier transform.

Definition

The Fourier Transform of a function f (x) on R is

(F f )(ξ) =

∫ ∞−∞

f (x)e−iξx dx .

and on Rn as

f (ξ) =

∫Rn

e−ix ·ξf (x) dx .

Domains? Ranges?Roughly, the Fourier Transform says how much of eachfrequency ξ is present in the function f .Extremely important transform with many interestingproperties.Invertible (as a map from L2(R)→ L2(R)).




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Projection-Slice Theorem

There is a simple relationship between the Fourier and Radontransforms

Theorem ∫ ∞−∞Rf (t, ω)e−itr dt = f (rω)

The 1D Fourier transform of Rf in the affine parameter t isthe 2D Fourier transform of f expressed in polar coordinates.




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Proof of the Projection-Slice Theorem

Proof. By definition∫ ∞−∞Rf (t, ω)e−itr dt =

∫R2

f (tω + sω⊥)e−itr dsdt.

Change variables: x = tω + sω⊥ and use the fact thatt =< x, ω > to obtain∫

R2

f (tω + sω⊥)e itr dsdt =

∫R2

f (x)e−i<x,ω>r dx = f (rω).




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Radon Inversion Formula

Theorem

Let f be a function such thatf ∈ L1(R2) ∩ domain of the Radon transform and f ∈ L1(R2),then

f (x) =1

(2π)2

∫ π

0

∫ ∞−∞

e ir<x,ω>FtRf (r , ω)|r | drdω.




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Proof of Radon Inversion Formula

Proof. Trace through definitions to see that Rf is even

FtRf (−r ,−ω) = FtRf (r , ω).

Then by the Fourier inversion theorem:

f (x) =1

(2π)2

∫R2

e i<x,ξ>f (ξ) dξ

=1

(2π)2

∫ 2π

0

∫ ∞0

e ir<x,ω>f (rω)r drdω

=1

(2π)2

∫ 2π

0

∫ ∞0

e ir<x,ω>FtRf (r , ω)r drdω

=1

(2π)2

∫ π

0

∫ ∞−∞





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Remarks on Radon Inversion Formula

f (x) =1

(2π)2

∫ π

0

∫ ∞−∞


Radial integral is a filter (|r |) applied to the Radon transform.Back-projection is the angular integral.

Formula is often called filtered back-projection formula.

Factor |r | suppresses low-frequency components and amplifieshigh frequency components.

Extreme care is needed to be more specific about domains andranges for Radon transform/Inversion formula. E.g. Fouriertransform of a delta function is 1, and 1 is not integrable.




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A second look at filtration

It is very interesting to note that if we replace |r | with just rin the formula

f (x) =1

(2π)2

∫ π

0

∫ ∞−∞

e ir<x,ω>FtRf (r , ω)|r | drdω

we could write

f (x) =1

2πi

∫ π

0

∫ ∞−∞

e ir<x,ω> Ft(∂tRf )(r , ω) drdω

=1

2πi

∫ π

0

∫ ∞−∞

∂tRf (t, ω)dω

We can derive a new, useful back-projection formula byintroducing the Hilbert transformation Hg = F−1(sgng):

f =1

4πR#H d

dt(Rf ).




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Numerical Methods

Given: real or simulated Radon transform data over a discreterange of affine parameter values and angles.

Our samples are approximately 500 pixels x 500 pixels. Wesample 200 evenly spaced angles in [0, 2π]. For each anglesample we approximate the line integral through the angle at130 different values of t. This gives about 26,000 sample datapoints.

Use the Hilbert transform back-projection formula:

f =1

4πR#H d

dt(Rf ).

Use differencing to approximate ddtRf .

Possible to approximate H ddt (Rf ) as a convolution

Integrate approximately to back-project.




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Reconstruction of “bump” phantom




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Reconstruction of Homer




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Important details: Domains, Stability

Domains: For what types of functions will the Radontransform be defined?

Need f (x , y) to be such that the restriction to any line will belocally integrableNeed some decay at infinity for convergence of integralsSmooth functions of compact support will certainly work.What else?

Stability: Do small errors in our measurement of Rf lead tosmall errors in the reconstruction of f ?

R turns out to be a smoothing operator: Rf has 1/2 morederivatives than f .




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Important details: Range

Range: What functions can be Radon transforms of otherfunctions?

Harder question.For a good choice of domain, R has zero kernel. In general,this means that many choices for R−1, the left inverse of R,exist.Need useful range conditions which produce uniqueness of theRadon inversion procedure.




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Practical details

Stability!

What angular range do we really need to use? Can we exploitsymmetries and take fewer measurements or view over smallerrange?

Resolution: What is the scale of detail in our reconstructions?

Contrast: Usually more interested in regions of sharptransition vs smoother textures. Can we emphasize these inour reconstruction?




References

References

Charles L. Epstein.Introduction to the mathematics of medical imaging.Society for Industrial and Applied Mathematics (SIAM),Philadelphia, PA, second edition, 2008.

F. Natterer.The mathematics of computerized tomography.B. G. Teubner, Stuttgart, 1986.


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