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Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
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
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
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].
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
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?
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
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)?
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Inverse Problem Schematic
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
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.
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
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.
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
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.
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
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
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
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?
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Radon TransformSinograms
The Inverse Problem for CT
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Radon TransformSinograms
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
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Radon TransformSinograms
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?
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Radon TransformSinograms
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.
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Radon TransformSinograms
A Phantom:
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Radon TransformSinograms
Sinogram
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Radon TransformSinograms
Whose sinogram is this?
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Radon TransformSinograms
Whose sinogram is this?
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Radon TransformSinograms
Whose sinogram is this?
Image Source http://images.hollywood.com/site/homer-simpson.jpg
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Radon TransformSinograms
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.
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
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.
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Back-ProjectionRadon InversionThe Reconstructions
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.
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Back-ProjectionRadon InversionThe Reconstructions
A Phantom:
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Back-ProjectionRadon InversionThe Reconstructions
Back-projection: Blurring
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Back-ProjectionRadon InversionThe Reconstructions
Back-projection: Fail
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Back-ProjectionRadon InversionThe Reconstructions
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)).
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Back-ProjectionRadon InversionThe Reconstructions
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.
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Back-ProjectionRadon InversionThe Reconstructions
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).
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Back-ProjectionRadon InversionThe Reconstructions
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.
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Back-ProjectionRadon InversionThe Reconstructions
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
∫ ∞−∞
e ir<x,ω>FtRf (r , ω)|r | drdω.
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Back-ProjectionRadon InversionThe Reconstructions
Remarks on Radon Inversion Formula
f (x) =1
(2π)2
∫ π
0
∫ ∞−∞
e ir<x,ω>FtRf (r , ω)|r | drdω.
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.
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Back-ProjectionRadon InversionThe Reconstructions
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 ).
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Back-ProjectionRadon InversionThe Reconstructions
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.
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Back-ProjectionRadon InversionThe Reconstructions
Reconstruction of “bump” phantom
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
Back-ProjectionRadon InversionThe Reconstructions
Reconstruction of Homer
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
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 .
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
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.
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
References
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?
Ryan Walker CT Scans and an Introduction to Inverse Problems
Inverse ProblemsCT Inverse Problem
ReconstructionAdvanced Issues
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.
Ryan Walker CT Scans and an Introduction to Inverse Problems