Mathematical Modelling of X-Ray CT

Mathematical Modellingof

X-Ray Computed Tomography

Sophia Bethany Coban

University of Manchester

May 20, 2013

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

Outline

1 IntroductionPhysical ModelMathematical Model

2 Continuous Data

3 Discrete Data

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 2/57


Discrete Data


X-Ray Computed Tomography

is a very important and a popular tool in imaging.

allows us to see insides of an object... without destroying it!

is commonly used in medical imaging and non-destructivematerial testing.



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Physical Model

The source shoots out X-rays ata given energy.

Rays travel through the objectand reach the detectors withless energy.

In other words, the rays areattenuated.

The detector records the intensity (energy) of the arriving rays.



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Physical Model







Discrete Data


Physical Model







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Physical Model







Discrete Data


Physical Model







Discrete Data


Mathematical Model

What we know:

The initial intensity of the X-ray beam, Iin, when it leaves thesource, and

The final intensity Iout at the detectors.

What we want to find out:

A map of grey values, resembling the insides of the object.



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Mathematical Model

What we know:







Discrete Data


Mathematical Model

What we know:







Discrete Data


Mathematical Model

What we know:







Discrete Data


Mathematical Model

What we know:







Discrete Data


Physical Model







Discrete Data


Mathematical Model

What we know:





How?:



Discrete Data


Formulation

Let us begin with a monochromatic beam (one energy).

Consider a small intensity dIout

at position dL. dIout is given by

dIout = − IinµdL.

Rearrange to get

dIout

Iin= −µdL.



Discrete Data


Formulation





Rearrange to get

dIout

Iin= −µdL.



Discrete Data


Formulation





Rearrange to get

dIout

Iin= −µdL.



Discrete Data


Formulation





Rearrange to get

dIout

Iin= −µdL.



Discrete Data


Formulation





Rearrange to get

dIout

Iin= −µdL.



Discrete Data


Formulation





Rearrange to get

dIout

Iin= −µdL.



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Formulation





Rearrange to get

dIout

Iin= −µdL.



Discrete Data


Formulation

Integrate both sides to get∫L

Iout

Iin= −

∫LµdL,

which implies

lnIout

Iin= −

∫LµdL.

Beer – Lambert Law

Light is attenuated exponentially as it travels through an object.Mathematically, this means absorption = − ln(Iout/Iin).



Discrete Data


Formulation


Iout

Iin= −

∫LµdL,

which implies

lnIout

Iin= −

∫LµdL.





Discrete Data


Formulation


Iout

Iin= −

∫LµdL,

which implies

lnIout

Iin= −

∫LµdL.





Discrete Data


Formulation


Iout

Iin= −

∫LµdL,

which implies

lnIout

Iin= −

∫LµdL.





Discrete Data


Formulation

So for a monochromatic beam we have

Iout = Iine−RL µdL.

Most X-ray sources produce a polychromatic beam (beamwith a range of energies), which means the attenuationcoefficient depends on energy, E at position x ,

Iout =

∫Iin(E )e−

RL µ(x ,E)dLdE .

Mathematically, the goal of X-ray CT is to recover theattenuation coefficient, µ, from the information at thedetectors, Iout .



Discrete Data


Formulation




Iout =

∫Iin(E )e−

RL µ(x ,E)dLdE .




Discrete Data


Formulation




Iout =

∫Iin(E )e−

RL µ(x ,E)dLdE .




Discrete Data

Forward ProjectionSinogramsBackward Projection

Outline

1 Introduction

2 Continuous DataForward ProjectionSinogramsBackward Projection

3 Discrete Data



Discrete Data


Continuous Data: Radon Transform

Recovering µ from only one projection is not easy!

We must scan the object at different angles to understandhow the linear attenuation coefficient varies along the line theX-rays travel.

This is called the forward problem, obtained by taking theRadon transform of µ,

R[µ](s,−→θ ) = ln(Iout/Iin).



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Discrete Data









Discrete Data


A Simple Sinogram Example



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More Sinograms!



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More Sinograms!



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More Sinograms!



Discrete Data



Forward problem using the Radon transform,


The transform is named after Johann Radon for his work in1917 (before X-Ray tomography was invented!).

Radon also provided an analytical inversion formula for thistransform (backward problem).

Exact methods aim to approximate the inverse Radontransform to get µ,

µ(x) = R−1[ln(Iout/Iin)].



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Discrete Data











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Discrete Data

Ax = b?Sparsity of AIssues with Solving Ax = b

Outline

1 Introduction

2 Continuous Data

3 Discrete DataAx = b?Sparsity of AIssues with Solving Ax = b



Discrete Data


Discrete Data

Reminder: The goal of X-ray CT is to recover theattenuation coefficient, µ, from the information at thedetectors, Iout .

This problem can be written in the form Ax = b.

How?:



Discrete Data


Discrete Data



How?:



Discrete Data


Discrete Data



How?:



Discrete Data


Discrete Data: Ax = b

Let us consider a slice of an objectbroken into 9 pixels.

We want to find the pixel values.

x = [x1, x2, . . . , x9]T.



Discrete Data





x = [x1, x2, . . . , x9]T.



Discrete Data





x = [x1, x2, . . . , x9]T.



Discrete Data



line 1: b1 = x1 + x2 + x3.



Discrete Data



A =

x1 x2 x3 x4 x5 x6 x7 x8 x9

line 1: 1 1 1 0 0 0 0 0 0



Discrete Data



line 2: b2 = x4 + x5 + x6.



Discrete Data



A =

x1 x2 x3 x4 x5 x6 x7 x8 x9

line 1: 1 1 1 0 0 0 0 0 0

line 2: 0 0 0 1 1 1 0 0 0



Discrete Data



line 3: b3 = x7 + x8 + x9.



Discrete Data



A =

x1 x2 x3 x4 x5 x6 x7 x8 x9

line 1: 1 1 1 0 0 0 0 0 0

line 2: 0 0 0 1 1 1 0 0 0

line 3: 0 0 0 0 0 0 1 1 1



Discrete Data


Discrete Data



Discrete Data


Discrete Data



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Discrete Data

So we can say that

The geometry matrix A is very large and sparse.

Rows of A correspond to the lines traveling through theobject.

Columns of A are the pixels of the object.

A is rarely square; usually we have an overdetermined system(i.e. m > n).

One final remark on sinograms!



Discrete Data


Issues with solving Ax = b

Let us consider an example with two projections sets.



Discrete Data



x = [x1, x2, . . . , x16]T.

After the first projection set, we have

b(1 : 4, 1) = 2; and

A(1, 1 : 4) = 1;A(2, 5 : 8) = 1;A(3, 9 : 12) = 1;A(4, 13 : 16) = 1;



Discrete Data



x = [x1, x2, . . . , x16]T.


b(1 : 4, 1) = 2; and

A(1, 1 : 4) = 1;A(2, 5 : 8) = 1;A(3, 9 : 12) = 1;A(4, 13 : 16) = 1;



Discrete Data



x = [x1, x2, . . . , x16]T.


b(1 : 4, 1) = 2; and

A(1, 1 : 4) = 1;A(2, 5 : 8) = 1;A(3, 9 : 12) = 1;A(4, 13 : 16) = 1;



Discrete Data



x = [x1, x2, . . . , x16]T.


b(1 : 4, 1) = 2; and

A(1, 1 : 4) = 1;A(2, 5 : 8) = 1;A(3, 9 : 12) = 1;A(4, 13 : 16) = 1;



Discrete Data



After the second projection set, we have

b(5 : 8, 1) = 2; and

A(5, [1 : 4 : 16]) = 1;A(6, [2 : 4 : 16]) = 1;A(7, [3 : 4 : 16]) = 1;A(8, [4 : 4 : 16]) = 1;



Discrete Data




b(5 : 8, 1) = 2; and

A(5, [1 : 4 : 16]) = 1;A(6, [2 : 4 : 16]) = 1;A(7, [3 : 4 : 16]) = 1;A(8, [4 : 4 : 16]) = 1;



Discrete Data




b(5 : 8, 1) = 2; and

A(5, [1 : 4 : 16]) = 1;A(6, [2 : 4 : 16]) = 1;A(7, [3 : 4 : 16]) = 1;A(8, [4 : 4 : 16]) = 1;



Discrete Data


MATLAB Commands

spy(A)

x=CGLS(A,b,zeros(16,1),1e-15);

x=reshape(x,[4,4]);

imagesc(x); colormap(gray)



Discrete Data


MATLAB Commands

spy(A)


x=reshape(x,[4,4]);




Discrete Data


MATLAB Commands

spy(A)


x=reshape(x,[4,4]);




Discrete Data


MATLAB Commands

spy(A)


x=reshape(x,[4,4]);




Discrete Data



The answer is not correct!

This is a least squares solution.

There is no unique answer!

We need more information!



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Another Example

ForwardProjection



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Another Example

ForwardProjection

ReconstructedImage



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Another Example

ForwardProjection

ReconstructedImage

PhantomImage



Discrete Data


References and Further Read

J. Radon, ”On the determination of functions from theirintegral values along certain manifolds”, IEEE Transactions onMedical Imaging, 5(4):170176. 1917. (Translated in 1986).

F. Natterrer,The Mathematics of Computerized Tomography,Classics in Applied Mathematics, Society for Industrial andApplied Mathematics.

R.L. Siddon, ”Fast calculation of the exact radiological pathfor a three-dimensional CT array,” Med. Phys. 12(2), 252258(1985).

Online notes by Marta Betcke and Bill Lionheart. Link:http://www.maths.manchester.ac.uk/ mbetcke/VCIPT/.


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Mathematical Modelling of X-Ray CT