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

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 3/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 4/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 4/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 4/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 4/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 4/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 5/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 5/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 5/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 5/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 5/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 6/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

How?:

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 7/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 8/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 8/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 8/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 8/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 8/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 8/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 8/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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).

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 9/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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).

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 9/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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).

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 9/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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).

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 9/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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 .

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 10/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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 .

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 10/57

IntroductionContinuous Data

Discrete Data

Physical ModelMathematical Model

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 .

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 10/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

Outline

1 Introduction

2 Continuous DataForward ProjectionSinogramsBackward Projection

3 Discrete Data

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 11/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

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).

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 12/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

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).

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 12/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

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).

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 12/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

A Simple Sinogram Example

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 13/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

A Simple Sinogram Example

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 14/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

A Simple Sinogram Example

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 15/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

A Simple Sinogram Example

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 16/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

A Simple Sinogram Example

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 17/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

A Simple Sinogram Example

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 18/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

A Simple Sinogram Example

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 19/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

A Simple Sinogram Example

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 20/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

A Simple Sinogram Example

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 21/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

More Sinograms!

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 22/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

More Sinograms!

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 23/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

More Sinograms!

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 24/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

Continuous Data: Radon Transform

Forward problem using the Radon transform,

R[µ](s,−→θ ) = ln(Iout/Iin).

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)].

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 25/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

Continuous Data: Radon Transform

Forward problem using the Radon transform,

R[µ](s,−→θ ) = ln(Iout/Iin).

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)].

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 25/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

Continuous Data: Radon Transform

Forward problem using the Radon transform,

R[µ](s,−→θ ) = ln(Iout/Iin).

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)].

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 25/57

IntroductionContinuous Data

Discrete Data

Forward ProjectionSinogramsBackward Projection

Continuous Data: Radon Transform

Forward problem using the Radon transform,

R[µ](s,−→θ ) = ln(Iout/Iin).

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)].

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 25/57

IntroductionContinuous Data

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

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 26/57

IntroductionContinuous Data

Discrete Data

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

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?:

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 27/57

IntroductionContinuous Data

Discrete Data

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

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?:

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 27/57

IntroductionContinuous Data

Discrete Data

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

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?:

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 27/57

IntroductionContinuous Data

Discrete Data

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

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 28/57

IntroductionContinuous Data

Discrete Data

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

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 28/57

IntroductionContinuous Data

Discrete Data

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

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.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 28/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data: Ax = b

line 1: b1 = x1 + x2 + x3.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 29/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data: Ax = b

A =

x1 x2 x3 x4 x5 x6 x7 x8 x9

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

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 30/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data: Ax = b

line 2: b2 = x4 + x5 + x6.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 31/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data: Ax = b

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

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 32/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data: Ax = b

line 3: b3 = x7 + x8 + x9.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 33/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data: Ax = b

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

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 34/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 35/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 36/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 37/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 38/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 39/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 40/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 41/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 42/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 43/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 44/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 45/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 46/57

IntroductionContinuous Data

Discrete Data

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

Discrete Data

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 47/57

IntroductionContinuous Data

Discrete Data

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

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!

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 48/57

IntroductionContinuous Data

Discrete Data

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

Issues with solving Ax = b

Let us consider an example with two projections sets.

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 49/57

IntroductionContinuous Data

Discrete Data

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

Issues with solving Ax = b

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;

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 50/57

IntroductionContinuous Data

Discrete Data

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

Issues with solving Ax = b

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;

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 50/57

IntroductionContinuous Data

Discrete Data

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

Issues with solving Ax = b

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;

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 50/57

IntroductionContinuous Data

Discrete Data

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

Issues with solving Ax = b

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;

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 50/57

IntroductionContinuous Data

Discrete Data

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

Issues with solving Ax = b

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;

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 51/57

IntroductionContinuous Data

Discrete Data

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

Issues with solving Ax = b

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;

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 51/57

IntroductionContinuous Data

Discrete Data

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

Issues with solving Ax = b

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;

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 51/57

IntroductionContinuous Data

Discrete Data

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

MATLAB Commands

spy(A)

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

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

imagesc(x); colormap(gray)

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 52/57

IntroductionContinuous Data

Discrete Data

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

MATLAB Commands

spy(A)

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

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

imagesc(x); colormap(gray)

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 52/57

IntroductionContinuous Data

Discrete Data

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

MATLAB Commands

spy(A)

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

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

imagesc(x); colormap(gray)

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 52/57

IntroductionContinuous Data

Discrete Data

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

MATLAB Commands

spy(A)

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

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

imagesc(x); colormap(gray)

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 52/57

IntroductionContinuous Data

Discrete Data

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

Issues with solving Ax = b

The answer is not correct!

This is a least squares solution.

There is no unique answer!

We need more information!

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 53/57

IntroductionContinuous Data

Discrete Data

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

Issues with solving Ax = b

The answer is not correct!

This is a least squares solution.

There is no unique answer!

We need more information!

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 53/57

IntroductionContinuous Data

Discrete Data

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

Issues with solving Ax = b

The answer is not correct!

This is a least squares solution.

There is no unique answer!

We need more information!

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 53/57

IntroductionContinuous Data

Discrete Data

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

Issues with solving Ax = b

The answer is not correct!

This is a least squares solution.

There is no unique answer!

We need more information!

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 53/57

IntroductionContinuous Data

Discrete Data

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

Another Example

ForwardProjection

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 54/57

IntroductionContinuous Data

Discrete Data

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

Another Example

ForwardProjection

ReconstructedImage

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 55/57

IntroductionContinuous Data

Discrete Data

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

Another Example

ForwardProjection

ReconstructedImage

PhantomImage

Sophia Bethany Coban Mathematical Modelling of X-Ray CT 56/57

IntroductionContinuous Data

Discrete Data

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

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/.

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