Tomography and Reconstruction

School of Engineering

Tomography and Reconstruction

Lecture Overview

•Applications

•Background/history of tomography

•Radon Transform

•Fourier Slice Theorem

•Filtered Back Projection

•Algebraic techniques

•Measurement of Projection data

•Example of flame tomography


Applications & Types of Tomography

Medical Applications Type of Tomography

Full body scan X-ray

Respiratory, digestive systems, brain scanning

PET Positron Emission Tomography

Respiratory, digestive systems.

Radio-isotopes

Mammography Ultrasound

Whole Body Magnetic Resonance (MRI, NMR)

PET scan on the brain showing Parkinson’s Disease

MRI and PET showing lesions in the brain.


Applications & Types of Tomography

Non Medical Applications Type of Tomography

Oil Pipe Flow

Turbine Plumes

Resistive/Capacitance Tomography

Flame Analysis Optical Tomography

ECT on industrial pipe flows


The History

• Johan Radon (1917) showed how a reconstruction from projections was possible.•Cormack (1963,1964) introduced Fourier transforms into the reconstruction algorithms.•Hounsfield (1972) invented the X-ray Computer scanner for medical work, (which Cormack and Hounsfield shared a Nobel prize).•EMI Ltd (1971) announced development of the EMI scanner which combined X-ray measurements and sophisticated algorithms solved by digital computers.


dxdytyxyxftP )sincos(),()(

linet

dsyxftP),(

),()(

tyx sincos

1sincos tyx

1)(tP

),( yxf

y

x

Line Integrals and Projections

The function is known as the Radon transform of the function f(x,y).

1)(tP


)(1 tP

),( yxf

y

x

)(2 tP

A projection is formed by combining a set of line integrals. Here the simplest projection, a collection of parallel ray integrals i.e constant θ, is shown.

Line Integrals and Projections

),( yxf

y

x

)(1 tP

)(2 tP

A simple diagram showing the fan beam projection


Fourier Slice Theorem

The Fourier slice theorem is derived by taking the one-dimensional Fourier transform of a parallel projection and noting that it is equal to a slice of the two-dimensional Fourier transform of the original object. It follows that given the projection data, it should then be possible to estimate the object by simply performing the 2D inverse Fourier transform.

dtetPwS wtj

2)()(

dxdyeyxfvuF vyuxj )(2),(),(

dxdyeyxfuF uxj 2),()0,(

dxedyyxfuF uxj 2),()0,(

dxexPuF uxj

20 )()0,(

dyyxfxP ),()(0

)()0,( 0 uSuF

Start by defining the 2D Fourier transform of the object function as

Define the projection at angle θ, Pθ(t) and its transform by

For simplicity θ=0 which leads to v=0

As the phase factor is no-longer dependent on y, the integral can be split.

The part in brackets is the equation for a projection along lines of constant x

Substituting in

Thus the following relationship between the vertical projection and the 2D transform of the object function:


)(1 tP

),( yxf

y

x

t

θ

v

u

Space Domain Frequency Domain

Fourier transform

The Fourier Slice theorem relates the Fourier transform of the object along a radial line.

The Fourier Slice Theorem

v

u

Collection of projections of an object at a number of angles

For the reconstruction to be made it is common to determine the values onto a square grid by linear interpolation from the radial points. But for high frequencies the points are further apart resulting in image degradation.


Filtered Back Projection

Filtered back projection is the most commonly used algorithm for straight ray tomography.

(a)The ideal Situation

(b) Fourier Slice Theorem

(c) The filter back projection takes the Fourier Slice and applies a weighting so that it becomes an approximation of that in (a).

The result of back projecting


The Array: Algebraic Reconstruction Technique (ART)

ART is used in indeterminate problems and was first used by Gordon et al in the reconstruction of biological material.

1 3

1 1

2

3

2 1 2 5

5

6

754Σy

Σx

Figure a. Initial 3 by 3 grid with ray sums and coefficients.

? ?

? ?

?

?

? ? ? 5

5

6

754Σy

Σx

Figure b. The indeterminate problem.

6/3 6/3

5/3 5/3

6/3

5/3

5/3 5/3 5/3 5

5

6

754Σy

Σx

Figure c. Step 1: All entries in unity, scaled by ray sum over number of row elements.

6/3 6/3

5/3 5/3

6/3

5/3

5/3 5/3 5/3 5

5

6

5.335.335.33Σy

Σx

Figure d. Step 2: Recalculated column sums.

1.5 1.88

1.25 1.57

2.63

2.19

1.25 1.57 2.19 5.01

5.01

6.01

7.015.024Σy

Σx

Figure e. Step 3. Recalculated row and column sums and elements.


Measurement of projection data

Attenuation of X-raysAssume no loss of intensity of the beam due to divergence, however the beam does attenuate due to photons either being absorbed or scattered by the object.

Photoelectric Absorption

This consists of an x-ray photon imparting all of its energy to an inner electron of an atom. The electron uses this energy to overcome the binding energy within its shell, and the rest appearing as kinetic energy in this freed electron.

Compton Scattering

This consists of the interaction of the photon with either the free electron or a loosely bound outer shell electron. As a result the x-ray is deflected from its original direction.


Measurement of projection data

Attenuation of X-rays

Consider N photons cross the lower boundary of this layer in some measured time interval, and N+ΔN emerge from the top side. (ΔN will be negative). ΔN follows the relationship,

xN

N 1

N

N

x

dxN

dN

0 0

dxdNN

1 xNN 0lnln

xeNxN 0)(μ=photon loss rate (per unit distance) of the Compton and photoelectric effects. In the limit Δx goes to zero so we get

Solving this across the thickness of the slab

Where N0 is the number of photons that enter the object. The number of photons as a function of the position within the slab is given by,

or


Signal entering flame

Signal leaving flame

Radiant intensity of

backlight, L1 Radiance emitted by gas, L3

Transmitted portion of back light radiation, L2

Mercury lamp

Burner

Fibre optic to spectrograph

Optical arrangement used to determine the optical thickness of a flame.

Background Lamp, L1 Flame, L3 Flame+Lamp, L

Counts 77.71 93.76 439.29 450.82

Minus Background

- 16.06 361.59 373.12

33.0ln2

1

L

LD

Interpretation of ResultsTransmitted portion of backlight radiation, L2: 11.53 counts

Radiation incident on fibre from backlight, L1: 16.06 counts

72% transmission at 309 nm Optical thickness at 309 nm,

Absorption Coefficient:

11* 079.0ln

mmxLL

Emission Coefficient:

1*

**1* 23.31

exp1

exp

mmx

xL

Flame Thickness, Emission and Absorption


3.8 Fibre optic

Acceptance cone of fibre

Tomographic array

The acceptance cone of the fibres fitted to the area

The Array: Fibre Geometry


Array Resolution:


The Array: Preliminary Results


Comparing Results

Single Photograph of OH modified for colour intensity

Single thermocouple scan

Averaged thermocouple result Average of three photographs

The burner has been modified by placing two coins on it’s base. The array result is shown, superimposed on a photograph of the modified burner.

Tomography and Reconstruction

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