Download - Tomographic Lecture

7/29/2019 Tomographic Lecture

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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


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Applications & Types of Tomography

Medical Applications Type of TomographyFull 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 Parkinsons

Disease

MRI and PET showing

lesions in the brain.


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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


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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 formedical 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.


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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


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)(1 tP

),( yxf

y

x

)(2 tP

A projection is formed by combining a set

of line integrals. Here the simplestprojection, a collection of parallel ray

integrals i.e constant , is shown.

Line Integrals and Projections

),( yxf

y

x

)(1 tP)(2 tP

A simple diagram showing the fan

beam projection


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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 2

0 )()0,(

dyyxfxP ),()(0

)()0,( 0 uSuF

Start by defining the 2D Fourier transform of the

object function as

Define the projection at angle , P(t) and its

transform by

For simplicity =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 constantx

Substituting in

Thus the following relationship between the

vertical projection and the 2D transform of theobject function:


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)(1 tP

),( yxf

y

x

t

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 objectat 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.


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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 FourierSlice and applies a

weighting so that it

becomes an approximation

of that in (a).

The result of back projecting


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The Array: Algebraic Reconstruction

Technique (ART)

ART is used in indeterminate problems

and was first used by Gordon et alin

the reconstruction of biological

material.

1 3

1 1

2

3

2 1 2 5

5

6

754y

x

Figure a. Initial 3 by 3 grid with

ray sums and coefficients.

? ?

? ?

?

?

? ? ? 5

5

6

754y

x

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

754y

x

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

5.335.335.33y

x

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

7.015.024y

x

Figure e. Step 3. Recalculated row

and column sums and elements.


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Measurement of projection data

Attenuation of X-rays

Assume 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 freedelectron.

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.


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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+N

emerge from the top side. (N will be

negative). N follows the relationship,

xN

N 1

N

N

x

dxN

dN

0 0

dxdNN

1 xNN 0lnln

xeNxN 0)(=photon loss rate (per unit distance)

of the Compton and photoelectric

effects. In the limit x goes to zero sowe get

Solving this across the thicknessof 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


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Signal enteringflame

Signal leaving

flame

Radiantintensity 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 Results

Transmitted 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

mm

x

L

L

Emission Coefficient:

1

*

**

1*

23.31exp1

exp

mmx

xL

Flame Thickness, Emission and

Absorption


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3.8 Fibre

optic

Acceptance

cone of fibre

Tomographic array

The acceptance cone of the fibres fitted to the area

The Array: Fibre Geometry


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Array Resolution:


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The Array: Preliminary Results


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Comparing Results

Single Photograph of OH modified for

colour intensitySingle thermocouple scan

Averaged thermocouple result Average of three photographs

The burner has been modified

by placing two coins on itsbase. The array result is

shown, superimposed on a

photograph of the modified

burner.