Upload
dax-shukla
View
224
Download
0
Embed Size (px)
Citation preview
7/29/2019 Tomographic Lecture
1/17
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
7/29/2019 Tomographic Lecture
2/17
School of Engineering
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.
7/29/2019 Tomographic Lecture
3/17
School of Engineering
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
7/29/2019 Tomographic Lecture
4/17
School of Engineering
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.
7/29/2019 Tomographic Lecture
5/17
School of Engineering
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
7/29/2019 Tomographic Lecture
6/17
School of Engineering
)(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
7/29/2019 Tomographic Lecture
7/17
School of Engineering
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:
7/29/2019 Tomographic Lecture
8/17
School of Engineering
)(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.
7/29/2019 Tomographic Lecture
9/17
School of Engineering
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
7/29/2019 Tomographic Lecture
10/17
School of Engineering
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.
7/29/2019 Tomographic Lecture
11/17
School of Engineering
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.
7/29/2019 Tomographic Lecture
12/17
School of Engineering
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
7/29/2019 Tomographic Lecture
13/17
School of Engineering
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
7/29/2019 Tomographic Lecture
14/17
School of Engineering
3.8 Fibre
optic
Acceptance
cone of fibre
Tomographic array
The acceptance cone of the fibres fitted to the area
The Array: Fibre Geometry
7/29/2019 Tomographic Lecture
15/17
School of Engineering
Array Resolution:
7/29/2019 Tomographic Lecture
16/17
School of Engineering
The Array: Preliminary Results
7/29/2019 Tomographic Lecture
17/17
School of Engineering
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.