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Hands on …Reconstruction From Projection
Shireen Y. ElhabianAmal A. FaragAly A. Farag
University of LouisvilleJanuary 2009
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An Analogy
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Agenda
• Reconstruction from projections (general)
– projection geometry and radon transform
• Reconstruction methodology
– Backprojection, (Fourier slice theorem), FilteredBackprojection.
• Reconstruction examples
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Introduction
• Only photography (reflection) and planar x-ray (attenuation) measure spatial propertiesof the imaged object directly.
• Otherwise, measured parameters are somehow related to spatial properties of imagedobject.
– CT, SPECT and PET (integral projections ofparallel rays), MRI (amplitude, frequency andphase) etc...
• Objective: We want to construct the object(image) which creates the measured parameters.
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Problem Statement
• Given a set of 1-Dprojections and the angles atwhich these projectionswere taken.
• How do we reconstruct the2-D image from which theseprojections were taken?
• Lets look at the nature ofthose projections … L
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Parallel Beams Projections
),( yxgy
x
)(tPq
θt
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Ray Geometry
• Let x and y be rectilinearcoordinates in a givenplane.
• A line in this plane at adistance t1 from the originis the given by:
where θ is the angle between a unitnormal to the line and the x-axis.
x
yt
Pq (t)
θ
t1
A
B
dsdx
dy
qq sincos1 yxt += g(x,y)
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What is Projection ?!!• Let g(x,y) be a 2-D function.
• A line running throughg(x,y) is called a ray.
• The integral of g(x,y) alonga ray is called ray integral.
• The set of ray integralsforms a projection definedas :
x
yt
Pq (t)
θ
t1
A
B
dsdx
dy
g(x,y)
( ) ( )ò ò¥
¥-
¥
¥--+= dxdytyxyxgtP 11 sincos,)( qqdq
Impulse sheath placed at the points constituting the ray
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Radon Transform
• Coordinate transformation:
• Radon transform x
yt
Pq (t)
θ
t1
A
B
dsdx
dy
g(x,y)
t
s
úû
ùêë
éúû
ùêë
é-
=úû
ùêë
éyx
st
cossin
sincos
( ) ( )
ò
ò ò¥
¥-
¥
¥-
¥
¥-
=
-+=
.),(
sincos,)(
dsstg
dxdytyxyxgtP qqdq
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Radon Space• Projections with different
angles are stored in sinogram(raw data).
• Each vertical line in asinogram is a projectionwith a different angle
Θ : Angles of projections
t : p
roje
ctio
n ra
ys
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The Myth Jf
),( yxgy
x
)(tPq
θt
),( yxgy
x
)(tPq
θt
50 100 150 200 250 300 350
50
100
150
200
250
300
350
400
Theta = 180
[ )pq ,0Î
Since Radon transform is a group of projections which are basically line integrals, the difference between the projection at θo and θo +πwill be the direction of the integration.
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The Fourier Slice Theorem• This theorem relates the 1D Fourier Transform of a projection and the 2D
Fourier transform of the object. It relates the Fourier transform of the objectalong a radial line.
1D Fourier Transform
)( fSq fu
v
),( vuG
uf =qcosvf =qsin
),( yxgy
x
)(tPq
θt
Frequency DomainSpace Domain
( ){ } ( )ò¥
¥-
-=Á= dtetPtPfS ftjD
pqqq
21)(
( ) ( )q,, fGvuG ºwhere
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The Fourier Slice Theorem• This theorem relates the 1D Fourier Transform of a projection and the 2D
Fourier transform of the object. It relates the Fourier transform of the objectalong a radial line.
1D Fourier Transform
),( vuG
),( yxgy
x
)(tPq
θt
Frequency DomainSpace Domain
v
u
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The Fourier Slice TheoremFrequency Domain
S(f,θ)Space Domain
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A Problem
• All projections contribute tolow freqencies
Solution:
use filtered projection
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Filtered Back-projection
Projections (raw data from the scanner)
Fourier Transform of all Projections
Filter Projections
Back-projection to uv-space
Inverse Fourier
Transform
( )!
( ) q
pp
q
q
ddfefSfyxg
yxttQ
ftj
sincos
0
)(
2
ResponseFilter
,
+=
¥
¥-ò ò ú
ú
û
ù
êê
ë
é=
"""" #"""" $%
Estimate of g(x,y)
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Tasks :-• Scanner simulation
– Phantom Generation
– Projections computation
• Reconstruction from projections
• Analysis:
– Experiment 1: the effect of filter type.
– Experiment 2: the effect of number of projections.
– Experiment 3: the effect of number of rays
Let’s do it …
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Scanner Simulation – Phantom Generation
• Given the spatial support of ourphantom.
• We assume that our phantom isconstructed of a set of ellipses, eachhas the following parameters:
– Intensity, ellipse center (x0,y0), ellipsemajor and minor axes length (a,b), andthe orientation (φ i.e. rotation angle)
X - direction
Y -
dire
ctio
nEllipse Center (x0,y0)
φ
Intensity
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Scanner Simulation – Phantom Generation
X - direction
Y -
dire
ctio
n
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Scanner Simulation – Phantom Generation
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Scanner Simulation – Phantom Generation
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Scanner Simulation – Projections Generation
• To be able to study different reconstruction techniques, we first neededto write a program that take projections of a known image.
• Basically, we take the image (which is just a matrix of intensities),rotate it, and sum up the intensities.
• In MATLAB this is easily accomplished with the 'imrotate' and 'sum'commands.
• But first, we zero pad the image so we don't lose anything when werotate.
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Scanner Simulation – Projections Generation from 0 to π
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Scanner Simulation – Projections Generation from 0 to 2π
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Reconstruction From Projections• Given the projections, we first filter them as shown
below.
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Reconstruction From Projections• Given the angles where the projections were taken, and the filtered
projections, the following will reconstruct an estimate of the original image.
θ
fPoint on the current radial line which corresponds to the 1D fourier transform of the current projection
( ) ( ) q
pp
q
q
ddfefSfyxg
yxttQ
ftj
sincos
0
)(
2,
+=
¥
¥-ò ò ú
û
ùêë
é=
!!! "!!! #$
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Experiment One
Studying the effect of using different filter types compared to the unfiltered
case.
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Reconstruction using unfiltered projections
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Reconstruction using Ramp filter
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Reconstruction using LPF filter
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Reconstruction using Butterworth filter
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Reconstruction using Sinusoidal filter
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Reconstruction using Ramp filter vs unfiltered caseOriginal image
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Reconstructed - unfiltered
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Reconstructed - ramp filter
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w
H(w)
Ramp filter frequency response
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Original image
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Reconstructed - unfiltered
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Reconstructed - LPF filter
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Low pass filter frequency response
Reconstruction using LPF filter vs unfiltered case
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Original image
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Reconstructed - unfiltered
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Reconstructed - butterworth filter
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H(w)
Butterworth filter frequency response
Reconstruction using Butterworth filter vs unfiltered case
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Original image
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Reconstructed - unfiltered
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Reconstructed - sinusoidal filter
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w
H(w)
Sinusoidal filter frequency response
Reconstruction using Sinusoidal filter vs unfiltered case
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Experiment Two
Studying the effect of reconstruction using different number of projections
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Reconstruction using different number of projections
Using sinusoidal filter and number of rays equal to image number columns
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Quantifying the reconstruction error
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1
2
3
4
5
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7
Number of Projections
Mea
n Sq
uare
Err
orMean square error using different number of projections
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Experiment Three
Studying the effect of reconstruction using different number of rays
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Reconstruction using different number of rays
Using sinusoidal filter and fixed number of projections
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Quantifying the reconstruction error
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Number of rays
Mea
n Sq
uare
Err
or
Mean square error using different number of rays
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Thank You