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Transcript of 1 Tomography Reconstruction : Introduction and new results on Region of Interest reconstruction...
1
Tomography Reconstruction : Introduction and new results on Region of
Interest reconstruction
-Catherine Mennessier- Rolf Clackdoyle-Moctar Ould Mohamed
Laboratoire Hubert Curien, St Etienne
Bucharest, May 2008
2
Table of contents
1. Introduction
2. Reconstruction in 2D tomography : standard algorithms
3. Reconstruction of a Region Of Interest from truncated data : new results.
3
1. Introduction
Computer Tomography : a non-destructive imaging technique for interior inspection.
Waste inspection CT scanner
Some applications…
4
1. Introduction
Domains of application:
• Medical image processing :– Anatomic imaging (CT, Image Guided Surgery, Diagnostic..) density– Functional imaging (SPECT, PET…search for tumour, heart muscle
viable…) radioactive tracer
• Industrial :– Non destructive techniques for characterization (drum nuclear waste..),
defect detection (on production lines)…
• Archaeology :– Interior reconstruction (of amphora…)
• Astronomy : – Doppler imaging
• Geology :– Seismic studies (wave tomography)
• …
5
1. Introduction
In transmission tomography, the X ray (or gamma ray…) are attenuated. The degree of attenuation depends on the density of the object. The absorption of the X-ray is measured, from different positions of the source/detector system.
6
1. Introduction
x
N0 f NxNfNNN 0
Beer-Lambert law:
X-ray and matter interaction:
• Photoelectric absorption
• Compton scattering
• Rayleigh scattering
X-ray attenuation
Macroscopic scale
Microscopic scale
The absorption coefficient f depends on the material. For instance, at 60KeV, water(0,203/cm), white matter(0,210/cm), gray matter(0,212/cm) …
7
1. Introduction
Lout
in
X
X
N
N
xfdxN
Nsodxxf
N
dN out
in
out
in
)(log,)(
xout
xin LX-ray source
X-ray sensor
Patient
X-ray and matter interaction :
9
2. Reconstruction in 2D tomography : standard algorithms
Notations
)sin,(cos),sin,(coswith ,)(),(
dtstfsp
s t
p(,s)
f(x)
10
2. Reconstruction in 2D tomography : standard algorithms
fp R
The Radon transform :
s t
p(,s)
f(x)
dtstfsp
SS
)(),(
)],([)(:R
02
We note :
11
2. Reconstruction in 2D tomography : the Fourier slice theorem
p(,s)
Fourier domainDirect domain
f(x)
F()
1D Fourier transform
2D Fourier transform
=
)(),( FP
F()
P(, )
12
2. Reconstruction in 2D tomography : the BackProjection
],[
).,()(
)()],([:R
0
20
dxpxb
SS
xx
p(1,s)
p(2,s)
p(3,s)
pb RWe note :
13
2. Reconstruction in 2D tomography : the BackProjection
Backprojection of the Radon transform of a centred disk of constant intensity :
N=
1
N=
2
N=
4
N=
180
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2. Reconstruction in 2D tomography : the FBP algorithm
1. Projection filtering
For k=1:N
pf(,s)=(pr ) (,s) where R()=| |
End
2. Backprojection
f=R* pf
)( with wherepf Rsrsgspf f )(),(),(R*
Ramp filter
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2. Reconstruction in 2D tomography : the FBP algorithm
)( with wherepf Rrppf f*R
Comments :
• To compute the single value f(x) at x, all the projections are needed as the filtering step is not local if one data is missing, all the reconstruction (for all x) is affected by the FBP algorithm.
• FBP is very efficient (standard from 30 years).
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3. Reconstruction of a ROI from truncated data : new results
Truncated data : only the lines that intersect the circle are measured
Not measuredmeasured
Is it possible to reconstruct exactly a part of the object from the incomplete set of data?
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3. Reconstruction of a ROI from truncated data : new results
Solution : the answer is
• no if FBP is used
• yes for some ROI using
- virtual fan-beam algorithm (2004)
-Differentiated Backprojection with truncated Hilbert Inverse (2004) (two-step, DBP, chord…)
Is it possible to reconstruct exactly a part of the object from an incomplete set of data?
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3. Reconstruction of a ROI from truncated data : new results
1. Virtual fan-beam
1. The ramp filter and the Hilbert transform
2. Fan-beam projection
3. Rebining (the Hilbert transform)
2. DBP
1. Differentiated Backprojection
2. Truncated Hilbert Inverse
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3. Reconstruction of a ROI from truncated data : virtual fan-beam
Inverse Radon transform and the Hilbert transform : the filtering step
Remind : )(with where Rrpppfs
ff **R
ansformHilbert tr theis s
1*pHp Where
where2
1
s
),(),(
),(),(
ss
sHps
sp f
Then
sssri
iR
1
2
12
2
1 )(
22
)())()sgn(
()sgn(
20
3. Reconstruction of a ROI from truncated data : virtual fan-beam
Rebinning formula:
Let us introduce :
dttafag )(),(
a
a.sat ),(),( agsp
a
s
21
3. Reconstruction of a ROI from truncated data : virtual fan-beam
Rebinning formula:
Let us define : )sin(
),(),( *
1agaHg
a
Hilbert rebinning formula :
a.sfor ),(),( aHgsHp
a
s
22
3. Reconstruction of a ROI from truncated data : new results
a.sat ),(),( aHgsHp
Not measuredmeasured
Is it possible to reconstruct exactly a part of the object from the incomplete set of data?
Yes, by selecting a switable virtual fan-beam projection
s
a
).,()( computeThen ],0[
dxHps
xf
23
3. Reconstruction of a ROI from truncated data : new results
The ROI that can be exactly reconstructed using the virtual fan-beam algorithm
24
3. Reconstruction of a ROI from truncated data : new results
),()()()(
),(R)( *
21220
0
11 where
then(x)s
Define
xxfxfxHfxb
spxb
xx
The DBP algorithm : Differentiated backprojection
Remind ),(),( sps
Hsp f
2
1
x1
xs
25
3. Reconstruction of a ROI from truncated data : new results
thenneeded, is for only t reconstruc To
Supp( Assume
2221
2
1
1
],[)(),(
],[))(
LLxxHfxf
LLxfxx
x
The DBP algorithm
fx1(x2) can be reconstructed where a vertical line, crossing the support of f, can be found, assuming backprojection of the line points is possible.
NB: Generalization for all the direction (not only the vertical line)
-L
+L
fx1(x2)
x2