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Image Reconstruction. Atam P Dhawan. y. b. Radiating Object f( a,b,g ). Image g(x,y,z). Image Formation System h. g. z. Image Domain. Object Domain. x. a. Image Formation. b. y. Radiating Object. Image. Image Formation System h. Selected Cross-Section. g. z. - PowerPoint PPT Presentation
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Image Reconstruction
Atam P Dhawan
Image Formation
z
Image Formation System
hObject Domain
Image Domain
x
y
Radiating Object f() Image g(x,y,z)
dddfzyxhzyxg ),,(),,,,,(),,(
dddfzyxhzyxg ),,(),,(),,(
Image Formation: External Source
Reconstructed Cross-Sectional Image
Radiation Source
z
Image Formation
Systemh
Object Domain
Image Domain
Selected Cross-Section
x
yRadiating Object
Image
Image Formation: Internal Source
z
Reconstructed Cross-Sectional Image
Image Formation
Systemh
Object Domain
Image Domain
Selected Cross-Section
x
y
Radiating Object
Image
Fourier Transform
dydxeyxgyxgFTvuG vyuxj ,),()},({),( )(2
Radon Transform
x
y
q
p
p
f(x,y)
P(p,)
Line integral projection P(p,) of the two-dimensional Radon transform.
Radon Transform
Projection p1
Projection p2
Projection p3
Reconstruction Space
A
B
dqqpqpfpJyxfR )cossin,sincos()()},({
Fourier Slice Theorem
• X-y coordinate system rotated to p-q
cossin
sincos
yxq
yxp
cossin
sincos
qpy
qpx
dpdqeqpqpfpJFyxfRF pj
2)cossin,sincos()}({)}},({{
dpepJS pj
2)()(
),(),()( )sincos(2 FdxdyeyxfS yxj
u = cos v= sin
Fourier Slice Theorem…
u
v
F(u,v)
Sk() S2()
S1()
Inverse Radon Transform
2 ( cos sin )
0
( , ) ( , ) j x yf r F e d d
),()},({),(ˆ )(21 dudvevuFvuFFyxf vyxuj
deSpJ
dpJ
ddeSrf
yxj
yxj
)sincos( 2*
0
*
0
)sincos( 2
)( )(
where
)(
)( ),(ˆ
Filtered Backprojection
)(
)(
)(
)( )(
1
1
2
)sincos( 2*
pJF
SF
deS
deSpJ
pj
yxj
to
),( yxf
The integration over the spatial frequency variable should be carried out from
But in practice, the projections are considered to be bandlimited.
This means that any spectral energy beyond a spatial frequency, say must be ignored. can be computed as
)()( 1
),(0
pphpJpddyxf
otherwise 0
if LRH
LRH )( ph LR is the Fourier transform of the filter kernel function in the spatial domain and is bandlimited.
)()( 1
),(0
pphpJpddyxf
otherwise 0
if LRH
LRH )( ph LR is the Fourier transform of the filter kernel function in the spatial domain and is bandlimited.
)( )( BH
otherwise 0
if 1)(
B
deHph pj2)()(
H()
1/2-1/2
1/2
If the projections are sampled with a time interval of t, the projections can be represented as )( kJ
Using the Sampling theorem and the bandlimited constraint, all spatial frequency components beyond are ignored such that
2
1
2
22 2/
)2/ ( sin
4
1 -
/
)/ ( sin
2
1)(
p
p
p
pph
For the bandlimited projections with a sampling interval of
Filter Function
H()
1/2-1/2
h(
hR-L(p)
HHamming(p)
The Final Algorithm: FBP
*
1
( ) ( ) ( )
( , ) * ( )i
L
i
J p J p h p p dp
f x y J pL
Iterative ART
MifwpN
jjjii ,...,1for
1,
MiwfqN
lli
kl
ki ,,,1 allfor
1,
1
jiN
lli
kiik
jkj w
w
qpff ,
1
2,
1
Raywith ray sum pi
f1 f2 f3
fN
Overlapping area for defining wi,j
PET ML Image ReconstructionLet us assume that the object to be reconstructed has an emission density function
tBzyx ],....,[),,( 21
with a Poisson process over a matrix of B pixels. The
emitted photons (in case of SPECT) or photon pairs (in case of PET) are detected by the
detectors with the measurement vector tDJJJJ ],...,,[ 21
with D measurements. The
problem is then to estimate Bbb ,...,1);( from the measurement vector.
Each emission in box b is detected by the detector d (SPECT) or the detector tube d
(PET) with probability p b d P( , ) (detection in d | photon emitted in b). The transition
matrix p b d( , ) is derived from the geometry of the detector array and the reconstruction
space.
Let ),( dbj denotes the number of emissions in box b detected in the detector or detector
tube d are independent Poisson variables with the expected value
),()(),()],([ dbpbdbdbjE .
A
DdBb
dbjdb
dbj
dbejPL
,...,1,...,1
),(),(
)!,(
),()|()(
ML-EM Algorithm
Bbdbpb
dbpdjbb
D
dB
b
old
oldnew ,...,1;),()(ˆ
),()()(ˆ)(ˆ
1
1
Multi-Grid EM Algorithm InitializeGrid level, k = 0Iterations, i = 0
i = i +1
NO
YES
NO
YES
i = 0
WaveletDecomposition
Final ReconstructedImage
NO
YESIs grid
optimizationmeasure
satisfied ?
Is intra-levelperformance
measuresatisfied ?
k = k + 1
Final ReconstructedImage
Is currentgrid resolution
>detector
resolution?
Initial Solution0
0
= EM( ,n)ki+1 i
k
= INTERP( )Wavelet Interpolation
i
kk+10
MGEM Reconstruction
Reconstruction in MRI
dxdydzezyxfMS zyxi zyx )(0 ),,( )(
Fourier Transform Reconstruction Method
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