MAPEM for Image Reconstruction Project

2.  Reconstruct an Image using Projection
 Maximum A Posteriori Expectation
Maximization (MAPEM)
 Optimizing the Iteration
 Reducing the time
 Performance Analysis –Amdahl’s Law.

3.  Mathematical Process
 Data obtained at various angles
 Medical imaging modalities
◦ CT scan
◦ PET
◦ MRI scan

4. Computed Tomography Scan
Positron Emission Tomography Magnetic Resonance imaging

5.  Analytical
◦ Filtered Back Projection (FBP)
 Iterative
◦ Algebraic Reconstruction Technique (ART)
◦ Statistical Reconstruction Technique
 Weighted Least Square
 Likelihood based iterative Expectation Maximization
 Maximum Likelihood Expectation Maximization
 Maximum A Posteriori Expectation Maximization

6.  Projection is a line integral along the path:


 cos
sin
where
y)ds
f(x,
)
( y
x
s
t
p
AB



 

7.  Projections with
different angles
are stored in
sinogram (raw
data)
 Each horizontal
line in sinogram is
a projection
with different
angle
)
(t
p
projection )
(t
p

8. Y is constant
posterior
likelihood
prior
X: reconstruction
Y: projection
Bayes:
MAP: maximize
p(Y|X) p(X) or ln p(Y|X) + ln p(X)

9.  Multi-core platform
◦ Thread level parallelism
 Simultaneous Multi-Threading (SMT)
 Chip Multi-Processor (CMP)
 Multi Core
 Many Core
◦ Two conclusions for applications running on multi-
core platform
 Conclusion 1. When multiple applications are executed,
the performance on multi-core platform is better than
single-core platform with the same clock frequency.
 Conclusion 2. Applications without parallelization can’t
make full use of computing power on multi-core
platform.
 OpenMP (OMP) directives is implemented

10.  Amdahl’s Law
◦ statement of the maximum theoretical speed-up
you casn ever hope to achieve.
time
execution
Parallel
time
execution
Sequential
Speedup
f
n
f
f
p
S
1
/
)
1
(
1
)
( 






11.  Shepp Logan Phantom
64 x 64 128 x 128 256 x 256

12.  Sinogram (Raw data)
A B C D E
1
2
3

13.  Iteration Optimized
Projections/
Sizes 30 20
1
5 12 10
64x64 20 18
1
0 23 19
128x128 27 27
3
5 23 41
256x256 61 29
4
5 39 61

14.  PSNR
10 12 15 20 30
64x64
53.43
72
53.37
98
53.09
74
53.26
82
53.36
85
128x1
28
61.67
12
61.23
3
62.13
11
62.09
66
62.15
82
256x2
56
69.58
12
69.70
39
70.25
1
70.31
48
71.01
51

15.  Image Reconstructed
A B C D E
1
2
3

16.  Time Complexity 64 x 64
10 12 15 20 30
1 core 3.09129 4.27891 2.09782 5.43004 8.33448
2 Core 2.41019 2.99262 1.47404 4.17731 5.21191
4 Core 1.92379 2.53712 1.24897 2.96779 4.50192
8 Core 1.5557 1.60578 0.946761 2.04252 3.55317
0
1
2
3
4
5
6
7
8
9
Reconstruction
Time
(s)
Time Complexity (64x64)
1 core
2 Core
4 Core
8 Core

17.  Time Complexity 128 x 128
10 12 15 20 30
1 core 47.9322 26.299 66.0366 66.9674 98.2584
2 Core 36.5989 24.8049 41.1161 49.6748 72.4108
4 Core 25.3501 20.7042 40.2762 42.9248 53.6694
8 Core 10.3271 11.9932 30.9506 20.7319 30.8544
0
20
40
60
80
100
120
Reconstruction
Time
(s)
Time complexity (128x128)
1 core
2 Core
4 Core
8 Core

18.  Time Complexity 256 x 256
10 12 15 20 30
1 core 568.4280 394.722 630.297 552.071 1758.98
2 Core 464.261 337.362 483.439 453.559 1415.51
4 Core 273.036 250.089 340.672 323.802 1019.31
8 Core 185.939 164.266 225.58 193.718 646.457
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Reconstruction
Time
(s)
Time complexity (256x256)
1 core
2 Core
4 Core
8 Core

19.  Performance Analysis
0
2
4
6
8
10
10 12 15 20 30
1
2
4
8 0
2
4
6
8
10
10 12 15 20 30
1
2
4
8
0
2
4
6
8
10
10 12 15 20 30
1
2
4
8
64 x 64 128 x 128
256 x 256