The document discusses reconstructing images from projection data using iterative reconstruction techniques like maximum a posteriori expectation maximization (MAPEM). It evaluates MAPEM's performance on multi-core platforms while varying the image size (64x64, 128x128, 256x256 pixels) and number of projections. Reconstruction time decreases and quality increases with more cores and projections, but more pixels increase time. Amdahl's law applies to parallelization on multi-core CPUs.

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