This document discusses image reconstruction using the maximum a posteriori expectation maximization (MAPEM) algorithm on a multi-core system. It aims to reduce reconstruction time through parallel computing. MAPEM is an iterative algorithm that incorporates prior knowledge to favor convergence. Implementation of the parallel MAPEM (pMAPEM) algorithm on a multi-core system showed improved reconstruction time over the serial MAPEM algorithm, with efficiency increasing with more cores. Reconstruction of Shepp Logan phantom images demonstrated the effectiveness of pMAPEM.

1. IMAGE RECONSTRUCTION USING MAXIMUM A
POSTERIORI EXPECTATION MAXIMIZATION
ALGORITHM ON A MULTI-CORE SYSTEM
30.06.2020

2. • Image Reconstruction Technique is suitable for identifying the disease
based on scanned image.
• Reconstructing an Image using Projection displays good quality
images
• To improve the accuracy level for medical applications, iterative
algorithms such as ART, MLEM and MAPEM are implemented for
limited number of angles.
• The number of iterations are optimized
• It is found that all three systems suffers from high computation time.
• To reduce reconstruction time a parallel computation method is
proposed.
• To prove the parallel computing performance Amdahl’s Law is used.
Objective

3. • Image acquisition is done by Medical
Modalities
• The object is placed inside the tunnel
• The scan area has a rotating emitter
and receiver in parallel that acquire at
different angle
• Positron emitting radio
pharmaceutical will be injected into
the object
• Positron combines with an electron
and converted into two photons
Introduction
CT scan PET
MRI Scan

4. • Analytical
• Filtered Back Projection (FBP)
• Iterative
• Algebraic Reconstruction Technique (ART)
• Solving a linear system of equations iteratively
• Statistical Reconstruction Technique
• Weighted Least Square
• Likelihood based iterative Expectation
Maximization
• Maximum Likelihood Expectation Maximization
• Maximum A Posteriori Expectation Maximization
Introduction

5. Introduction
• Estimated Image – Image acquired using projection
• Computed Projection and Measured projections are compared
• If discrepancy occurs back projection is applied to form updated image
• Updated image will be considered as Estimated Image during next
iteration
• Iteration will be repeated till the discrepancies has been reduced.
• Depending on data size and number of iterations the reconstruction
time will be more.
• To reduce the reconstruction time parallel computing is introduced.

6. • Serial Computing
• Problem is broken into number of instructions
• Instructions are executed sequentially one after other
• Parallel Computing
• Problem is divided into sub problems
• Each Sub problem is broken into number of
instructions
• Instructions are executed sequentially under a
processor
• Implemented with many computers generally
referred as parallel computers or multi-core
processors
• Programming Languages, Operating System,
Parallel algorithm and compiler that support
parallel programming
Parallel Computing

7. • The multiprocessor can be distributed system or shared memory.
• Distributed System
• Designed by placing the components on different networked computers
• The coordination and communication is done by passing message to one another
• Shared Memory
• Multiprocessor that has more than one processor inside a single computer itself
• Each processor is treated as Thread
• Thread reads, writes and processes on data concurrently using common memory
• Data, Task and Pipeline are the categories of parallel computing
• API used to implement parallel programs are Open Multi-Processing
(OMP) and Message Passing Interface (MPI)
• MPI – Distributed Systems
• OMP – Shared Memory
Parallel Computing

8. • The program starts as a single master thread.
• As parallel region begins the master thread forks the region into specified
number of slave threads and perform simultaneously.
• When parallel region ends, slave threads joins back to the master thread.
• A Thread is an execution entity and associated static memory is called as
thread private memory.
• Special directives are used to run the code in parallel.
• The number of threads is mentioned using environment variable.
• “omp.h” header file holds OpenMP prototypes and macro definitions.
OpenMP

9. • C++ omp is used in this research work.
• Environmental Variable
• OMP_DYNAMIC
• OMP_NUM_THREADS
• OMP_NUM_THREADS=num_threads
• OMP_SCHEDULE
• Directive
• #pragma omp name_of_directives[clause]
• Parallel
• Used to denote a region to be executed in parallel
• If(exp)
• Shared(list)
• Reduction(operator:list)
• for
OpenMP

10. • Amdahl’s Law
• statement of the maximum theoretical speed-up you can
ever hope to achieve.
S(n) = ts/tp
• Quinn’s Notation
Speedup and Efficiency
time
execution
Parallel
time
execution
Sequential
Speedup 
f
n
f
f
p
S
1
/
)
1
(
1
)
( 






11. Maximum A Posteriori Expectation Maximization
(MAPEM)
• MAPEM is introduced with a prior knowledge as a constraint that
favors convergence of the expectation maximization algorithm process
called as regularization
• The prior is usually chosen to penalize the noisy images.
• The goal of the required criterion is simultaneously maximized which
leads to a scheme called One Step Late (OSL) algorithm.
• The priori term is the derivative of an energy function chosen to
enforce smoothing.
• At the initial condition the reconstruction method guesses the estimate
value that resembles the internal structure, by feeding projection data
as input.

12. 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)
Maximum A Posteriori Expectation Maximization
(MAPEM)

13. • E-Step Procedure: Estimates the
expectation of the missing value i.e.
unlabeled class information. This step
corresponds to performing classification of
each unlabeled document. Probability
distribution is calculated using current
parameter.
• M-Step Procedure: Calculates the
maximum likelihood parameters for the
current estimate of the complete data.
Maximum A Posteriori Expectation Maximization
(MAPEM)

14. Dataset
• Phantom imaging is specially designed object to
evaluate, analyze and tune performance of
devices
• The research has used Shepp Logan Phantom as
dataset
• It serves as the model of a human head in the
development and testing of image reconstruction
algorithms
• Various sizes of 64, 128 and 256 is used to
perform image reconstruction.

15. Maximum A Posteriori Expectation Maximization
(MAPEM)

16. Parallel Maximum A Posteriori Expectation
Maximization (pMAPEM)

17. GUI Implementaion

18. Results
64x64 size 128x128 size

19. Results
Image Reconstructed
Using MAPEM for
256 x256 size

20. Results
Sizes 10 12 15 20 30
64 x 64 19 23 10 18 20
128 x 128 41 23 35 27 27
256 x 256 61 39 45 29 61
Optimized number of Iterations

21. PSNR
1 12 15 20 30
FBP 49.497 50.0059 50.7319 51.4147 51.9243
SIRT 52.2959 52.5904 52.2921 53.0862 53.3409
SART 52.3645 52.6329 52.5692 53.2317 53.4897
ART 53.2312 53.1949 53.4115 53.3965 53.3756
MLEM 53.505 53.5182 53.5731 53.5845 53.6517
MAPEM 53.5372 53.5798 53.5974 53.5882 53.3685
pMAPEM 53.5372 53.5798 53.5974 53.5882 53.3685
10 12 15 20 30
FBP 49.497 54.6397 55.8001 57.0232 58.7456
SIRT 59.0597 59.231 59.8758 60.306 60.9639
SART 59.0982 59.2974 59.9331 60.4298 61.1544
ART 61.4595 61.4505 62.0032 61.7996 62.1416
MLEM 61.7051 61.7653 62.142 62.1385 62.2634
MAPEM 61.6712 61.233 62.1311 62.0966 62.1582
pMAPEM 61.6712 61.233 62.1311 62.0966 62.1582
10 12 15 20 30
FBP 58.0899 59.0863 60.4143 62.0325 64.4252
SIRT 65.3235 65.6486 66.345 66.9003 68.5452
SART 65.351 65.6801 66.3955 67.0409 68.6266
ART 69.3787 69.5936 70.676 70.3131 71.0605
MLEM 69.5807 69.7042 70.5335 70.4742 71.0236
MAPEM 69.5812 69.7039 70.251 70.3148 71.0151

22. Time Taken to reconstruct
10 12 15 20 30
FBP 0.002 0.008099 0.006877 0.00332 0.011966
SIRT 0.74811 0.62328 0.55388 1.385 1.9381
SART 0.48602 0.6842 0.42174 1.0409 1.6463
ART 2.75688 1.85126 1.6436 1.03605 1.63309
MLEM 3.29546 3.12173 5.51383 5.78669 9.76802
MAPEM 2.66441 3.79177 2.09434 5.26602 8.26126
2 Core 2.25607 3.20836 1.3518 3.20718 6.32703
4 Core 1.33296 2.15583 1.14649 3.01207 4.43421
8 Core 1.12781 1.57789 0.86358 2.00983 3.07759
10 12 15 20 30
FBP 0.005284 0.0037 0.004395 0.005571 0.009635
SIRT 8.2162 4.5569 11.0323 8.057 18.671
SART 6.9792 4.9261 8.5043 9.9196 13.096
ART 90.4598 67.6586 85.3753 36.8356 30.1496
MLEM 39.0187 34.6487 55.9709 58.7498 96.2617
MAPEM 43.4195 29.3706 56.6291 54.7103 84.1047
2 Core 39.0638 23.9133 47.5634 48.2149 67.2989
4 Core 26.928 18.3259 44.5746 36.3727 50.3595
8 Core 21.8957 11.1107 21.8165 22.1902 31.908
10 12 15 20 30
FBP 0.008455 0.007704 0.00781 0.015839 0.021083
SIRT 75.6588 76.6664 91.628 56.3881 176.8353
SART 50.5161 57.6855 56.3243 56.3881 202.067
ART 1609.1 1699.73 1889.8 918.3131 723.983
MLEM 522.894 462.973 750.215 709.861 2134.4
MAPEM 726.522 532.309 727.098 532.317 771.465
2 Core 502.087 332.341 502.65 332.347 463.192
4 Core 398.953 297.146 399.495 297.143 447.483
8 Core 198.488 145.926 199.045 145.934 259.513

24. Results
Size Cores 10 12 15 20 30
64 x 64
1 Core 1 1 1 1 1
2 Cores 1.47544 1.97114 1.98946 1.98291 1.97556
4 Cores 1.95798 3.80297 3.94194 3.88316 3.88814
8 Cores 2.30998 7.0474 7.69289 7.39583 7.16673
128 x 128
1 Core 1 1 1 1 1
2 Cores 1.38349 1.97346 1.99118 1.97773 1.99661
4 Cores 1.73617 3.87638 3.95761 3.93252 3.93043
8 Cores 1.94444 7.62123 7.76133 7.58582 7.86187
256 x 256
1 Core 1 1 1 1 1
2 Cores 1.27736 1.97445 1.99581 1.98687 1.99471
4 Cores 1.51692 3.91267 3.95574 3.8835 3.91221
8 Cores 1.6212 7.50019 7.74031 7.91742 7.71174

25. • Performance Analysis
Results
64 x 64 128 x 128
256 x 256