1. SSIP 2008 Binary Tomography
Project work-Team 9
Binary Tomography

2. Team 9-Binary Tomographers
Attila Kozma, University of Szeged
Tibor Lukic, University of Novi Sad
Erik Wernersson, Uppsala University
Vladimir Curic, University of Novi Sad

3. Outline Binary Tomography The Problem Optimization techniques
Evaluation of proposed methods

4. Binary Tomography Tomography is imaging by sections.
Binary Tomography is a subset of Tomography.
Image is binary.

5. The problem
Problem-How to (re) construct image if we know a few projection vectors.

6. Modeling the problem Horizontal and vertical projections
Different projections, different angles
One ray=one equation

7. General overview Unique No Noise Noise Not Unique
Prior information has to be used.

8. Simulated Annealing Pseuocode outline Set Initial Temperature, T=2
Generate Initial Solution
WHILE T>0 DO
1) Create A New Possible Solution
2) Choose The Best Solution According To The Objective Function Or Choose The Worst With Probability ~exp(delta E / T)
3) Lower The Energy According To Scheme
END

9. Three Projections

10. Four Projections

11. Deterministic Binary Tomography
Combinatorial optimization problem.
Convex relaxation.
where the binary factor, μ>0 and vector e=(1,1,…,1). Starting with
zero value of μ, we iteratively increase μ to enforce binary solutions.
An optimization problem is solved by application of SPG algorithm.

12. SPG Algorithm
The Spectral Projected Gradient (SPG) algorithm is a deterministic
optimization for solving convex-constrained problem ,
where Ω is a closed convex set. Introduced by Birgin, Martinez
and Raydan (2000).
Requirements.
f is defined and has continuous partial derivatives on Ω;
The projection of an arbitrary point onto a set Ω is defined.

13. Experiments
Reconstruction from projections without any noise.

14. Experiments
Reconstructions from projections with Gaussian noise (mean:0, variance: 0.01).

15. Branch and Bound Original problem Associated problem
Relaxation of associated problem

16. Branching

17. Bounding Too many branches. We have to cut.
Solve the relaxation of the actual problem.
The optimum of the relaxation (Z) gives a lower boundary.
In the whole subtree only bigger values than Z are possible for optimal solutions.

18. Experiments

19. Experiments

20. Evaluation of the proposed methods
Original
B & B
S. A.
SPG
Reconstructions from 2 projections by different methods.

21. Evaluation of the proposed methods
Original
S. A.
SPG
Reconstructions from 4 projections in comparable time

22. Thank you!