Project work-Team 9

1. 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 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. Original problem Associated problem Branch and Bound 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 S. A. SPG Original Reconstructions from 4 projections in comparable time

22. Thank you!