SSIP 2008 Binary Tomography Project work-Team 9 Binary Tomography

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

Outline Binary Tomography The Problem Optimization techniques Evaluation of proposed methods

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

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

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

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

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

Three Projections

Four Projections

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.

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.

Experiments Reconstruction from projections without any noise.

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

Branch and Bound Original problem Associated problem Relaxation of associated problem

Branching

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.

Experiments

Experiments

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

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

Thank you!