Project workTeam 9

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Project work-Team 9

• Binary Tomography

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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

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Outline

• Binary Tomography
• The Problem
• Optimization techniques
• Evaluation of proposed methods

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Binary Tomography

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

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The problem

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

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Modeling the problem

• Horizontal and vertical projections
• Different projections, different angles
• One rayone equation

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General overview
Prior information has to be used.
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Simulated Annealing
Pseuocode outline Set Initial Temperature,
T2 Generate Initial Solution WHILE Tgt0 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
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Three Projections
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Four Projections
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Deterministic Binary Tomography
Combinatorial optimization problem.
Convex relaxation.
where the binary factor, µgt0 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.
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SPG Algorithm
The Spectral Projected Gradient (SPG) algorithm
is a deterministic optimization for solving
convex-constrained problem
,
where O is a closed convex set. Introduced by
Birgin, Martinez and Raydan (2000).
Requirements.

• f is defined and has continuous partial
  derivatives on O
• The projection of an arbitrary point onto a set O
  is defined.

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SPG based Algorithm for Binary Tomography
.
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Experiments
Reconstruction from projections without any noise.
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Experiments
Reconstructions from projections with Gaussian
noise (mean0, variance 0.01).
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Branch and Bound

• Original problem

• Associated problem

Relaxation of associated problem
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Branching
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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.

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Experiments
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Experiments
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Evaluation of the proposed methods
Original
B B
SPG
S. A.
Reconstructions from 2 projections by different
methods.
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Evaluation of the proposed methods
SPG
S. A.
Original
Reconstructions from 4 projections in comparable
time
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Thank you!