Multi-source Absolute Phase Estimation: A Multi-precision Approach Based on Graph Cuts. José M. Bioucas-Dias Instituto Superior Técnico Instituto de Telecomunicações Portugal. Graph Cuts and Related Discrete or Continuous Optimization Problems - IPAM 08. Phase Denoising (PD).
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José M. Bioucas-Dias
Instituto Superior Técnico
Instituto de Telecomunicações
Graph Cuts and Related Discrete or Continuous Optimization Problems - IPAM 08
Phase Unwrapping (PU)
Absolute Phase Estimation
Absolute Phase Estimation in InSAR (Interferometric SAR)
InSAR Problem: Estimate 2- 1 from signals read by s1 and s2
Long’s Peak, Colorado
Simulated Interferograms Images of
Prior (1st order MRF):
clique potential (pairwise interaction)
Enforce piecewise smoothness
(discontinuity preserving)Bayesian Approach
PU ! summing over walksPhase Unwrapping: Path Following Methods
Why isn’t PU a trivial problem?
High phase rate
[Flynn, 97] (exact)! Sequence of positive cycles on a graph
[Costantini, 98] (exact)! min-cost flow on a graph
[Bioucas-Dias & Leitao, 01] (exact)! Sequence of positive cycles on
[Frey et al., 01] (approx)! Belief propagation on a 1st order MRF
[Bioucas-Dias & Valadao, 05] (exact)! Sequence of K min cuts ( )
[Ghiglia, 96]! LPN0 (continuous relaxation)
[Bioucas-Dias & G. Valadao, 05, 07] ! Sequence of min cuts ( )Phase Unwrapping Algorithms
while success == false
success == truePUMA (Phase Unwrapping MAx-flow)
Finds a sequence of steepest descent binary images
has the complexity of a min cut
[Veksler, 99] (1-jump moves )
[Murota, 03] (steepest descent algorithm for L-convex functions)
[Ishikawa, 03] (MRFs with convex priors)
[Kolmogorov & Shioura, 05,07], [Darbon. 05](Include unary terms)
[Ahuja, Hochbaum, Orlin, 03] (convex dual network flow problem)PUMA: Convex Priors
Convex priors does not preserve discontinuities
Models Gaussian noise
Tentative suboptimal solutions:
(Probing [Boros et al., 2006], Improving [Rother et al., 2007] )
Majorizing nonsubmodular terms
Majorization Minimization (MM) [Lange & Fessler, 95]
[Rother et al., 05] ! similar approach for alpha expansion moves
no. of nonsubmodular terms
Jumps 2 [1 2 3 4]
Compute using the algorithm [Darbon, 07] for 1st order
submodular priors (complexity )
[Ahuja, Hochbaum, Orlin, 04]
High phase rate
Major degradation mechanism in PU and APE
We can infer
Multi-source Absolute Phase Estimation
total variation (TV)
Optimization: Non-convex data term + TV
Exact solution: Levelable functions [Darbon, 07],
[Ahuja et al, 04], [Zalesky, 03], [Ishikawa, 03],
2. Run PUMA in a multiscale fashion with the schedule:
success == false
while success == false
success == trueAbsolute Phase (1-PU+v-PU + Denoising)
Vladimir Katkovnik, Tampere University of Technology)
and phase denoising methods based on integer
phase estimation based on integer optimization
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