Multi-source Absolute Phase Estimation: A Multi-precision Approach Based on Graph Cuts

1. 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

2. Phase Denoising (PD) Phase Unwrapping (PU) Estimation of Estimation of (wrapped phase) Absolute Phase Estimation

3. Applications • Synthetic aperture radar/sonar • Magnetic resonance imaging • Doppler weather radar • Doppler echocardiography • Optical interferometry • Diffraction tomography

4. Absolute Phase Estimation in InSAR (Interferometric SAR) InSAR Problem: Estimate 2- 1 from signals read by s1 and s2

5. Mountainous terrain around Long’s Peak, Colorado Interferogram

6. Differential Interferometry Height variation 7 mm/year -17 mm/year

7. Magnetic Resonance Imaging - MRI Wrapped phase Intensity Interferomeric Phase • measure temperature • visualize veins in tissues • water-fat separation • mapthe principal magnetic field

8. Outline • Forward problem (sensor model) • Absolute phase estimation: Bayesian formulation • Computing the MAP estimate via integer optimization • Multi-source absolute phase estimation • Phase unwrapping • Convex and non-convex priors • Unambiguous interval increasing • Phase unwrapping • Convex and non-convex priors

9. Forward Problem: Sensor Model

10. Simulated Interferograms Images of

11. Data density: Prior (1st order MRF): clique set clique potential (pairwise interaction) non-convex convex Enforce smoothness Enforce piecewise smoothness (discontinuity preserving) Bayesian Approach

12. posterior density • Phase unwrapping: Maximum a Posteriori Estimation Criterion

13. Assume that Then PU ! summing over walks Phase Unwrapping: Path Following Methods Why isn’t PU a trivial problem? Discontinuities High phase rate Noise

14. [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 a graph [Frey et al., 01] (approx)! Belief propagation on a 1st order MRF convex [Bioucas-Dias & Valadao, 05] (exact)! Sequence of K min cuts ( ) non-convex [Ghiglia, 96]! LPN0 (continuous relaxation) [Bioucas-Dias & G. Valadao, 05, 07] ! Sequence of min cuts ( ) Phase Unwrapping Algorithms

15. while success == false then success == true PUMA (Phase Unwrapping MAx-flow) Finds a sequence of steepest descent binary images

16. is submodular: each binary optimization has the complexity of a min cut • Related algorithms [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 • A local minimum is a global minimum • Takes at most K iterations

17. Results ( )

18. Results ( ) Convex priors does not preserve discontinuities

19. PUMA: Non-convex priors Ex: Models discontinuities Models Gaussian noise Shortcomings: • Local minima is no more a global minima • Energy contains nonsubmodular terms (NP-hard) Tentative suboptimal solutions: • Majorization Minimization • Quadratic Pseudo Boolean Optimization (Probing [Boros et al., 2006], Improving [Rother et al., 2007] )

20. Non-increasing property Majorizing nonsubmodular terms Majorization Minimization (MM) [Lange & Fessler, 95] [Rother et al., 05] ! similar approach for alpha expansion moves

21. Interferogram no. of nonsubmodular terms iter us MM QOBOP QPBOI QPBOP MM QPBOI 1 590/0 2,5 e-2 590/0 590 326/0 1,0 e-2 2 326/0 410 263/0 1,0 e-2 263/0 271 3 154/0 6,0 e-3 154/0 179 4 123/0 4,0 e-3 123/0 141 5 94/0 4,0 e-2 6 94/0 117 88/0 2,5 e-3 88/0 91 7 57/15000 1,0e-3 57/15000 57 8 T 1 s 120 s 2 s Results

22. Interferogram  MM QOBOP QPBOI Results

23. Multi-jump version of PUMA Jumps 2 [1 2 3 4]

24. PUMA + dyadic scaling then • Unary terms may be non-convex Compute using the algorithm [Darbon, 07] for 1st order submodular priors (complexity ) Absolute Phase (PU + Denoising) • Related algorithms: [Zalesky, 03], [Ishikawa, 03], [Ahuja, Hochbaum, Orlin, 04]

28. Multi-source Absolute Phase Estimation

29. Noise High phase rate Major degradation mechanism in PU and APE

30. Use more than one observation with different frequencies Two sources We can infer • noise is an issue • unwrap phase images with range larger than Multi-source Absolute Phase Estimation

31. Two sources

33. Absolute phase estimation: • Phase v-unwrapping: Computing the MAP estimate

34. Initialization: 1-unwrapp in the interval using total variation (TV) Optimization: Non-convex data term + TV Exact solution: Levelable functions [Darbon, 07], [Ahuja et al, 04], [Zalesky, 03], [Ishikawa, 03], (takes time) 2. Run PUMA in a multiscale fashion with the schedule: • scale v ! v-unwrapping] • scales ! denoising Proposed Algorithm

35. for t=0:tmax success == false while success == false then success == true Absolute Phase (1-PU+v-PU + Denoising)

37. High phase rate + noise

38. Parabolic surface

39. Future Directions • High order interactions • Denoise (first) + Unwrap • Local adaptive models (collaboration with Vladimir Katkovnik, Tampere University of Technology) • Huge images (ex: 10000£10000)

40. Concluding Remarks • Addressed discontinuity preserving phase unwrapping and phase denoising methods based on integer optimization • Addressed multi-source absolute phase estimation • Introduced the concept of v-phase unwrapping • Introduced a new algorithm for multi-source absolute phase estimation based on integer optimization

41. References • J. Dias and J. Leitao, “The ZM algorithm for interferometric image reconstruction in SAR/SAS”, IEEE Transactions on Image processing, vol. 11, no. 4, pp. 408-422, 2002. • J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts", IEEE Transactions on Image processing, vol. 16, no. 3, pp. 698-709, 2007. • V. Kolmogorov and A. Shioura, “New algorithms for the dual of the convex cost network flow problem with applications to computer vision", Technical Report, June, 2007 • J. Darbon, Composants logiciels et algorithmes de minimisation exacte d’energies dedies au traitement des images, PhD thesis, Ecole Nationale Superieure des Telecommunications, 2005. • Y. Boykov and Vladimir Kolmogorov, An Experimental Comparison of Min-Cut/Max-Flow, IEEE Transactions on Pattern Analysis and Machine Intelligence, September 2004.

42. References • C. Rother, V. Kolmogorov, V. Lempitsky, and M. Szummer, “Optimizing binary MRFs via extended roof duality”, in IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2007. • E. Boros, P. L. Hammer, and G. Tavares. Preprocessing of unconstrained quadratic binary optimization. Technical Report RRR 10-2006, RUTCOR, Apr. 2006. • J. Darbon and M. Sigelle, “Image restoration with discrete constrained total variation Part II: Levelable functions, convex and non-convex cases”, Journal of Mathematical Imaging and Vision. Vol. 26 no. 3, pp. 277-291, December 2006. • B. Zalesky, “Efficient determination of Gibbs estimators with submodular energy functions”, arXiv:math/0304041v1 [math.OC] 3 Apr 2003.

43. Acknowledgements Gonçalo Valadão Yuri Boykov Vladimir Kolmogorov