**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30184

##### Maximizer of the Posterior Marginal Estimate of Phase Unwrapping Based On Statistical Mechanics of the Q-Ising Model

**Authors:**
Yohei Saika,
Tatsuya Uezu

**Abstract:**

We constructed a method of phase unwrapping for a typical wave-front by utilizing the maximizer of the posterior marginal (MPM) estimate corresponding to equilibrium statistical mechanics of the three-state Ising model on a square lattice on the basis of an analogy between statistical mechanics and Bayesian inference. We investigated the static properties of an MPM estimate from a phase diagram using Monte Carlo simulation for a typical wave-front with synthetic aperture radar (SAR) interferometry. The simulations clarified that the surface-consistency conditions were useful for extending the phase where the MPM estimate was successful in phase unwrapping with a high degree of accuracy and that introducing prior information into the MPM estimate also made it possible to extend the phase under the constraint of the surface-consistency conditions with a high degree of accuracy. We also found that the MPM estimate could be used to reconstruct the original wave-fronts more smoothly, if we appropriately tuned hyper-parameters corresponding to temperature to utilize fluctuations around the MAP solution. Also, from the viewpoint of statistical mechanics of the Q-Ising model, we found that the MPM estimate was regarded as a method for searching the ground state by utilizing thermal fluctuations under the constraint of the surface-consistency condition.

**Keywords:**
Bayesian inference,
maximizer of the posterior
marginal estimate,
phase unwrapping,
Monte Carlo simulation,
statistical mechanics

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1334177

**References:**

[1] D. L. Fried, "Least-square fitting of a wave-front distortion estimate to an array of phase differences measurements," J. Opt. Soc. Am, vol. 67, pp. 370-375, 1977.

[2] R. H. Hudgin, "Wave-front reconstruction for compensated imaging," J. Opt. Soc. Am, vol. 67, pp. 375-378, 1977.

[3] R. J. Noll, "Phase estimation from slope-type wave-front sensors," J. Opt. Soc. Am, vol. 68, 139-140, 1978.

[4] R. M. Goldstein and H. A. Zebker, "Interferometric radar mapping of ocean currents," Nature, vol. 328, pp. 707-709, 1987.

[5] D. C. Ghiglia and. L. A. Romero, "Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative method," J. Opt. Soc. Am. A, vol. 11, pp. 107-117, 1994.

[6] D. C. Ghiglia and M. D. Pritt, "Two-Dimensional Phase Unwrapping Theory Algorithms and Software," New York: Wiley, 1998.

[7] J. L. Marroquin and M. Rivera, "Quadratic regularization functionals for phase unwrapping", J. Opt. Soc. Am. A, vol. 12, pp. 2393-2400, 1995.

[8] L. Guerriero, G. Nico, G. Pasquariello and Stramaglia, "A new regularization scheme for phase unwrapping," Appl. Opt. vol. 37, no. 14, pp. 3053-3058, 1998.

[9] G. Nico, G. Palubinskas and M. Datcu, "Bayesian Approaches to Phase Unwrapping: Theoretical Study", IEEE Trans. Signal Processing, vol. 48(4), pp.2545-2556, 2000.

[10] A. P. Shanker and H. Zebker, "Edgelist phase unwrapping algorithm for time series InSAR analysis," J. Opt. Soc. Am. A, vol. 27(3), pp. 605-612, 2010.

[11] H. Nishimori, "Theory of spin glasses and information; An introduction," Oxford, London, 2001.

[12] K. Tanaka, "Statistical mechanical approach to image processing," Journal of Physics A: Mathematical and General, vol. 35(37), R31-R150, 2002.

[13] K. Tanaka, J. Inoue and D. M. Titterington, "Probabilistic image processing by means of the Bethe approximation for the Q-Ising model," Journal of Physics A: Mathematical and General, vol. 36(43), pp. 11023-11035, 2003.

[14] Y. Saika and H. Nishimori, "Statistical mechnics of image restoration using the plane rotator model," Journal of the Physical Society of Japan, vol. 71(4), pp. 1052-1058, 2002.

[15] Y. Saika, J. Inoue, H. Tanaka and M. Okada, "Bayes-optimal solution to inverse halftoning based on statistical mechanics of the Q-Ising model", Central European Journal of Physics, Vol. 7(3), pp. 444-456, 2009.

[16] Y. Saika, "Statistical Mechanics of inverse halftoning", Numerical Simulations and Engineering Process (Book-s Chapter), pp. 525-540, 2011.

[17] Y. Saika and T. Aoki, "Thermodynamics-inspired inverse halftoning via multiple halftone images", CAAI-Transactions on Intelligent Systems, vol. 7(1), pp. 86-94, 2012.

[18] Y. Saika and Y. Haraguchi, "Maximizer of the Posterior Marginal Estimate for Noise Reduction of JPEG-compressed Image", International Journal of Computer and Information Technology, vol. 6(1), pp. 13-17, 2012.

[19] Y. Saika and H. Nishimori, "Statistical mechanical approach to phase retrieval using the Q-Ising model", Progress Theoretical Physics Supplement, vol. 157, pp. 292-295, 2005.