Absolute phase image reconstruction: a stochastic nonlinear filtering approach

This paper formulates and proposes solutions to the problem of estimating/reconstructing the absolute (not simply modulo-2 ) phase of a complex random field from noisy observations of its real and imaginary parts. This problem is representative of a class of important imaging techniques such as interferometric synthetic aperture radar, optical interferometry, magnetic resonance imaging, and diffraction tomography. We follow a Bayesian approach; then, not only a probabilistic model of the observation mechanism, but also prior knowledge concerning the (phase) image to be reconstructed, are needed. We take as prior a nonsymmetrical half plane autoregressive (NSHP AR) Gauss–Markov random field (GMRF). Based on a reduced order state-space formulation of the (linear) NSHP AR model and on the (nonlinear) observation mechanism, a recursive stochastic nonlinear filter is derived. The corresponding estimates are compared with those obtained by the extended Kalman–Bucy filter, a classical linearizing approach to the same problem. A set of examples illustrate the effectiveness of the proposed approach.