The ZπM algorithm: a method for interferometric image reconstruction in SAR/SAS

Abstract

Abstract-This paper presents an effective algorithm for absolute phase (not simply modulo-2 ) estimation from incomplete, noisy and modulo-2 observations in interferometric aperture radar and sonar (InSAR/InSAS). The adopted framework is also representative of other applications such as optical interferometry, magnetic resonance imaging and diffraction tomography. The Bayesian viewpoint is adopted; the observation density is 2 -periodic and accounts for the interferometric pair decorrelation and system noise; the a priori probability of the absolute phase is modeled by a compound Gauss-Markov random field (CGMRF) tailored to piecewise smooth absolute phase images. We propose an iterative scheme for the computation of the maximum a posteriori probability (MAP) absolute phase estimate. Each iteration embodies a discrete optimization step ( -step), implemented by network programming techniques and an iterative conditional modes (ICM) step ( -step). Accordingly, the algorithm is termed , where the letter stands for maximization. An important contribution of the paper is the simultaneous implementation of phase unwrapping (inference of the 2 -multiples) and smoothing (denoising of the observations). This improves considerably the accuracy of the absolute phase estimates compared to methods in which the data is low-pass filtered prior to unwrapping. A set of experimental results, comparing the proposed algorithm with alternative methods, illustrates the effectiveness of our approach.

Phase Unwrapping via Graph Cuts

“…Although energy (2) was introduced deterministically in a natural way, it can also be derived under a Bayesian perspective as in [7]. Consider a first-order Markov random field prior on the absolute phase image given by p(φ)…”

“…The need for phase estimation is common to many imaging techniques, from which we point up interferometric synthetic aperture radar and sonar (InSAR/InSAS) [3], [4], [5], [6], [7], [8], [9], magnetic resonance imaging (MRI) [10], [11], and optical interferometry [12]. In InSAR, as in InSAS, two or more antennas measure the phase between them and the terrain; the topography may then be inferred from the difference between those phases, relying on simple geometric reasoning.…”

“…Itoh condition might be violated if the true phase surface is discontinuous, or if the wrapped phase is noisy. In either cases, PU becomes a very difficult problem, to which much attention has been devoted [17], [18], [19], [20], [5], [21], [7].…”

“…Phase unwrapping approaches belong mainly to one of the following classes: path following [17], [22], [23], minimum L p norm [24], [25], [19], [20], [26], [7], Bayesian/regularization [18], [27], [28], [29], [30], [7], [31], [32], and parametric modelling [33], [34].…”

“…With p = 2, we have a least squares method [35]. The exact solution with p = 2 is developed in [7] using network programming techniques. An approximation to the least squares solution can be obtained by relaxing the discrete domain Z MN to R MN , where M and N are, respectively, the number of lines and columns, and applying FFT or DCT based techniques [5,Chap.…”