Quadratic regularization functionals for phase unwrapping

Marroquín, José L.; Rivera, Mariano

doi:10.1364/josaa.12.002393

Cited by 119 publications

(53 citation statements)

References 10 publications

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“…Regarding the latter, as the problem relies on the nonregularity of some energy terms E ij (δ i , δ j ), i.e., they do not verify (18), our procedure consists in approximating them by regular ones. We do that by leaning on majorize minimize MM [47] concepts.…”

Section: Nonconvex Potentialsmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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Phase Unwrapping via Graph Cuts

Bioucas-Dias

Valadão

2007

IEEE Trans. on Image Process.

473

220

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Abstract-Phase unwrapping is the inference of absolute phase from modulo-2π phase. This paper introduces a new energy minimization framework for phase unwrapping. The considered objective functions are first-order Markov random fields. We provide an exact energy minimization algorithm, whenever the corresponding clique potentials are convex, namely for the phase unwrapping classical L p norm, with p ≥ 1. Its complexity is KT (n, 3n), where K is the length of the absolute phase domain measured in 2π units and T (n, m) is the complexity of a max-flow computation in a graph with n nodes and m edges. For nonconvex clique potentials, often used owing to their discontinuity preserving ability, we face an NP-hard problem for which we devise an approximate solution. Both algorithms solve integer optimization problems, by computing a sequence of binary optimizations, each one solved by graph cut techniques. Accordingly, we name the two algorithms PUMA, for phase unwrapping max-flow/min-cut. A set of experimental results illustrates the effectiveness of the proposed approach and its competitiveness in comparison with state-of-the-art phase unwrapping algorithms.

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Section: Nonconvex Potentialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Phase Unwrapping via Graph Cuts

Bioucas-Dias

Valadão

2007

IEEE Trans. on Image Process.

473

220

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“…To handle this problem, both the path following and the minimum-norm procedures have incorporated quality maps as a measure of confidence on the observed data at each site (see, e.g., [18] for path following, [19] for weighted least-squares and [15] for weighted norm). In a quite different vein and recognizing that phase estimation is an ill-posed problem, papers [20]- [23] have adopted the regularization framework to impose smoothness on the solution. The same objective has been pursued in papers [24]- [29] by adopting a Bayesian viewpoint.…”

Section: Introductionmentioning

confidence: 99%

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

Dias

Leitao

2002

IEEE Trans. on Image Process.

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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.

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“…In a quite different vein, and recognizing that the absolute phase estimation is an ill-posed problem, papers [4], [5], [6], [7] have adopted the regularization framework to impose smoothness on the solution. The same objective has been pursued in papers [8], [9], [10], [11] by adopting a Bayesian viewpoint.…”

Section: Introductionmentioning

confidence: 99%

A Discrete/Continuous Minimization Method in Interferometric Image Processing

Dias

Leitao

2001

Lecture Notes in Computer Science

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Abstract. The 2D absolute phase estimation problem, in interferometric applications, is to infer absolute phase (not simply modulo-2π) from incomplete, noisy, and modulo-2π image observations. This is known to be a hard problem as the observation mechanism is nonlinear. In this paper we adopt the Bayesian approach. The observation density is 2π-periodic and accounts for the observation noise; the a priori probability of the absolute phase is modeled by a first order noncausal Gauss Markov random field (GMRF) tailored to smooth absolute phase images. We propose an iterative scheme for the computation of the maximum a posteriori probability (MAP) estimate. Each iteration embodies a discrete optimization step (Z-step), implemented by network programming techniques, and an iterative conditional modes (ICM) step (π-step). Accordingly, we name the algorithm ZπM, where letter M stands for maximization. A set of experimental results, comparing the proposed algorithm with other techniques, illustrates the effectiveness of the proposed method.

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Quadratic regularization functionals for phase unwrapping

Cited by 119 publications

References 10 publications

Phase Unwrapping via Graph Cuts

Phase Unwrapping via Graph Cuts

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

A Discrete/Continuous Minimization Method in Interferometric Image Processing

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