New regularization scheme for phase unwrapping

Phase unwrapping for phase imaging using the plug-and-play proximal algorithm

Phase unwrapping (PU) is essential for various scientific optical applications. This process aims to estimate continuous phase values from acquired wrapped values, which are limited to the interval (-π,π]. However, the PU process can be challenging due to factors such as insufficient sampling, measurement errors, and inadequate equipment calibration, which can introduce excessive noise and unexpected phase discontinuities. This paper presents a robust iterative method based on the plug-and-play (PnP) proximal algorithm to unwrap two-dimensional phase values while simultaneously removing noise at each iteration. Using a least-squares formulation based on local phase differences and reformulating it as a partially differentiable equation, it is possible to employ the fast cosine transform to obtain a closed-form solution for one of the subproblems within the PnP framework. As a result, reliable phase reconstruction can be achieved even in scenarios with extremely high noise levels.

Automated phase unwrapping in digital holography with deep learning

Digital holography can provide quantitative phase images related to the morphology and content of biological samples. After the numerical image reconstruction, the phase values are limited between -π and π; thus, discontinuity may occur due to the modulo 2π operation. We propose a new deep learning model that can automatically reconstruct unwrapped focused-phase images by combining digital holography and a Pix2Pix generative adversarial network (GAN) for image-to-image translation. Compared with numerical phase unwrapping methods, the proposed GAN model overcomes the difficulty of accurate phase unwrapping due to abrupt phase changes and can perform phase unwrapping at a twice faster rate. We show that the proposed model can generalize well to different types of cell images and has high performance compared to recent U-net models. The proposed method can be useful in observing the morphology and movement of biological cells in real-time applications.

SPUD: simultaneous phase unwrapping and denoising algorithm for phase imaging

Recent methods for phase unwrapping in the presence of noise include denoising algorithms to filter out noise as a preprocessing stage. However, including a denoising stage increases the overall computational complexity resulting in long execution times. In this paper, we present a noniterative simultaneous phase unwrapping and denoising algorithm for phase imaging, referred to as SPUD. The proposed method relies on the least squares discrete cosine transform (DCT) solution for phase unwrapping with an additional sparsity constraint on the DCT coefficients of the unwrapped solution. Simulation results with different levels of noise and wrapped phase fringe density reveal the suitability of the proposed method for accurate phase unwrapping and restoration. When compared to the 2D windowed Fourier transform filter, SPUD performs better in terms of phase error and execution times. The processing of experimental data from synthetic aperture radar showed the capability for processing real images, including removing phase dislocations. An implementation of the proposed algorithm can be accessed and executed through a Code Ocean compute capsule.

Fast phase unwrapping algorithm for interferometric applications

A wide range of interferometric techniques recover phase information that is mathematically wrapped on the interval (-pi, pi). Obtaining the true unwrapped phase is a longstanding problem. We present an algorithm that solves the phase unwrapping problem, using a combination of Fourier techniques. The execution time for our algorithm is equivalent to the computation time required for performing eight fast Fourier transforms and is stable against noise and residues present in the wrapped phase. We have extended the algorithm to handle data of arbitrary size. We expect the state of the art of existing interferometric applications, including the possibility for real-time phase recovery, to benefit from our algorithm.

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

This paper presents an effective algorithm for absolute phase (not simply modulo-2-pi) estimation from incomplete, noisy and modulo-2pi 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-pi-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 (Z-step), implemented by network programming techniques and an iterative conditional modes (ICM) step (pi-step). Accordingly, the algorithm is termed ZpiM, where the letter M stands for maximization. An important contribution of the paper is the simultaneous implementation of phase unwrapping (inference of the 2pi-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.

Two-dimensional phase unwrapping by quad-tree decomposition

One problem to be tackled when interferometric phase-shifting techniques are used is the method in which the phase can be reconstructed. Because an inverse trigonometric function appears in the formulation, the final data are not the phase, but the phase modulo 2pi. A new phase-unwrapping algorithm based on a two-step procedure is presented. In the first step, the digital image to be analyzed is divided into continuous patches by a quad-tree-like recursive procedure; in the second step, the same level patches are joined together by an error-norm-minimizing approach to obtain larger, almost continuous ones. The basic idea of the procedure is to simplify the problem by factoring the complete image into square, variable-size, homogeneous areas (i.e., regions with no internal phase jump) so that only interfaces need to be dealt with. By hierarchically recombining the so-obtained subimages, an unwrapped phase field can be obtained. After a complete description of the algorithm, some examples of its use on synthesized digital images are illustrated. As the algorithm can be used with and without quality masks and the error-minimizing step can use different norms, a full class of unwrapping algorithms can be implemented by this approach.

Noise Reduction in Phase Maps with 2 pi Phase Jumps by Means of the Heat Equation

A new algorithm for filtering noise in phase maps that contain 2pi discontinuities is presented. The algorithm is based on a thermal model that uses the heat equation to perform low-pass filtering. A similar approach is used in image processing for filtering noise, but the edges are generally distorted because of their inherent high frequencies. A solution that consists in redefining the spatial derivatives is proposed here. Simulation results are presented and discussed.

Noise-residue filtering of interferometric phase images

The effect of noise-induced phase inconsistency on interferometric phase information content is studied. Phase inconsistencies, or residues, hinder a correct unwrapping of the phase signal. The probability of noise-induced phase inconsistencies is obtained as a function of the signal-to-noise ratio. Moreover, a two-dimensional noise-residue filter, intended to be applied as a preprocessing step before phase unwrapping, is proposed. The method is based on the observation that the noise creates adjacent phase inconsistencies mainly in interferometric phase images. A local analysis leads to a set of rules to be applied to reduce noise-induced phase inconsistencies. The filter performances are tested on noisy synthetic and real phase data.

Mean-field annealing for phase unwrapping

Use of mean-field annealing theory is proposed for solving the phase-unwrapping (PU) problem. PU is formulated as a constrained optimization problem for the field of integer corrections to be added to the wrapped gradient field. A deterministic algorithm is described to provide an approximation of the average of the correction field over the global minima of the cost function. The proposed algorithm can be applied for any choice of the cost function. Using a cost function based on second-order differences, we obtain results close to those from simulated annealing and spend less computational time.