Parallel Branch-Cut Algorithm Based on Simulated Annealing for Large-Scale Phase Unwrapping

As is well known, multibaseline phase unwrapping (PU) is put forward to overcome single-baseline PU in discontinuous-terrain-height estimation. This letter presents a refined algorithm based on the cluster analysis (CA)-based noise-robust efficient multibaseline PU algorithm proposed by H. Yu. The basic idea is to combine multiple interferometric synthetic aperture radar interferograms with different baseline lengths by a linear combination. The new interferograms after the linear combination increase the ambiguity heights. The number of resulting groups on the envelope of the intercept histogram is decreased and the distance between different intercept groups is widened. Compared with the conventional CA method, the significant advantage of the refined CA (RCA) algorithm is that it improves noise robustness when the intercept groups are densely distributed. The proposed RCA algorithm is validated using the simulated interferometric data. The results demonstrate that the noise robustness performance is better than that of the CA method.

In SAR interferometry, the periodicity of the phase must be removed using two-dimensional phase unwrapping. The goal of the procedure is to find a smooth surface in which large spatial phase differences, called discontinuities, are restricted to places where their presence is reasonable. The pioneering work of Goldstein et al. identified points of local unwrap inconsistency called residues, which must be connected by discontinuities. This paper presents an overview of recent work that treats phase unwrapping as a discrete optimization problem with the constraint that residues must be connected. Several algorithms use heuristic methods to reduce the total number of discontinuities. Constantini has introduced the weighted sum of discontinuity magnitudes as a criterion of unwrap error and shown how algorithms from optimization theory are used to minimize it. Pixels of low quality are given low weight to guide discontinuities away from smooth, high-quality regions. This method is generally robust, but if noise is severe it underestimates the steepness of slopes and the heights of peaks. This problem is mitigated by subtracting (module 2/spl pi/) a smooth estimate of the unwrapped phase from the data, then unwrapping the resulting residual phase. The unwrapped residual is added to the smooth estimate to produce the final unwrapped phase. The estimate can be computed by lowpass filtering of an existing unwrapped phase; this makes possible an iterative algorithm in which the result of each iteration provides the estimate for the next. An example illustrates the results of optimal discontinuity placement and the improvement from unwrapping of the residual phase.

Two-dimensional phase unwrapping (2-D PU) is a crucial step in interferometric signal processing. The accuracy of phase unwrapping (PU) will significantly affect subsequent processing. Existing algorithms perform poorly in areas with rapid terrain changes or with low coherence. In order to solve the above problems, this letter proposes a method of using a neural network to predict residuals, followed by the minimum <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L^{1}$ </tex-math></inline-formula> -norm algorithm for phase reconstruction. The network for predicting residuals is called residual prediction network (RPNet). To construct more complex loss terms, the continuous relaxation strategy is applied to its training. The proposed method overcomes the effects of noise and terrain changes due to the accurate prediction of residuals. The compensated phase gradients enable the minimum <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L^{1}$ </tex-math></inline-formula> -norm algorithm to reconstruct phases more accurately. The experimental results performed on simulation and interferometric synthetic aperture radar (InSAR) data demonstrate the effectiveness of the proposed method.

Strong turbulence causes phase discontinuities known as branch points in an optical field. These discontinuities complicate the phase unwrapping necessary to apply phase corrections onto a deformable mirror in an adaptive optics (AO) system. This paper proposes a non-optimal but effective and implementable phase unwrapping method for optical fields containing branch points. This method first applies a least-squares (LS) unwrapper to the field which isolates and unwraps the LS component of the field. Four modulo-2pi-equivalent non-LS components are created by subtracting the LS component from the original field and then restricting the result to differing ranges. 2pi phase jumps known as branch cuts are isolated to the non-LS components and the different non-LS realizations have different branch cut placements. The best placement of branch cuts is determined by finding the non-LS realization with the lowest normalized cut length and adding it to the LS component. The result is an unwrapped field which is modulo-2pi -equivalent to the original field while minimizing the effect of phase cuts on system performance. This variable-range 'phi LS +phi non phi LS' unwrapper, is found to outperform other unwrappers designed to work in the presence of branch points at a reasonable computational burden. The effect of improved unwrapping is demonstrated by comparing the performance of a system using a fixed-range phi 'LS + phi non--LS' realization unwrapper against the variable-range 'phi LS +phi non--LS' unwrapper in a closed-loop simulation. For the 0.5 log-amplitude variance turbulence tested, the system Strehl performance is improved by as much as 41.6 percent at points where fixed-range 'phi LS + phi non phi LS' unwrappers result in particularly poor branch cut placement. This significant improvement in previously poorly performing regions is particularly important for systems such as laser communications which require minimum Strehl ratios to operate successfully.

Two-dimensional phase unwrapping (PU) is always affected by errors. In this paper we propose and implement a new method that locates the large and small-scale PU errors. In addition, by using ground control points, we eliminate the largescale errors. The method is tested on the real and simulated SAR interferometric data. Results of the tests indicate a high efficiency of the proposed method.

The branch-cut algorithm is a powerful noise-immune algorithm for two-dimensional phase unwrapping. Shorter branch-cuts lead to better results for the unwrapped phase. Herein, we propose a branch-cut algorithm with fast search ability for the shortest branch-cuts based on modified GA. First, the local-nearest-neighbor algorithm is used to pair the positive and negative residues so as to optimize the initial population. This step is fast but gives a local optimum. Next, a branch-cut algorithm based on modified GA is used to globally search the paired residues giving branch-cuts with the shortest total length. Finally, the phase is unwrapped while avoiding branch-cuts. The performance of the proposed algorithm is tested by both simulation and experiment, and the results demonstrate that it can rapidly find the branch-cuts with shortest total length and a high solution accuracy.

This paper explains the underestimation of phase slope and the consequent distortion of the phase surface, observed in two-dimensional (2-D) phase unwrapping by linear estimators, like least squares methods applied to synthetic aperture radar (SAR) interferometry. These methods minimize the difference between the gradient of the unwrapped phase and the wrapped differences of the measured wrapped phase. Using the probability distributions of phase noise and phase differences for a given coherence, the probability of a phase gradient error giving rise to a nonconservative vector field is derived. It is shown that this phase gradient error has nonzero mean in the presence of phase slopes. Linear phase estimators cannot distinguish the mean phase gradient error from a true phase slope; hence, the unwrapped phase shows a slope bias. This bias is quantified as a function of coherence and the number of independent samples that are averaged. The theoretical results are confirmed by simulations.

Nowadays, SAR Interferometry (InSAR) is a technique to generate digital elevation models from space.[1] The phase image unwrapping is the most important step in processing the interferometric SAR data. Goldstein's branch-cut algorithm is the most classical one. It is about the connection of residues with branch-cut lines to neutralize their charges so that the integral path can't stride over the branches in the process of phase unwrapping, thus avoiding pervasion of error. However, this connection always brings about numerous, lengthy and even closed branch-cut lines that can not be unwrapped, especially when many residues exist. This paper presents an optimized minimum spanning tree phase unwrapping algorithm, and described the implementation in detail. It is expected to remove the redundancies of the branch-cut tree accurately and ensure the effectiveness of phase unwrapping.

Identifying the residues is the most significant process in Goldstein's Branch Cut algorithm. The path-independent-integral criterion is a sufficient and unnecessary condition of residue identification. To overcome this shortcoming, this essay aims to analyze the phase quality of the residues. We find that on the aspect of position, the positions of the residues can typically show the distribution of the phase quality of the graph generally. We also find that on the aspect of the quality value, the quality values of the residues can help us figure out the phase quality of local regions. In a nutshell, analyzing the quality values of the residues can guide us to unwrap phases in a completely correct way.

Interferometric Synthetic Aperture Radar (InSAR) is a microwave imaging system with all-weather earth observation. Because of its wide observation area, high precision, day and night operational capabilities, it has a significant advantage in monitoring large-scale surface deformation. Phase unwrapping is a key step in InSAR data processing. The core of Goldstein traditional branch unwrapping method is the establishment of branch-cutting lines. The shorter the overall length of the branch line, the solution is more accurate. Based on the idea of solving the shortest path of Travelling Salesman Problem (TSP), a new algorithm combining ant colony algorithm and genetic algorithm is proposed. Based on the advantages of these two algorithms, we avoid the disadvantages of excessive parameters, slow calculation speed and low efficiency in ant colony algorithm, reduce the time consumed by ant colony algorithm in solving local optimal solution, improve the local search ability of ant colony algorithm, establish shorter branch line than traditional branch method, prevent islanding effect and improve the precision of phase unwrapping. The tests performed on real and simulated interferometric SAR data confirm the validity of the proposed method.

A sparse-representation-based direct minimum Lp-norm algorithm is proposed for a two-dimensional MRI phase unwrapping. First, the algorithm converts the weighted-Lp-norm-minimization-based phase unwrapping problem into a linear system problem whose system (coefficient) matrix is a large, symmetric one. Then, the coefficient-matrix is represented in the sparse structure. Finally, standard direct solvers are employed to solve this linear system. Several wrapped phase datasets, including simulated and MR data, were used to evaluate this algorithm's performance. The results demonstrated that the proposed algorithm for unwrapping MRI phase data is reliable and robust.

We present a block least-squares (BLS) method for two-dimensional (2-D) phase unwrapping. The method works by tessellating the input image into small square blocks with only one phase wrap. These blocks are unwrapped using a simple procedure, and the unwrapped blocks are merged together using one of two proposed block merging algorithms. By specifying a suitable mask, the method can easily handle objects of any shape. This approach is compared with the Ghiglia-Romero method and the Marroquin-Rivera method. On synthetic images with different noise levels, the BLS method is shown to be superior, both with respect to the resulting gray values in the unwrapped image as well as visual inspection. The method is also shown to successfully unwrap synthetic and real images with shears, fiber-optic interferometry images, and medical magnetic resonance images. We believe the new method has the potential to improve the present quality of phase unwrapped images of several different image modalities.

Like time complexity models that have significantly contributed to the analysis and development of fast algorithms, energy complexity models for parallel algorithms are desired as crucial means to develop energy efficient algorithms for ubiquitous multicore platforms. Ideal energy complexity models should be validated on real multicore platforms and applicable to a wide range of parallel algorithms. However, existing energy complexity models for parallel algorithms are either theoretical without model validation or algorithm-specific without ability to analyze energy complexity for a wide-range of parallel algorithms. This paper presents a new general validated energy complexity model for parallel (multithreaded) algorithms. The new model abstracts away possible multicore platforms by their static and dynamic energy of computational operations and data access, and derives the energy complexity of a given algorithm from its work, span and I/O complexity. The new model is validated by different sparse matrix vector multiplication (SpMV) algorithms and dense matrix multiplication (matmul) algorithms running on high performance computing (HPC) platforms (e.g., Intel Xeon and Xeon Phi). The new energy complexity model is able to characterize and compare the energy consumption of SpMV and matmul kernels according to three aspects: different algorithms, different input matrix types and different platforms. The prediction of the new model regarding which algorithm consumes more energy with different inputs on different platforms, is confirmed by the experimental results. In order to improve the usability and accuracy of the new model for a wide range of platforms, the platform parameters of ICE model are provided for eleven platforms including HPC, accelerator and embedded platforms.

Phase unwrapping is a process to reconstruct the absolute phase from a wrapped phase map whose range is (−&pi;, &pi;]. As the absolute phase cannot be directly extracted from the fringe pattern, phase unwrapping is therefore required by phasemeasure techniques. Currently, many phase unwrapping algorithms have been proposed. In this paper, four popular phase unwrapping algorithms, including the Goldstein’s branch cut method, the quality-guided method, the Phase Unwrapping via Max Flow (PUMA) method, and the phase estimation using adaptive regularization based on local smoothing method (PERALS), are reviewed and discussed. Detailed accuracy comparisons of these methods are provided as well.

SUMMARY Two-dimensional phase unwrapping (PU) process usually causes a noise-induced distortion in the geographical information of a wrapped phase image obtained by, for example, interferometric synthetic aperture radar (InSAR). This paper presents a novel method to reduce the phase-unwrapping distortion by being based on two-dimensional fractional Brownian motion (fBm) theory. The method incorporates fractal geometry estimation with conventional global-transform PU. For the spatialfrequency spectrum of an observed phase image, we estimate the fractal dimension by assuming an almost constant dimension over the image. Then, according to the estimation, we compensate the distorted spectrum of the tentatively computed global PU result. We obtain a better topographical map as the inverse Fourier transform of the compensated spectrum. It is demonstrated that the proposed method increases the signal-to-noise ratio of PU results for simulated data with various noise levels. Evaluations on an actual InSAR phase image also show that the method significantly improves the quality of the conventional global-transform PU result, in par