Reliable phase unwrapping algorithm based on rotational and direct compensators

Phase unwrapping still plays an important role in many data-processing chains based on phase information. Here, we introduce a new phase unwrapping approach for noisy wrapped phase maps of continuous objects to improve the accuracy and computational time requirements of phase unwrapping using a rotational compensator (RC) method. The proposed algorithm is based on compensating the singularity of discontinuity sources. It uses direct compensation for adjoining singular point (SP) pairs and uses RC for other SP pairs. The performance of the proposed method is tested through both simulated and real wrapped phase data. The proposed algorithm is faster than the original algorithm with the RC and has proved efficiency compared to other phase unwrapping methods.     

Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms

Two-dimensional (2-D) phase unwrapping, that is, deducing unambiguous phase values from a 2-D array of values known only modulo 2pi, is a key step in interpreting data acquired with synthetic aperture radar interferometry. Noting the recent network formulation of the phase unwrapping problem, we apply here some well-established ideas of network theory to formalize the problem, analyze its complexity, and derive algorithms for its solution. It has been suggested that the objective of phase unwrapping should be to minimize the total number of places where unwrapped and wrapped phase gradients differ. Here we use network constructions to show that this so-called minimum L0-norm problem is NP-hard, or one that complexity theory suggests is impossible for efficient algorithms to solve exactly. Therefore we must instead find approximate solutions; we present two new algorithms for doing so. The first uses the network ideas of shortest paths and spanning trees to improve on the Goldstein et al. residue-cut algorithm [Radio Sci. 23, 713 (1988)]. Our improved algorithm is very fast, provides complete coverage, and allows user-defined weights. With our second algorithm, we extend the ideas of linear network flow problems to the nonlinear L0 case. This algorithm yields excellent approximations to the minimum L0 norm. Using interferometric data, we demonstrate that our algorithms are highly competitive with other existing algorithms in speed and accuracy, outperforming them in the cases presented here.     

Phase unwrapping through demodulation by use of the regularized phase-tracking technique

Most interferogram demodulation techniques give the detected phase wrapped owing to the arctangent function involved in the final step of the demodulation process. To obtain a continuous detected phase, an unwrapping process must be performed. Here we propose a phase-unwrapping technique based on a regularized phase-tracking (RPT) system. Phase unwrapping is achieved in two steps. First, we obtain two phase-shifted fringe patterns from the demodulated wrapped phase (the sine and the cosine), then demodulate them by using the RPT technique. In the RPT technique the unwrapping process is achieved simultaneously with the demodulation process so that the final goal of unwrapping is therefore achieved. The RPT method for unwrapping the phase is compared with the technique of least-squares integration of wrapped phase differences to outline the substantial noise robustness of the RPT technique.     

Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms.

Phase contrast magnetic resonance velocity imaging is a powerful technique for quantitative in vivo blood flow measurement. Current practice normally involves restricting the sensitivity of the technique so as to avoid the problem of the measured phase being 'wrapped' onto the range -pi to +pi. However, as a result, dynamic range and signal-to-noise ratio are sacrificed. Alternatively, the true phase values can be estimated by a phase unwrapping process which consists of adding integral multiples of 2pi to the measured wrapped phase values. In the presence of noise and data undersampling, the phase unwrapping problem becomes non-trivial. In this paper, we investigate the performance of three different phase unwrapping algorithms when applied to three-dimensional (two spatial axes and one time axis) phase contrast datasets. A simple one-dimensional temporal unwrapping algorithm, a more complex and robust three-dimensional unwrapping algorithm and a novel velocity encoding unwrapping algorithm which involves unwrapping along a fourth dimension (the 'velocity encoding' direction) are discussed, and results from the three are presented and compared. It is shown that compared to the traditional approach, both dynamic range and signal-to-noise ratio can be increased by a factor of up to five times, which demonstrates considerable promise for a possible eventual clinical implementation. The results are also of direct relevance to users of any other technique delivering time-varying two-dimensional phase images, such as dynamic speckle interferometry and synthetic aperture radar.     

Two-dimensional phase unwrapping using a hybrid genetic algorithm

A novel hybrid genetic algorithm (HGA) is proposed to solve the branch-cut phase unwrapping problem. It employs both local and global search methods. The local search is implemented by using the nearest-neighbor method, whereas the global search is performed by using the genetic algorithm. The branch-cut phase unwrapping problem [a nondeterministic polynomial (NP-hard) problem] is implemented in a similar way to the traveling-salesman problem, a very-well-known combinational optimization problem with profound research and applications. The performance of the proposed algorithm was tested on both simulated and real wrapped phase maps. The HGA is found to be robust and fast compared with three well-known branch-cut phase unwrapping algorithms.     

New regularization scheme for phase unwrapping

A new, to our knowledge, algorithm for the phase unwrapping (PU) problem that is based on stochastic relaxation is proposed and analyzed. Unlike regularization schemes previously proposed to handle this problem, our approach dispells the following two assumptions about the solution: a Gaussian model for noise and the magnitude of the true phase-field gradient's being less than pi everywhere. We formulate PU as a constrained optimization problem for the field of integer multiples of 2pi, which must be added to the wrapped phase gradient to recover the true phase gradient. By solving the optimization problem using simulated annealing with constraints, one can obtain a consistent solution under difficult conditions resulting from noise and undersampling. Results from synthetic test images are reported.     

A simple spatial domain algorithm to increase the residues of wrapped phase maps

In phase unwrapping, the locations and densities of the residues are indicative of the severity of the unwrapping problem. Gdeisat et al. proposed an algorithm to increase the number of residues in a wrapped phase map to improve the results of phase unwrapping. The method proposed, though, is extremely time-consuming involving the computation of a Fourier transform along with the selection and shift of its spectral components. Moreover, there is no theoretical analysis on the reasons why the frequency shift should be able to increase the number of residues. In an attempt to address these problems, we propose a simple algorithm to increase the number of residues in a wrapped phase map. The algorithm uses a simple multiply operation in spatial domain to realize the frequency shift exploiting the frequency shift property of Fourier transform. We also discuss the relationship between the number of residues and frequency shift. Experimental results have demonstrated that the proposed method can speed up by more than 50% of the calculation time.     

Threshold automatic selection hybrid phase unwrapping algorithm for digital holographic microscopy

Conventional quality-guided (QG) phase unwrapping algorithm is hard to be applied to digital holographic microscopy because of the long execution time. In this paper, we present a threshold automatic selection hybrid phase unwrapping algorithm that combines the existing QG algorithm and the flood-filled (FF) algorithm to solve this problem. The original wrapped phase map is divided into high- and low-quality sub-maps by selecting a threshold automatically, and then the FF and QG unwrapping algorithms are used in each level to unwrap the phase, respectively. The feasibility of the proposed method is proved by experimental results, and the execution speed is shown to be much faster than that of the original QG unwrapping algorithm.     

Phase-unwrapping algorithm for images with high noise content based on a local histogram

We present a robust algorithm of phase unwrapping that was designed for use on phase images with high noise content. We proceed with the algorithm by first identifying regions with continuous phase values placed between fringe boundaries in an image and then phase shifting the regions with respect to one another by multiples of 2pi to unwrap the phase. Image pixels are segmented between interfringe and fringe boundary areas by use of a local histogram of a wrapped phase. The algorithm has been used successfully to unwrap phase images generated in a three-dimensional shape measurement for noninvasive quantification of human skin structure in dermatology, cosmetology, and plastic surgery.     

Adaptive Quality Guided Phase Unwrapping Algorithm for Whole‐Field Digital Photoelastic Parameter Estimation of Complex Models

Abstract: With advancements in digital image processing and data acquisition, a separate branch of photoelasticity namely digital photoelasticity came into existence. Here, intensity information of the acquired image is used for the evaluation of whole‐field photoelastic parameters. Digital photoelasticity provides only wrapped phasemaps of isoclinics and isochromatics and they have to be unwrapped in different ways for getting the continuous‐phase values. In the case of the isochromatic phasemap, ambiguity removal prior to unwrapping is essential. In this paper, a 10‐step phase‐shifting methodology is proposed and a new strategy for obtaining the isochromatic phasemap free of ambiguity is demonstrated. Isoclinic unwrapping is performed by a new adaptive quality guided algorithm. Adaptive in the sense that isoclinic phase unwrapping is done autonomously even in the presence of isotropic points/π jumps occurring in the isoclinic phasemap. The isochromatic phasemap is also unwrapped using the quality guided path follower. The methodology is validated for the problem of a ring under diametral compression and later shown for three other models which have complex stress fields. Wherever possible, the parameters obtained by the new methodology are compared with analytical or numerical methods and the comparison is quite good.     

Disk-growing algorithm for phase-map unwrapping: application to speckle interferograms

Interferometric techniques combined with phase shifting allow computation of the phase that is linked to the displacement of the object under study. The phases before and after displacement are computed from three or more interferograms (called specklegrams when speckle is used as the information carrier). Subtraction of these two phase patterns leads to a raw phase map. Phase unwrapping restores the 2π discontinuities and gives a continuous phase map. The disk-growing algorithm presented allows the inner and the outer propagation of the unwrapping from a growing disk and so avoids the main problem of anisotropic error propagation for noisy phase maps. It works successfully in speckle interferometry.     

一种新的质量图导引路径积分相位展开算法

本文首先简要介绍了路径积分相位展开方法的原理，针对路径积分法在处理轮廓不连续包裹相位图时存在的缺陷，提出了一种新的质量图用来导引相位展开。算法以包裹相位质量为依据，最先展开质量最高的像素，最后展开质量最低的像素，以防止相位展开误差的传播，把不可避免的相位展开误差局限在最小的区域内。实验结果表明，该算法计算速度快，能有效地处理轮廓不连续相位展开问题。     

Image-inpainting and quality-guided phase unwrapping algorithm

For the wrapped phase map with regional abnormal fringes, a new phase unwrapping algorithm that combines the image-inpainting theory and the quality-guided phase unwrapping algorithm is proposed. First, by applying a threshold to the modulation map, the valid region (i.e., the interference region) is divided into the doubtful region (called the target region during the inpainting period) and the reasonable one (the source region). The wrapped phase of the doubtful region is thought to be unreliable, and the data are abandoned temporarily. Using the region-filling image-inpainting method, the blank target region is filled with new data, while nothing is changed in the source region. A new wrapped phase map is generated, and then it is unwrapped with the quality-guided phase unwrapping algorithm. Finally, a postprocessing operation is proposed for the final result. Experimental results have shown that the performance of the proposed algorithm is effective.     

Path-independent phase unwrapping of subsampled phase maps

A technique for unwrapping subsampled phase maps is presented. The subsampled phase map is obtained by standard phase-shifting methods that use subsampled interferograms. The technique then estimates the wrapped local curvature of the subsampled phase map. This local curvature is then low-pass filtered with a free-boundary low-pass filter to reduce phase noise. Finally the estimated local curvature of the wave front is integrated by the use of a least-squares technique to obtain the searched continuous wave front.     

Unwrapping of Digital Speckle-Pattern Interferometry Phase Maps by use of a Minimum L(0)-Norm Algorithm

The performance of a minimum L(0)-norm unwrapping algorithm is investigated by use of synthetic digital speckle-pattern interferometry (DSPI) wrapped phase maps that simulate experimentally obtained data. This algorithm estimates its own weights to mask inconsistent pixels. Particular features usually included in DSPI wrapped phase distributions, such as shears, speckle noise, fringe cuts, object physical limits, and superimposed phase maps, are analyzed. Some adequate approaches to solving these features are discussed. Finally, it is shown that a complex case in which shears and fringe cuts coexist in the wrapped phase cannot be solved satisfactorily with the minimum L(0)-norm algorithm by itself. To cope with this problem, we propose a new scheme.     

Phase recovery from a single interferogram with closed fringes by phase unwrapping

We describe a new algorithm for phase determination from a single interferogram with closed fringes based on an unwrapping procedure. Here we use bandpass filtering in the Fourier domain, obtaining two wrapped phases with sign changes corresponding to the orientation of the applied filters. An unwrapping scheme that corrects the sign ambiguities by comparing the local derivatives is then proposed. This can be done, assuming that the phase derivatives do not change abruptly among adjacent areas as occurs with smooth continuous phase maps. The proposed algorithm works fast and is robust against noise, as demonstrated in experimental and simulated data.     

Unified approach for rough phase measurement without phase unwrapping by changing the sensitivity factor

Phase evaluation is extraordinarily important in optical, acoustic and radar techniques where coherent signals are employed as information carriers. In most of the cases, phase information are obtained from an inverse trigonometric function and wrapped into ?π to π. A phase unwrapping process is thus required to obtain the final unwrapped phase which represents the ultimate physical quantity to be measured. One-dimensional phase unwrapping is easily achieved by adding or subtracting an integer multiple of 2π to the wrapped phase to establish a smooth phase map. Two-dimensional phase unwrapping, however, is quite troublesome and an elegant unwrapping routing should be chosen in most of the cases to deal with phase residues caused by noise and other error sources. It would be valuable if two-dimensional phase unwrapping can be avoided and the physical quantity obtained directly. In the past, researchers have proposed other methods such as the multiple wavelengths approach which incorporates information from multiple wavelengths to eliminate the need for phase unwrapping. In this study, we extend the multiple wavelengths approach by varying the sensitivity factor, which is more convenient and cost-effective, to achieve the aim of requiring no phase unwrapping. Furthermore, an elegant phase derivative approach is used to solve the phase ambiguity problem in the multiple wavelengths method. Both simulation results and real experiment data of shadow moiré and shearography demonstrate the usefulness of this method, especially for discontinuous surface profile measurements such as steps. Advantages and disadvantages for the proposed method are also discussed in this paper.     

Full-field automated photoelasticity by fourier polarimetry with three wavelengths

The search for fast, precise, and robust testing techniques remains an important problem in automated full-field photoelasticity. The polarizer-sample-analyzer (PSA)-based three-wavelength polarimetric method presented here employs discrete Fourier analysis and a spectral content unwrapping algorithm to provide completely automatic, simple, fast, and accurate determination of both photoelastic parameters. Fourier analysis of experimental data and a three-wavelength approach reduce the effect of noise and efficiently cope with poor accuracy in regions of both isochromatic and isoclinic maps. Because any polarimetric technique yields the phase value in the principal range of the corresponding trigonometric function, the final step in data processing is phase unwrapping. Because of the good quality of the wrapped phase map and because each point is processed independently, our suggested three-wavelength unwrapping algorithm exhibits a high level of robustness. Unlike some other PSA three-wavelength techniques, the given algorithm here solves the problem of phase unwrapping completely. Specifically, it converts experimentally obtained arccosine-type phase maps directly into full phase value distributions, skipping the step of generating an arctangent-type ramped phase map and resorting to other unwrapping routines for final data processing. The accuracy of the new technique has been estimated with a Babinet-Soleil compensator. Test experiments with the disk in diametric compression and a quartz plate have proved that the technique can be used for precise determination of the isoclinic angle and relative retardation, even for large values of the latter.