Noise-immune phase unwrapping by use of calculated wrap regions

We implement a proposed method for noise-immune phase unwrapping through the calculation of what may be called wrap regions. The theory of this method is reformulated and discussed with regard to random wrapping, a phenomenon that occurs with diffusely reflecting objects if the optical phase change caused by a deformation is calculated by subtraction of the phases calculated before and after the deformation. Two methods of eliminating random wrapping are presented. The algorithm for phase unwrapping is described, and experimental results are presented.     

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.     

Object-image-based method to construct an unweighted quality map for phase extraction and phase unwrapping

A method to construct an unweighted quality map for phase extraction and phase unwrapping is proposed, based on an object image pattern. The object image pattern must be recorded under the same conditions as that of the corresponding interference patterns, except that the lights coming from the reference arm of the interferometer are hidden. An unweighted quality map that can represent the valid and invalid regions in the interference patterns is completed successfully, based on two factors: the fact that the object region in the object image pattern is homologous with the valid region (i.e., the interference region) in the interference patterns, and on distinguishing between the object and background regions in the object image pattern using neighbor window threshold filtering and fitting the boundary of the object image. The application of the proposed method to the real measurement shows its feasibility and correctness. This paper might provide an alternative method for constructing an unweighted quality map for phase extraction and phase unwrapping.     

Two-dimensional phase unwrapping with use of statistical models for cost functions in nonlinear optimization

Interferometric radar techniques often necessitate two-dimensional (2-D) phase unwrapping, defined here as the estimation of unambiguous phase data from a 2-D array known only modulo 2pi rad. We develop a maximum a posteriori probability (MAP) estimation approach for this problem, and we derive an algorithm that approximately maximizes the conditional probability of its phase-unwrapped solution given observable quantities such as wrapped phase, image intensity, and interferogram coherence. Examining topographic and differential interferometry separately, we derive simple, working models for the joint statistics of the estimated and the observed signals. We use generalized, nonlinear cost functions to reflect these probability relationships, and we employ nonlinear network-flow techniques to approximate MAP solutions. We apply our algorithm both to a topographic interferogram exhibiting rough terrain and layover and to a differential interferogram measuring the deformation from a large earthquake. The MAP solutions are complete and are more accurate than those of other tested algorithms.     

Two-dimensional phase unwrapping of subsampled phase-shifted interferograms

Abstract

A novel procedure for phase recovery from undersampled phaseshifted interferograms is described. First, synthetic interferograms are used to calculate the wrapped phase differences along two orthogonal directions. Second, they are unwrapped to recover the true Laplacian of the phase. The final step is to integrate the unwrapped phase differences to obtain the required phase. This method may be used with either path-independent or path-dependent phase unwrapping algorithms. The technique overcomes the sensitivity to noise of previous algorithms when low pass filtering techniques are applied during the calculation of the phase differences and least square methods are employed in the final step of phase integration.     

Temporal phase unwrapping of digital hologram sequences

A method for recording and evaluating digital image-plane holograms is presented. Hundreds of holograms of an object that has been subjected to dynamic deformation (e.g., vibrations) are recorded. The phase of the wave front is calculated from the recorded holograms by use of a two-dimensional digital Fourier-transform method. By temporal phase unwrapping it is possible to determine the absolute deformation (included the direction of motion) of the object. Experimental results are presented, and the advantages of temporal phase unwrapping compared with spatial phase unwrapping are discussed.     

Performance evaluation of two-dimensional phase unwrapping algorithms

We present a performance evaluation of eight two-dimensional phase unwrapping methods with respect to correct phase unwrapping and execution times. The evaluated methods are block least squares (BLS), adaptive integration (AI), quality guided path following (QUAL), mask cut (MCUT), multigrid (MGRID), preconditioned conjugate gradient (PCG), Flynn's (FLYNN), and Liang's (LIANG). This set included integration- (path following), least-squares-, L(1)-, and model-based methods. The methods were tested on several synthetic images, on two magnetic resonance images, and on two interferometry images. The synthetic images were designed to demonstrate different aspects of the phase unwrapping problem. To test the noise robustness of the methods, independent noise was added to the synthetic images to yield different signal-to-noise ratios. Each experiment was performed 50 times with different noise realizations to test the stability of the methods. The results of the experiments showed that the congruent minimum L(1) norm FLYNN method was best overall and the most noise robust of the methods, but it was also one of the slowest methods. The integration-based QUAL method was the only method that correctly unwrapped the two interferometry images. The least-squares-based methods (MGRID, PCG) gave worse results on average than did the integration- (or path following) based methods (BLS, AI, QUAL, MCUT) and were also slower. The model-based LIANG method was sensitive to noise and resulted in large errors for the magnetic resonance images and the interferometry images. In conclusion, for a particular application there is a trade-off between the quality of the unwrapping and the execution time when we attempt to select the most appropriate method.     

Flexible error-reduction method for shape measurement by temporal phase unwrapping: phase averaging method

Temporal phase unwrapping is an important method for shape measurement in structured light projection. Its measurement errors mainly come from both the camera noise and nonlinearity. Analysis found that least-squares fitting cannot completely eliminate nonlinear errors, though it can significantly reduce the random errors. To further reduce the measurement errors of current temporal phase unwrapping algorithms, in this paper, we proposed a phase averaging method (PAM) in which an additional fringe sequence at the highest fringe density is employed in the process of data processing and the phase offset of each set of the four frames is carefully chosen according to the period of the phase nonlinear errors, based on fast classical temporal phase unwrapping algorithms. This method can decrease both the random errors and the systematic errors with statistical averaging. In addition, the length of the additional fringe sequence can be changed flexibly according to the precision of the measurement. Theoretical analysis and simulation experiment results showed the validity of the proposed method.     

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.     

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.     

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.     

Assessment of autonomous phase unwrapping of isochromatic phase maps in digital photoelasticity

Photoelasticity is the only whole-field experimental technique which can analyse both 2-D and 3-D elasticity problems. In digital photoelasticity one gets two phase maps, one corresponding to principal stress direction (isoclinic) and the other corresponding to principal stress difference (isochromatic). The phase maps for both isoclinics and isochromatics are to be unwrapped differently for obtaining continuous phase values. Autonomous phase unwrapping is one of the challenging issues and this paper focuses on recent advances on isochromatic phase map unwrapping. A comparative study of different autonomous phase unwrapping algorithm is done by solving a bench mark problem and a stress frozen slice with cut-outs. The need for domain delimiting in addition to domain masking is brought out while using autonomous phase unwrapping algorithms.     

Modulo 2pi fringe orientation angle estimation by phase unwrapping with a regularized phase tracking algorithm

The fringe orientation angle provides useful information for many fringe-pattern-processing techniques. From a single normalized fringe pattern (background suppressed and modulation normalized), the fringe orientation angle can be obtained by computing the irradiance gradient and performing a further arctangent computation. Because of the 180 degrees ambiguity of the fringe direction, the orientation angle computed from the gradient of a single fringe pattern can be determined only modulo pi. Recently, several studies have shown that a reliable determination of the fringe orientation angle modulo 2pi is a key point for a robust demodulation of the phase from a single fringe pattern. We present an algorithm for the computation of the modulo 2pi fringe orientation angle by unwrapping the orientation angle obtained from the gradient computation with a regularized phase tracking method. Simulated as well as experimental results are presented.     

Residue vector, an approach to branch-cut placement in phase unwrapping: theoretical study

What we believe to be a novel technique of branch-cut placement in the phase unwrapping is proposed. This approach is based on what we named residue vector, which is generated by a residue in a wrapped phase map and has an orientation that points out toward the balancing residue of opposite polarity. The residue vector can be used to guide the manner in which branch cuts are placed in phase unwrapping. Also, residue vector can be used for the determination of the weighting values used in different existing phase unwrapping methods such as minimum cost flow and least squares. The theoretical foundations of the residue-vector method are presented, and a branch-cut method using its information is developed and implemented. A general comparison is made between the residue-vector map and other existing quality maps.     

Phase recovering without phase unwrapping in phase-shifting interferometry by cubic and average interpolation

A simple phase estimation employing cubic and average interpolations to solve the oversampling problem in smooth modulated phase images is described. In the context of a general phase-shifting process, without phase-unwrapping, the modulated phase images are employed to recover wavefront shapes with high fringe density. The problem of the phase reconstruction by line integration of its gradient requires a form appropriate to the calculation of partial derivatives, especially when the phase to recover has higher-order aberration values. This is achieved by oversampling the modulated phase images, and many interpolations can be implemented. Here an oversampling procedure based on the analysis of a quadratic cost functional for phase recovery, in a particular case, is proposed.     

The use of models of mixture for analysis of shock-wave experiments with incomplete phase transformation

It is suggested to use the known models of mixture for determining phase concentrations and temperature by the data of shock-wave experiments with incomplete phase transformation. Calculation results are given for the case of some known experiments involving graphite and C₆₀ fullerite, and the dependence of the accuracy of calculation on the error of measurement is studied. The possibility is discussed of using the obtained degree of phase transformation in shock wave for constructing hydrodynamic models of kinetics of phase transitions at high pressures and temperatures, which adequately fit the experiment.     

Robustness of reduced temporal phase unwrapping in the measurement of shape

The predictions of success rate and depth uncertainty for the negative exponential sequence used for temporal phase unwrapping of shape data are generalized to include the effect of a reduced sequence and speckle noise in single-channel and multichannel systems, respectively. To cope with the reduction of the sequence, a scaling factor is introduced. A thorough investigation is made of the performance of this algorithm, called the reduced temporal phase-unwrapping algorithm. Two different approaches are considered: a single-channel approach in which all the necessary images are acquired sequentially in time and a multichannel approach in which the three channels of a color CCD camera are used to carry the phase-stepped images for each fringe density in parallel. The performance of these two approaches are investigated by numerical simulations. The simulations are based on a physical model in which the speckle contrast, the fringe modulation, and random noise are considered the sources of phase errors. Expressions are found that relate the physical quantities to phase errors for the single-channel and the multichannel approaches. In these simulations the single-channel approach was found to be the most robust. Expressions that relate the measurement accuracy and the unwrapping reliability, respectively, with the reduction of the fringe sequence were also found. As expected, the measurement accuracy is not affected by a shorter fringe sequence, whereas a significant reduction in the unwrapping reliability is found as compared with the complete negative exponential sequence.     

Temporal phase unwrapping: application to surface profiling of discontinuous objects

The recently proposed technique of temporal phase unwrapping has been used to analyze the phase maps from a projected-fringe phase-shifting surface profilometer. A sequence of maps is acquired while the fringe pitch is changed; the phase at each pixel is then unwrapped over time independently of the other pixels in the image to provide an absolute measure of surface height. The main advantage is that objects containing height discontinuities are profiled as easily as smooth ones. This contrasts with the conventional spatial phase-unwrapping approach for which the phase jump across a height discontinuity is indeterminate to an integral multiple of 2pi. The error in height is shown to decrease inversely with the number of phase maps used.