A Robust and Simple Measure for Quality-Guided 2D Phase Unwrapping Algorithms.

Quality-based 2D phase unwrapping algorithms provide one of the best tradeoffs between speed and quality of results. Their robustness depends on a quality map, which is used to build a path that visits the most reliable pixels first. Unwrapping then proceeds along this path, delaying unwrapping of noisy and inconsistent areas until the end, so that the unwrapping errors remain local. We propose a novel quality measure that is consistent, technically sound, effective, fast to compute, and immune to the presence of a carrier signal. The new measure combines the benefits of both the quality-guided and the residue-based phase unwrapping approaches. The quality map is justified from the two different theoretical points of view. Exhaustive tests on a variety of artificially generated and real 2D wrapped phase signals illustrate its potential usefulness in the field of fringe projection profilometry.

Aiding phase unwrapping by increasing the number of residues in two-dimensional wrapped-phase distributions.

In phase unwrapping residues are points of locally inconsistent phase that occur within a wrapped-phase map, which are usually regarded as being problematic for phase-unwrapping algorithms. Real phase maps typically contain a number of residues that are approximately proportional to the subsequent difficulty in unwrapping the phase distribution. This paper suggests the radical use of the discrete Fourier transform to actually increase the number of residues in 2D phase-wrapped images that contain discontinuities. Many of the additional residues that are artificially generated by this method are located on these discontinuities. For example, in fringe projection systems, such phase discontinuities may come from physical discontinuity between different parts of the object, or by shadows cast by the object. The suggested technique can improve the performance of path independent phase-unwrapping algorithms because these extra residues simplify the process of setting the branch cuts in the wrapped image based on the distance to the nearest residue. The generated residues can also be used to construct more reliable quality maps and masks. The paper includes an initial analysis upon simulated phase maps and goes on to verify the results on a real experimental wrapped-phase distribution.

Hybrid robust and fast algorithm for three-dimensional phase unwrapping.

We present a hybrid three-dimensional (3D) unwrapping algorithm that combines the strengths of two other fast and robust existing techniques. In particular, a branch-cut surface algorithm and a path-following method have been integrated in a symbiotic way, still keeping execution times within a range that permits their use in real-time applications that need a relatively fast solution to the problem. First, branch-cut surfaces are calculated, disregarding partial residue loops that end at the boundary of the 3D phase volume. These partial loops are then used to define a quality for each image voxel. Finally, unwrapping proceeds along a path determined by a minimum spanning tree (MST). The MST is built according to the quality of the voxels and avoids crossing the branch-cut surfaces determined at the first step. The resulting technique shows a higher robustness than any of the two methods used in isolation. On the one hand, the 3D MST algorithm benefits from the branch-cut surfaces, which endows it with a higher robustness to noise and open-ended wraps. On the other hand, incorrectly placed surfaces due to open loops at the boundaries in the branch-cut surface approach disappear.

Fast and robust three-dimensional best path phase unwrapping algorithm.

What we believe to be a novel three-dimensional (3D) phase unwrapping algorithm is proposed to unwrap 3D wrapped-phase volumes. It depends on a quality map to unwrap the most reliable voxels first and the least reliable voxels last. The technique follows a discrete unwrapping path to perform the unwrapping process. The performance of this technique was tested on both simulated and real wrapped-phase maps. And it is found to be robust and fast compared with other 3D phase unwrapping algorithms.

Spatial fringe pattern analysis using the two-dimensional continuous wavelet transform employing a cost function.

We present a novel ridge extraction algorithm for use with the two-dimensional continuous wavelet transform to extract the phase information from a fringe pattern. A cost function is employed for the detection of the ridge. The results of the proposed algorithm on simulated and real fringe patterns are illustrated. Moreover, the proposed algorithm outperforms the maximum ridge extraction algorithm and it is found to be robust and reliable.

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.

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.

Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform.

A novel technique that uses a fan two-dimensional (2D) continuous wavelet transform (CWT) to phase demodulate fringe patterns is proposed. The fan 2D CWT algorithm is tested by using computer generated and real fringe patterns. The result of this investigation reveals that the 2D CWT technique is capable of successfully demodulating fringe patterns. The proposed algorithm demodulates fringe patterns without the requirement of removing their background illumination prior to the demodulation process. Also, the algorithm is exceptionally robust against speckle noise. The performance of the 2D CWT technique in fringe pattern demodulation is compared with that of the 1D CWT algorithms. This comparison indicates that the 2D CWT outperforms its 1D counterpart for this application.

Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path.

We describe what is to our knowledge a novel technique for phase unwrapping. Several algorithms based on unwrapping the most-reliable pixels first have been proposed. These were restricted to continuous paths and were subject to difficulties in defining a starting pixel. The technique described here uses a different type of reliability function and does not follow a continuous path to perform the unwrapping operation. The technique is explained in detail and illustrated with a number of examples.

Robust, fast, and effective two-dimensional automatic phase unwrapping algorithm based on image decomposition.

We describe what is to our knowledge a novel approach to phase unwrapping. Using the principle of unwrapping following areas with similar phase values (homogenous areas), the algorithm reacts satisfactorily to random noise and breaks in the wrap distributions. Execution times for a 512 x 512 pixel phase distribution are in the order of a half second on a desktop computer. The precise value depends upon the particular image under analysis. Two inherent parameters allow tuning of the algorithm to images of different quality and nature.

Fringe pattern demodulation with a two-frame digital phase-locked loop algorithm.

A novel technique called a two-frame digital phase-locked loop for fringe pattern demodulation is presented. In this scheme, two fringe patterns with different spatial carrier frequencies are grabbed for an object. A digital phase-locked loop algorithm tracks and demodulates the phase difference between both fringe patterns by employing the wrapped phase components of one of the fringe patterns as a reference to demodulate the second fringe pattern. The desired phase information can be extracted from the demodulated phase difference. We tested the algorithm experimentally using real fringe patterns. The technique is shown to be suitable for noncontact measurement of objects with rapid surface variations, and it outperforms the Fourier fringe analysis technique in this aspect. Phase maps produced withthis algorithm are noisy in comparison with phase maps generated with the Fourier fringe analysis technique.

Fringe pattern demodulation with a two-dimensional digital phase-locked loop algorithm.

A novel technique called a two-dimensional digital phase-locked loop (DPLL) for fringe pattern demodulation is presented. This algorithm is more suitable for demodulation of fringe patterns with varying phase in two directions than the existing DPLL techniques that assume that the phase of the fringe patterns varies only in one direction. The two-dimensional DPLL technique assumes that the phase of a fringe pattern is continuous in both directions and takes advantage of the phase continuity; consequently, the algorithm has better noise performance than the existing DPLL schemes. The two-dimensional DPLL algorithm is also suitable for demodulation of fringe patterns with low sampling rates, and it outperforms the Fourier fringe analysis technique in this aspect.