Residue calibrated least-squares unwrapping algorithm for noisy and steep phase maps

Branch-cut algorithm with second-order residues in phase unwrapping

This Letter presents a novel, to the best of our knowledge, branch-cut algorithm for unwrapping phase maps acquired through fringe projection systems. The algorithm exposes second-order residues with vortex phase structures, computed from the original wrapped phase distribution using a second-order derivative that considers the wrap cycle. Remarkably, it reveals a consistent vorticity direction associated with these vortex structures, significantly facilitating branch-cut connections between residues, regardless of whether they carry the same or opposite charges. This innovative approach challenges a longstanding misconception in conventional branch-cut methods, which traditionally connected matched residues only when they had opposite charges. As a result, the algorithm provides a highly accurate methodology for explaining how phase behavior relates to surface characteristics. Its effectiveness is demonstrated through both computer simulations and practical experiments.

Simulated wrapped phase optimizes phase retrieval in phase-shifting interferometry

Phase retrieval is crucial in phase-shifting interferometry and other phase measurement techniques. However, in noisy wrapped phase maps with high steepness, discontinuities arise and cause phase unwrapping errors. To solve this problem, this Letter presents a phase retrieval method based on a simulated wrapped phase. By establishing the correspondence between the simulated and measured interferograms, the difference in wrapped phases between them can be obtained. The difference in wrapped phase map, which has sparse and wide interference fringes, has a higher reliability of phase unwrapping. The proposed method not only possesses high phase retrieval accuracy but it also simplifies the processing of interferograms. Furthermore, the layout of all interferometric systems, the parameters of optical components, and the model of the measured object are known, so the proposed method can be used as a reference for phase retrieval.

Discontinuous phase unwrapping based on the minimization of Zernike gradient polynomial residual

In many certain optical metrology cases, the pupil is usually divided into multiple connected domains by secondary mirror spiders, thus producing segment piston errors and leaving a false phase unwrapping result. In this paper, a method based on minimization of Zernike gradient polynomial residual (MZGR) is proposed to estimate segment piston errors and correct erroneous phase unwrapping results. Simulations and experiments demonstrated that this method can obtain the segment piston errors precisely under complex aberration forms and varied obscurations, indicating reliable practicality. Comparison to the 4D commercial solution, the RMS (root-mean-square) of the residual decreased from 0.154 λ to 0.020 λ.

Phase unwrapping algorithm for a segmented phase based on iterative pseudo-phase inpainting

Segmented phase unwrapping is an intractable problem in the phase-shifting technique. To solve the problem, this Letter presents an iterative pseudo-phase inpainting algorithm (IPPI). By means of image inpainting, the IPPI can be used to realize the pseudo-phases connecting each other among these phase islands. The error points in the pseudo-phases can be reduced by iterations of phase inpainting with the assistance of the reference pseudo-phase obtained by introducing the numerical carrier frequency and using the 2D Fourier transform. Compared with other methods, the proposed algorithm does not have to do any processing on the effective area of the wrapped phase, which ensures the authenticity of the result. The simulated and experimental verifications show that the proposed method not only possesses high precision, but also can be applied to a segmented phase with severe noise.

Robust particle-Kalman filtering phase unwrapping algorithm for wrapped fringe patterns

This paper presents a robust phase unwrapping algorithm based on a particle-Kalman filter for wrapped fringe patterns by combining a particle filter and an extended Kalman filter, which formulates the phase unwrapping problem of wrapped fringe patterns as an optimal state estimation problem under the frame of the particle-Kalman filter. First, a state space equation for state variables is extended to the second order of Taylor series, and a local phase gradient estimator based on a modified matrix pencil model is used to obtain the first-order and second-order phase gradient information required by the extended state space equation, which is conducive to enhancing the phase unwrapping accuracy of the proposed procedure. Second, the initial estimate of unwrapped phase is obtained through applying an efficient phase unwrapping program based on a particle filter to unwrap noisy wrapped pixels. Finally, the initial estimate of unwrapped phase obtained by the particle filter is taken as the predicted estimate of state variables and further processed by the extended Kalman filter to obtain the final estimate of unwrapped phase. In addition, an efficient quality-guided strategy that has been demonstrated well is used to guarantee that the particle-Kalman filter efficiently and accurately unwraps wrapped pixels along a suitable path. Results obtained with synthetic data and experimental data demonstrate the effectiveness of the proposed method and show that this new approach can obtain more acceptable solutions from noisy wrapped fringe patterns, with respect to some of the most commonly used methods.