Demodulation of a single-image interferogram using a Zernike-polynomial-based phase-fitting technique with a differential evolution algorithm

Fast and robust two-frame random phase-shifting interferometry without pre-filtering

To obtain higher phase accuracy with less computation time in phase-shifting interferometry, a random phase-shifting algorithm based on principal component analysis and least squares iteration (PCA&LSI) is proposed. The algorithm does not require pre-filtering, and only requires two-frame phase-shifted interferograms and less computation time to obtain a relatively accurate phase distribution. This method can still extract the phase with high precision when there are few fringes in the interferogram. Moreover, it eliminates the limitation that the PCA algorithm needs more than three frames of interferograms with uniform phase shift distribution to accurately extract the phase. Numerical simulations and experiments confirm that the method is suitable for complex situations with different fluctuations in background intensity and modulation amplitude. And it can still achieve accurate phase extraction compared with other methods under different noise conditions.

Fringe pattern demodulation using Zernike polynomials and a l1-norm regularized extended Kalman filter

A novel algorithm for closed fringe demodulation for an absolute phase estimation, to the best of our knowledge, is proposed. The two-dimensional phase is represented as a weighted linear combination of a certain number of Zernike polynomials (ZPs). Essentially, the problem of phase estimation is converted into the estimation of ZP coefficients. The task of ZP coefficient estimation is performed based on a state space model. Due to the nonlinear dependence of the fringe intensity measurement model on the ZP coefficients, the extended Kalman filter (EKF) is used for the state estimation. A pseudo-measurement model is considered based on the state vector sparsity constraint to improve the convergence performance of the EKF. Simulation and experimental results are provided to demonstrate the noise robustness and the practical applicability of the proposed method.

Variable mesh optimization applied to fringe pattern demodulation using a Bézier surface

We present a parametric method to carry out a demodulation process in complex fringe pattern images with either open or closed fringes; this method is based on the parallel demodulation algorithm and introduces a novel way, to the best of our knowledge, to approximate the phase map using the Bezier surface control points. For this study, a recently developed population meta-heuristic called Variable Mesh Optimization is introduced to implement the optimization process. The results of the proposed method improve the phase error and the run time scores in comparison with other global optimization algorithms, which address this type of problem.

Two-frame fringe pattern phase demodulation using Gram-Schmidt orthonormalization with least squares method

Gram-Schmidt (GS) orthogonal normalization is a fast and efficient two-frame fringe phase demodulation method. However, the precision of the GS method is limited due to the residual background terms and noise, as well as several approximation operations in the GS method. To obtain a phase map with higher accuracy, we propose an algorithm combining GS orthogonal normalization and least squares iterative (LSI) phase shift algorithm (GS&LSI). In our method, the phase was first obtained using GS method, and then a refinement operation using LSI was adopted to get the final wrapped phase map. Because of the LSI process, the demodulation result is greatly improved in many cases. Simulation and experimental result are presented to validate the potential of the proposed method.

Fast and accurate wavefront reconstruction in two-frame phase-shifting interferometry with unknown phase step

A fast and accurate wavefront reconstruction method for two-frame phase-shifting interferometry is proposed. The unknown phase step between the two interferograms is estimated directly by solving a quartic polynomial equation, and then the phase map is readily reconstructed after obtaining the phase step. The whole phase reconstruction process is nearly analytical and thus very fast and easy to realize. Good performance of the proposed method is demonstrated by reconstructing the phase maps from simulated and real fringes along with comparisons to several existing well-established algorithms.

Two-frame phase-shifting interferometry for testing optical surfaces

Standard phase-shifting interferometry (PSI) generally requires collecting at least three phase-shifted interferograms to extract the physical quantity being measured. Here, we propose the application of a simple two-frame PSI for the testing of a range of optical surfaces, including flats, spheres, and aspheres. The two-frame PSI extracts modulated phase from two randomly phase-shifted interferograms using a Gram-Schmidt algorithm, and can work in either null testing or non-null testing modes. Since only two interferograms are used for phase demodulation and the phase shift amount can be random, requirements on environmental conditions and phase shifter calibration are greatly relaxed. Experimental results of three different mirrors suggest that the two-frame PSI can achieve comparable measurement precision with conventional multi-frame PSI, but has faster data acquisition speed and less stringent hardware requirements. The proposed two-frame PSI expands the flexibility of PSI and holds great potential in many applications.

Closed fringe demodulation using phase decomposition by Fourier basis functions

We report a new technique for the demodulation of a closed fringe pattern by representing the phase as a weighted linear combination of a certain number of linearly independent Fourier basis functions in a given row/column at a time. A state space model is developed with the weights of the basis functions as the elements of the state vector. The iterative extended Kalman filter is effectively utilized for the robust estimation of the weights. A coarse estimate of the fringe density based on the fringe frequency map is used to determine the initial row/column to start with and subsequently the optimal number of basis functions. The performance of the proposed method is evaluated with several noisy fringe patterns. Experimental results are also reported to support the practical applicability of the proposed method.

Demodulation of two-shot fringe patterns with random phase shifts by use of orthogonal polynomials and global optimization

We propose a simple and robust phase demodulation algorithm for two-shot fringe patterns with random phase shifts. Based on a smoothness assumption, the phase to be recovered is decomposed into a linear combination of finite terms of orthogonal polynomials, and the expansion coefficients and the phase shift are exhaustively searched through global optimization. The technique is insensitive to noise or defects, and is capable of retrieving phase from low fringe-number (less than one) or low-frequency interferograms. It can also cope with interferograms with very small phase shifts. The retrieved phase is continuous and no further phase unwrapping process is required. The method is expected to be promising to process interferograms with regular fringes, which are common in optical shop testing. Computer simulation and experimental results are presented to demonstrate the performance of the algorithm.

Modal wavefront reconstruction in radial shearing interferometry with general aperture shapes

Wavefront reconstruction in radial shearing interferometry with general aperture shapes is challenging because the problem may be ill-conditioned. Here we propose a Gram-Schmidt orthogonalization method to cope with off-axis wavefront reconstruction with any aperture type. The proposed method constructs a set of orthogonal basis functions and computes the corresponding expansion coefficients, which are converted into another set of expansion coefficients to reproduce the original wavefront. The method can effectively alleviate the ill-conditioning of the problem, and is numerically stable compared with the classic least-squares method, especially for non-circular apertures and in the presence of noise. Computer simulation and experimental results are presented to demonstrate the performance of the algorithm.

Carrier peak isolation from single interferogram using spectrum shift technique

This paper presents a new method to obtain a wrapped phase distribution from a single interferogram with a spatial carrier modulation. The Fourier transform of the interferogram has three peaks: one is a dc peak around the origin in the Fourier domain, and the other two are carrier peaks that have information of phase modulation by an object placed in the interferometer. Since the wrapped phase can be evaluated by one of the two carrier peaks, the dc peak and the adjoint peak that is the other peak of two carrier peaks should be removed by filters. The proposed filtering process consists of two stages: dc peak filtering and adjoint peak filtering. A spectrum shift filter based on symmetrical characteristics of the spectrum is applied in both stages as a basic filter that can remove most of the undesired spectrum. An additional two filters are applied to remove the remaining spectrum. The new method can automatically isolate the carrier peak, even when the boundary of peaks is not very clear. Numerical evaluations of simulation data and experimental data demonstrate that the proposed method can successfully isolate the carrier peak.