J. C. Estrada

Two-step demodulation based on the Gram-Schmidt orthonormalization method

This Letter presents an efficient, fast, and straightforward two-step demodulating method based on a Gram-Schmidt (GS) orthonormalization approach. The phase-shift value has not to be known and can take any value inside the range (0,2π), excluding the singular case, where it corresponds to π. The proposed method is based on determining an orthonormalized interferogram basis from the two supplied interferograms using the GS method. We have applied the proposed method to simulated and experimental interferograms, obtaining satisfactory results. A complete MATLAB software package is provided at http://goo.gl/IZKF3.

The general theory of phase shifting algorithms.

We have been reporting several new techniques of analysis and synthesis applied to Phase Shifting Interferometry (PSI). These works are based upon the Frequency Transfer Function (FTF) and how this new tool of analysis and synthesis in PSI may be applied to obtain very general results, among them; rotational invariant spectrum; complex PSI algorithms synthesis based on simpler first and second order quadrature filters; more accurate formulae for estimating the detuning error; output-power phase noise estimation. We have made our cases exposing these aspects of PSI separately. Now in the light of a better understanding provided by our past works we present and expand in a more coherent and holistic way the general theory of PSI algorithms. We are also providing herein new material not reported before. These new results are on; a well defined way to combine PSI algorithms and recursive linear PSI algorithms to obtain resonant quadrature filters.

Two-step interferometry by a regularized optical flow algorithm

A two-step phase-shifting method, that can demodulate open- and closed-fringed patterns without local sign ambiguity is presented. The proposed method only requires a constant phase-shift between the two interferograms. This phase-shift does not need to be known and can take any value inside the range (0, 2π), excluding the singular case where it corresponds to π. The proposed method is based on determining first the fringe direction map by a regularized optical flow algorithm. After that, we apply the spiral phase transform (SPT) to one of the fringe patterns and we determine its quadrature signal using the previously determined direction. The proposed technique has been applied to simulated and experimental interferograms obtaining satisfactory results. A complete MATLAB software package is provided in [http://goo.gl/Snnz7].

Fast two-dimensional simultaneous phase unwrapping and low-pass filtering.

Here, we present a fast algorithm for two-dimensional (2D) phase unwrapping which behaves as a recursive linear filter. This linear behavior allows us to easily find its frequency response and stability conditions. Previously, we published a robust to noise recursive 2D phase unwrapping system with smoothing capabilities. But our previous approach was rather heuristic in the sense that not general 2D theory was given. Here an improved and better understood version of our previous 2D recursive phase unwrapper is presented. In addition, a full characterization of it is shown in terms of its frequency response and stability. The objective here is to extend our previous unwrapping algorithm and give a more solid theoretical foundation to it.

Two-step self-tuning phase-shifting interferometry

A two-step self-tuning phase-shifting method is presented. The phase-step between the two interferograms is not known when the experiment is performed. Our demodulating method finds, in a robust way, this unknown phase-step. Once the phase-step is estimated we proceed to phase demodulate the interferograms. Moreover our method only requires the fringe patterns to have a constant unknown phase-shift between them. As a consequence, this technique can be used to demodulate open and closed-fringed patterns without phase-sign ambiguity. The method may be regarded as a self-tuning quadrature filter, which determines the phase-shift between the two fringe patterns and finally estimates the demodulated phase map. The proposed technique has been tested with simulated and real interferograms obtaining satisfactory results.

Noise robust linear dynamic system for phase unwrapping and smoothing

Phase unwrapping techniques remove the modulus ambiguities of wrapped phase maps. The present work shows a first-order feedback system for phase unwrapping and smoothing. This system is a fast sequential unwrapping system which also allows filtering some noise because in deed it is an Infinite Impulse Response (IIR) low-pass filter. In other words, our system is capable of low-pass filtering the wrapped phase as the unwrapping process proceeds. We demonstrate the temporal stability of this unwrapping feedback system, as well as its low-pass filtering capabilities. Our system even outperforms the most common and used unwrapping methods that we tested, such as the Flynns method, the Goldstains method, and the Ghiglia least-squares method (weighted or unweighted). The comparisons with these methods shows that our system filters-out some noise while preserving the dynamic range of the phase-data. Its application areas may cover: optical metrology, synthetic aperture radar systems, magnetic resonance, and those imaging systems where information is obtained as a demodulated wrapped phase map.

Noise in phase shifting interferometry

We present a theoretical analysis to estimate the amount of phase noise due to noisy interferograms in Phase Shifting Interferometry (PSI). We also analyze the fact that linear filtering transforms corrupting multiplicative noise in Electronic Speckle Pattern Interferometry (ESPI) into fringes corrupted by additive gaussian noise. This fact allow us to obtain a formula to estimate the standard deviation of the noisy demodulated phase as a function of the spectral response of the preprocessing spatial filtering combined with the PSI algorithm used. This phase noise power formula is the main result of this contribution.

Phasorial analysis of detuning error in temporal phase shifting algorithms

Phase error analysis in Temporal Phase Shifting (TPS) algorithms due to frequency detuning has been to date only performed numerically. In this paper, we show an exact analytical expression to obtain this phase error due to detuning using the spectral TPS response. The new proposed method is based on the phasorial representation of the output of the TPS quadrature filter. Doing this, the detuning problem is reduced to a ratio of two symmetrical spectral responses of the quadrature filter at the detuned frequency. Finally, some popular cases of TPS algorithms are analyzed to show the usefulness of the proposed method.

Easy and straightforward construction of wideband phase-shifting algorithms for interferometry

We show a practical way for building wideband phase-shifting algorithms for interferometry. The idea presented combines first- and second-order quadrature filters to obtain wideband phase-shifting algorithms. These first- and second-order quadrature filters are analogous to the first- and second-order filters commonly used in communications engineering, named building blocks. We present a systematic way to develop phase-shifting algorithms with large detuning robustness or large bandwidth. In general, the approach presented here gives a powerful frequency analysis and design tool for phase-shifting algorithms robust to detuning for interferometry.

A self-tuning phase-shifting algorithm for interferometry.

In Phase Stepping Interferometry (PSI) an interferogram sequence having a known, and constant phase shift between the interferograms is required. Here we take the case where this constant phase shift is unknown and the only assumption is that the interferograms do have a temporal carrier. To recover the modulating phase from the interferograms, we propose a self-tuning phase-shifting algorithm. Our algorithm estimates the temporal frequency first, and then this knowledge is used to estimate the interesting modulating phase. There are several well known iterative schemes published before, but our approach has the unique advantage of being very fast. Our new temporal carrier, and phase estimator is capable of obtaining a very good approximation of their temporal carrier in a single iteration. Numerical experiments are given to show the performance of this simple yet powerful self-tuning phase shifting algorithm.