Practical eight-frame algorithms for fringe projection profilometry.

In this paper we present several eight-frame algorithms for their use in phase shifting profilometry and their application for the analysis of semi-fossilized materials. All algorithms are obtained from a set of two-frame algorithms and designed to compensate common errors such as phase shift detuning and bias errors.

Algorithm for phase extraction from a set of interferograms with arbitrary phase shifts.

The generalized analytical quadrature filter from a set of interferograms with arbitrary phase shifts is obtained. Both symmetrical and non symmetrical algorithms for any order are reported. The analytic expression is obtained through the convolution of a set of two-frame algorithms and expressed in terms of the combinatorial theory. Finally, the solution is applied to obtain several generalized tunable quadrature filters.

Two-frame algorithm to design quadrature filters in phase shifting interferometry.

The main purpose of this paper is to present a method to design tunable quadrature filters in phase shifting interferometry. From a general tunable two-frame algorithm introduced, a set of individual filters corresponding to each quadrature conditions of the filter is obtained. Then, through a convolution algorithm of this set of filters the desired symmetric quadrature filter is recovered. Finally, the method is applied to obtain several tunable filters, like four and five-frame algorithms.

Calculus of exact detuning phase shift error in temporal phase shifting algorithms.

The detuning phase shift error is a common systematic error observed in temporal phase shifting (TPS) algorithms. This error, generally due to miscalibration of the phase shifter, is solved by using a quadrature filter insensitive to this detuning error. To compare algorithms, this error is frequently analyzed numerically. However, in this work we present an exact and analytical expression to calculate such error which is applicable to any kind of filters with real or complex frequency response. Finally, a table with the detuning error for several algorithms is reported.

A method to design tunable quadrature filters in phase shifting interferometry.

The main purpose of this paper is to present a method to design tunable quadrature filters in phase shifting interferometry. The algorithm is obtained from a generalized Fourier transform of a symmetrical quadrature filter. This formalism allows us to represent the detuning phase shift error and bias modulation as geometrical conditions. Therefore, the design of the filter becomes a set of solvable linear equations. Hence, to prove our method, several general tunable filters, like three and four frame algorithms, are obtained. Finally, from our results we reproduce particular symmetrical four frame algorithms reported in literature.

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.

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.