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If the best phase measurements are to be achieved, phase-stepping methods need algorithms that are 112 insensitive to the harmonic content of the sampled waveform and 122 insensitive to phase-shift miscalibration. A method is proposed that permits the derivation of algorithms that satisfy both requirements, up to any arbitrary order. It is based on a one-to-one correspondence between an algorithm and a polynomial. Simple rules are given to permit the generation of the polynomial that corresponds to the algorithm having the prescribed properties. These rules deal with the location and multiplicity of the roots of the polynomial. As a consequence, it can be calculated from the expansion of the products of monomials involving the roots. Novel algorithms are proposed, e.g., a six-sample one to eliminate the effects of the second harmonic and a 10-sample one to eliminate the effects of harmonics up to the fourth order. Finally, the general form of a self-calibrating algorithm that is insensitive to harmonics up to an arbitrary order is given.

Abstract. Fringe analysis is the process of extracting quantitative measurement data from fringe -or line -patterns. It usually consists of phase detection and phase unwrapping. Phase detection is the calculation of the fringes phase from the recorded intensity patterns, and this issue represents the major part of the material in this review. Different techniques for this phase calculation are presented, with special emphasis on the characteristic polynomial method, allows us permits to easily design customized algorithms coping with many error sources. For reference, a table presenting the properties of almost all algorithms which have been in recent years is provided in the Appendix. A generic method allowing us to quantitatively evaluate the phase errors and the effect of noise is presented here for the first time. Finally, some elements regarding the complex problem of phase unwrapping are given.

A new (N + 1)-bucket algorithm is proposed for phase-stepping systems. It eliminates most of the errors caused by the phase-shifter miscalibration.

The characteristic polynomials associated with the algorithms used in digital phase detection are used to investigate the effects of additive noise on phase measurements. First, it is shown that a loss factor eta can be associated with any algorithm. This parameter describes the influence of the algorithm on the global signal-to-noise ratio (SNR). Second, the variance of the phase error is shown to depend mainly on the global SNR. The amplitude of a modulation of this variance at twice the signal frequency depends on a single parameter beta. The material presented here extends previously published results, and as many as 19 algorithms are analyzed.