Phase-shifting algorithms (PSAs) are usually derived for static or quasi-static conditions, where the temporal phase step is the only significant variation expected between successive frames. When these assumptions are valid, choosing the right algorithm often translates into faster acquisition times or robustness against systematic errors (such as detuning, random noise, and distorting harmonics). In practice, however, one may need to cope with dynamic conditions that require more complex phase-demodulation approaches. In this work, we present a PSA designed for robust quadrature filtering assuming temporal variations of the background and contrast functions. The frequency transfer function (FTF) formalism allows us to design its spectral response and to assess its robustness against systematic errors. This procedure is conceptually and computationally easy to generalize for many-step algorithms. Finally, a work-in-progress application for high-dynamic range (HDR) in fringe-projection profilometry is presented as proof of concept.
Here we describe a 2-projectors and 1-camera setup for profilometry of discontinuous solids by means of co-phased demodulation of projected fringes and red, green, and blue (RGB) multichannel operation. The dual projection configuration for this profilometer is proposed to solve efficiently specular regions and self-occluding shadows due to discontinuities, which are the main drawbacks for a 1-projector 1-camera configuration. This is because the regions where shadows and specular reflections are generated, and the fringe contrast drops to zero, are in general different for each projection direction; thus, the resulting fringe patterns will have complementary phase information. Multichannel RGB operation allows us to work simultaneously with both projectors and to record independently the complementary fringe patterns phase-modulated by the 3D profile of the object under study. In other words, color encoding/decoding reduces the acquisition time respect to one-at-a-time grayscale operation and, in principle, enables the study of dynamic phenomena. The co-phased demodulation method implemented in this work benefits from the complex (analytic) nature of the output signals estimated with most phase demodulation methods (such as the Fourier method, and temporal phaseshifting algorithms). This allowed us to straightforwardly generate a single phase-map well-defined for the entire area of interest. Finally we assessed our proposed profilometry setup by measuring a fractured spherical cap made of (uncoated) expanded polystyrene. The results were satisfactory but in the authors’ opinion this must be considered a preliminary report.
Pixelated phase-mask (PPM) interferometers have become an industry standard for instantaneous
phase-shifting interferometry. In commercially available PPM interferometers, an array with 2x2
unit-cells is used, which codify up-to 4 phase-steps within a single PPM interferogram. Recently we
have shown that such 2x2 unit-cell arrays allows a harmonic rejection as good as the 4-step leastsquares
phase-shifting algorithm (LS-PSA); this harmonics rejection is relatively-low and may not
be enough to correctly demodulate some severely intensity distorted fringe patterns. In previous
works we have proposed a new PPM with a 3x3 unit-cell to improve the harmonics rejection of the
2x2 array. With this new 3x3 unit-cell one is able to reject as many harmonics as with a 9-step LS-PSA<sup>10</sup>.
In this paper we are extending the analysis of MxN unit-cell synchronous demodulation of
PPM. The new results allow us to answer some important open questions about the method: for a
given configuration, which harmonics cannot be rejected and why? Why, prior to low-pass filtering,
we observe multiple copies of the interferogram’s spectrum and what does this imply? We believe
these preliminary results are important contributions towards a formulation of a general theory MxN
unit-cell pixelated carrier interferometry.