In this work, we deduce explicit conditions to design phase shifting algorithms (PSAs) in order to suppress errors in phase measurement, the double-frequency ripple distortion and the spurious piston. These errors arise when non-uniform phase-shifting interferograms are processed with conventional PSAs which assume uniform phase shifts. By modeling the non-uniform phase shifts as a polynomial of the unperturbed phase shift value !0, we show that the conditions for eliminating those errors are associated to the m-th derivative of the PSA's frequency transfer function (FTF). Thus, we propose an approach to design robust algorithms based on the FTF formalism, and we present a ready-to-apply eight-frame PSA. Finally, our conclusions are supported by computer simulations.
In this work, we introduce the windowed generalized phase-shifting algorithms (WG-PSAs) using static and dynamic weighting functions/widows. These algorithms are derived from a weighted least square fitted to the monochromatic temporal fringe, thereby, the selection of the window plays an important role due to the fact that it shall reduce the influence of those intensities jeopardizing the phase estimation. In order to make the best selection, we propose to employ an adaptive/dynamic window which has the ability to detect the fringe patterns that jeopardize the phase retrieval. This window is computed iteratively by analyzing the error between the measured and fitted intensities. Furthermore, we provide the analysis of our scheme using the frequency transfer function (FTF) formalism for phaseshifting algorithms. Finally, we executed numerical experiments with synthetic data in which we compare the performance of the dynamic window versus several static ones from the state-of-the-art; although our scheme is more computationally expensive due to the iterative procedure, it works better than the traditional generalized PSAs with a window included or not.
We propose a fringe-projection profilometry technique for shape defects measurement, which can be employed for three-dimensional (3-D) quality inspection. The proposal consists of using a template surface to design phase-shifting algorithms with special-purpose phase response via the frequency transfer function. These algorithms can jointly and directly estimate the spatial phase deviations in a 3-D inspection. Phase deviations correspond to shape differences between the template shape and a testing one. The phase-unwrapping procedure is unnecessary when phase differences are small, as is usual in quality inspections. Experimental results show that our technique is so sensitive that the ripples in a fingerprint can be retrieved.
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-PSA10.
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.
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