arXiv subjects

Moises Padilla

Publications and source records attributed to Moises Padilla.

12 recordsLinked to original sources

arXiv2023

Wavelength-stepping algorithm for testing thickness, front and back surfaces of optical plates with high signal-to-noise ratio

We propose a least-squares phase-stepping algorithm (LS-PSA) consisting of only 14 steps for high-quality optical plate testing. Optical plate testing produces an infinite number of simultaneous fringe patterns due to multiple reflections. However, because of the small reflection of common optical materials, only a few simultaneous fringes have amplitudes above the measuring noise. From these fringes, only the variations of the plate's surfaces and thickness are of interest. To measure these plates, one must use wavelength-stepping, which corresponds to phase-stepping in standard digital interferometry. The designed PSA must phase-demodulate a single fringe sequence and filter out the remaining temporal fringes. In the available literature, researchers have adapted PSAs to the dimensions of particular plates. As a consequence, there are as many PSAs published as different testing plate conditions. Moreover, these PSAs are designed with too many phase-steps to provide detuning robustness well above the required level. Instead, we mathematically prove that a single 14-step LS-PSA can adapt to several testing setups. As is well known, this 14-step LS-PSA has a maximum signal-to-noise ratio (SNR) and the highest harmonics rejection among any other 14-step PSA. Due to optical dispersion and experimental length measuring errors, the fringes may have a slight phase detuning. Using propagation error theory, we demonstrate that measuring distances with around 1% uncertainty produces a small and acceptable detuning error for the proposed 14-step LS-PSA.

physics.optics↗

arXiv2019

High-precision temporal interferometry from nonlinearly-spaced phase-shifted interferograms through spatial-filtering

We present a high-precision temporal-spatial phase-demodulation algorithm for phase-shifting interferometry (PSI) affected by random/systematic phase-stepping errors. Laser interferometers in standard optical-shops suffer from several error sources including random phase-shift deviations. Even calibrated phase-shifters do not achieve floating-point linear accuracy, as routinely obtained in multimedia video-projectors for fringe-projection profilometry. In standard optical-shops, calibrated phase-shifting interferometers suffer from nonlinearities due to vibrations, turbulence, and environmental fluctuations (temperature, pressure, humidity, air composition) still under controlled laboratory conditions. These random phase-step errors (even if they are small), increases the uncertainty of the phase measurement. This is particularly significant if the wavefront tolerance is tightened to high precision optics. We show that these phase-step errors precludes high-precision wavefront measurements because its uncertainty increases to around lambda/10. We develop an analytical expression based on optical-wavefront formalism showing that these phase-step nonlinearities appear as a spurious conjugate signal degrading the desired wavefront. Removing this spurious conjugate constitutes the central objective of the proposed nonlinear phase-shifting algorithm (nPSA). Using this nPSI algorithm we demodulate experimental interferograms subject to small vibrations and phase-shifter nonlinearities, obtaining a high-precision spurious-free, demodulated wavefront. We show that our artifact-free, temporal-spatial quadrature filtering, accomplishes an equivalent wavefront precision as the one obtained from floating-point linear phase-shifting interferometry.

eess.SP↗

arXiv2019

Fourier spectra for nonuniform phase-shifting algorithms based on principal component analysis

We develop an error-free, nonuniform phase-stepping algorithm (nPSA) based on principal component analysis (PCA). PCA-based algorithms typically give phase-demodulation errors when applied to nonuniform phase-shifted interferograms. We present a straightforward way to correct those PCA phase-demodulation errors. We give mathematical formulas to fully analyze PCA-based nPSA (PCA-nPSA). These formulas give a) the PCA-nPSA frequency transfer function (FTF), b) its corrected Lissajous figure, c) the corrected PCA-nPSA formula, d) its harmonic robustness, and e) its signal-to-noise-ratio (SNR). We show that the PCA-nPSA can be seen as a linear quadrature filter, and as consequence, one can find its FTF. Using the FTF, we show why plain PCA often fails to demodulate nonuniform phase-shifted fringes. Previous works on PCA-nPSA (without FTF), give specific numerical/experimental fringe data to "visually demonstrate" that their new nPSA works better than competitors. This often leads to biased/favorable fringe pattern selections which "visually demonstrate" the superior performance of their new nPSA. This biasing is herein totally avoided because we provide figures-of-merit formulas based on linear systems and stochastic process theories. However, and for illustrative purposes only, we provide specific fringe data phase-demodulation, including comprehensive analysis and comparisons.

eess.SP↗

arXiv2018

Design of non-uniformly spaced phase-stepped algorithms using their frequency transfer function

Here we show how to design phase-shifting algorithms (PSAs) for nonuniform phase-shifted fringe patterns using their frequency transfer function (FTF). Assuming that the nonuniform/nonlinear (NL) phase-steps are known, we introduce the desired zeroes in the FTF to obtain the specific NL-PSA formula. The advantage of designing NL-PSAs based on their FTF is that one can reject many distorting harmonics of the fringes. We can also estimate the signal-to-noise ratio (SNR) for interferograms corrupted by additive white Gaussian noise (AWGN). Finally, for non-distorted noiseless fringes, the proposed NL-PSA retrieves the modulating phase error-free, just as standard/linear PSAs do.

eess.SP↗

arXiv2018

Phase-stepping algorithms for synchronous demodulation of nonlinear phase-shifted fringes

Standard phase-stepping algorithms (PSAs) estimate the measuring phase of linear carrier temporal-fringes with respect to a linear-reference. Linear-carrier fringes are normally obtained using feedback, closed-loop, optical phase-shifting devices. On the other hand, open-loop, phase-shifting devices, usually give fringe patterns with nonlinear phase-shifts. The Fourier spectrum of linear-carrier fringes is composed by Dirac deltas only. In contrast, nonlinear phase-shifted fringes are wideband, spread-spectrum signals. It is well known that using linear-phase reference PSA to demodulate nonlinear phase-shifted fringes, one obtains an spurious-piston. The problem with this spurious-piston, is that it may wrongly be taken as a real optical thickness. Here we mathematically find the origin of this spurious-piston and design nonlinear phase-stepping PSAs to cope with open-loop, nonlinear phase-shifted interferometric fringes. We give a general theory to tailor nonlinear phase-stepping PSAs to demodulate nonlinear phase-shifted wideband fringes.

eess.SP↗

arXiv2017

Shannon information storage in noisy phase-modulated fringes and fringe-data compression by phase-shifting algorithms

Optical phase-modulated fringe-patterns are usually digitized with XxY pixels and 8 bits/pixel (or higher) gray-levels. The digitized 8 bits/pixel are raw-data bits, not Shannon information bits. Here we show that noisy fringe-patterns store much less Shannon information than the capacity of the digitizing camera. This means that high signal-to-noise ratio (S/N) cameras may waste to noise most bits/pixel. For example one would not use smartphone cameras for high quality phase-metrology, because of their lower (S/N) images. However smartphones digitize high-resolution (12 megapixel) images, and as we show here, the information storage of an image depends on its bandwidth and its (S/N). The standard formalism for measuring information are the Shannon-entropy H, and the Shannon capacity theorem (SCT). According to SCT, low (S/N) images may be compensated with a larger fringe-bandwidth to obtain high-information phase measurements. So broad bandwidth fringes may give high quality phase, in spite of digitizing low (S/N) fringe images. Most real-life images are redundant, they have smooth zones where the pixel-value do not change much, and data compression algorithms are paramount for image transmission/storage. Shannon's capacity theorem is used to gauge competing image compression algorithms. Here we show that phase-modulated phase-shifted fringes are highly correlated, and as a consequence, phase-shifting algorithms (PSAs) may be used as fringe-data compressors. Therefore a PSA may compress a large number of phase-shifted fringes into a single complex-valued image. This is important in spaceborne optical/RADAR phase-telemetry where downlink is severely limited by huge distance and low-power downlink. That is, instead of transmitting M phase-shifted fringes, one only transmit the phase-demodulated signal as compressed sensing data.

eess.SP↗

arXiv2017

Extended depth-range profilometry using the phase-difference and phase-sum of two close-sensitivity projected fringes

We propose a high signal-to-noise extended depth-range three-dimensional (3D) profilometer projecting two linear-fringes with close phase-sensitivity. We use temporal phase-shifting algorithms (PSAs) to phase-demodulate the two close sensitivity phases. Then we calculate their phase-difference and their phase-sum. If the sensitivity between the two phases is close enough, their phase-difference is not-wrapped. The non-wrapped phase-difference as extended-range profilometry is well known and has been widely used. However as this paper shows, the closeness between the two demodulated phases makes their difference quite noisy. On the other hand, as we show, their phase-sum has a much higher phase-sensitivity and signal-to-noise ratio but it is highly wrapped. Spatial unwrapping of the phase-sum is precluded for separate or highly discontinuous objects. However it is possible to unwrap the phase-sum by using the phase-difference as first approximation and our previously published 2-step temporal phase-unwrapping. Therefore the proposed profilometry technique allows unwrapping the higher sensitivity phase-sum using the noisier phase-difference as stepping stone. Due to the non-linear nature of the extended 2-steps temporal-unwrapper, the harmonics and noise errors in the phase-difference do not propagate towards the unwrapping phase-sum. To the best of our knowledge this is the highest signal-to-noise ratio, extended depth-range, 3D digital profilometry technique reported to this date.

physics.optics↗

arXiv2017

Profilometry with digital fringe-projection at the spatial and temporal Nyquist frequencies

A phase-demodulation method for digital fringe-projection profilometry using the spatial and temporal Nyquist frequencies is presented. It allows us to digitize tridimensional surfaces using the highest spatial frequency (π radians per pixel) and consequently with the highest sensitivity for a given digital fringe projector. Working with the highest temporal frequency (π radians per temporal sample), the proposed method rejects the DC component and all even-order distorting harmonics using a bare-minimum 2-step phase shift. The proposed method is suitable for digitization of piece-wise continuous surfaces because it does not require spatial low-pass filtering. Gamma calibration is unnecessary because the projected fringes are binary, and the harmonics produced by the binary profile can be easily attenuated with a slight defocusing on the digital projector. Viability of the proposed method is supported by experimental results showing complete agreement with the predicted behavior.

physics.ins-det↗

arXiv2017

Temporal-multiplexing interferometry applied to co-phased profilometry

Fringe-projection profilometry with 1 camera and 1 fringe-projector is a well-known and widely used technique in optical metrology. Spatial-frequency multiplexing interferometry with several spatial-carriers having non-overlapping spatial-spectra is also well known and productive in optical metrology. In this paper we propose temporal-multiplexing phase-shifting interferometry applied to profilometry. That is, instead of having fringe-patterns with well separated spatial-spectra, we propose instead to separate the fringe information in the temporal-spectra. In other words, we may have overlapping spatial-spectra, but separated in the temporal-spectra by frequency multiplexing. Using 1-camera and several fringe-projectors one minimizes the object shadows and specular reflections from the digitizing solid. Temporal multiplexing profilometry allows us to illuminate the object from several projectors turned-on simultaneously. In previous phase-shifting co-phased profilometry, the projectors were turned-on and off sequentially. As seen in this work temporal-multiplexing allow us to demodulate the several fringe-patterns without crosstalk from other simultaneously projected fringes. This is entirely analogous to having several television stations broadcasting simultaneously, each TV-transmitter having its own broadcasting frequency. A given TV-receiver tunes into a single TV-station and filter-out all other broadcasters. Following this analogy, each fringe-projector must have its own temporal broadcasting frequency to remain well separated from all other projectors in the time-spectra domain. In addition to the general theory presented, we assess its feasibility with experimental results.

physics.optics↗

arXiv2016

Fourier analysis of RGB fringe-projection profilometry and robust phase-demodulation methods against crosstalk distortion

In this paper we apply the frequency transfer function (FTF) formalism to analyze the red, green and blue (RGB) phase-shifting fringe-projection profilometry technique. The phase-shifted fringe patterns in RGB fringe projection are typically corrupted by crosstalk because the sensitivity curves of most projection-recording systems overlap. Crosstalk distortion needs to be compensated in order to obtain high quality measurements. We study phase-demodulation methods for null/mild, moderate, and severe levels of RGB crosstalk. For null/mild crosstalk, we can estimate the searched phase-map using Bruning's 3-step phase-shifting algorithm (PSA). For moderate crosstalk, the RGB recorded data is usually preprocessed before feeding it into Bruning's PSA; alternatively, in this paper we propose a computationally more efficient approach, which combines crosstalk compensation and phase-demodulation into a single process. For severe RGB crosstalk, we expect non-sinusoidal fringes' profiles (distorting harmonics) and significant uncertainties on the crosstalk calibration (which produces pseudo-detuning error). Analyzing these distorting phenomena, we conclude that squeezing interferometry is the most robust demodulation method for RGB fringe-projection techniques. We support our conclusions with numerical simulations and experimental results.

physics.optics↗

arXiv2016

Synthesis of multi-wavelength temporal phase-shifting algorithms optimized for high signal-to-noise ratio and high detuning robustness using the frequency transfer function

Synthesis of single-wavelength temporal phase-shifting algorithms (PSA) for interferometry is well-known and firmly based on the frequency transfer function (FTF) paradigm. Here we extend the single-wavelength FTF-theory to dual and multi-wavelength PSA-synthesis when several simultaneous laser-colors are present. The FTF-based synthesis for dual-wavelength PSA (DW-PSA) is optimized for high signal-to-noise ratio and minimum number of temporal phase-shifted interferograms. The DW-PSA synthesis herein presented may be used for interferometric contouring of discontinuous industrial objects. Also DW-PSA may be useful for DW shop-testing of deep free-form aspheres. As shown here, using the FTF-based synthesis one may easily find explicit DW-PSA formulae optimized for high signal-to-noise and high detuning robustness. To this date, no general synthesis and analysis for temporal DW-PSAs has been given; only had-hoc DW-PSAs formulas have been reported. Consequently, no explicit formulae for their spectra, their signal-to-noise, their detuning and harmonic robustness has been given. Here for the first time a fully general procedure for designing DW-PSAs (or triple-wavelengths PSAs) with desire spectrum, signal-to-noise ratio and detuning robustness is given. We finally generalize DW-PSA to higher number of wavelength temporal PSAs.

physics.optics↗

arXiv2015

Fourier phase-demodulation applied to strip-light 360-degrees profilometry of 3D solids; theoretical principles

360-degrees digitalization of three-dimensional (3D) solids using a projected light-strip is a well established technique. These profilometers project a light-strip over the solid under analysis while the solid is rotated a full revolution. Then a computer program typically extracts the centroid of this light-strip, and by triangulation one obtains the shape of the solid. Here instead of using intensity-based strip centroid estimation, we propose to use Fourier phase-demodulation. This 360-degrees profilometer first constructs a carrier-frequency fringe-pattern by closely adding individual light-strip images. Secondly this high-density fringe-pattern is phase-demodulated using the standard Fourier technique.

physics.optics↗