Wavefront reconstruction of discontinuous phase objects from optical deflectometry.

One of the challenges of phase measuring deflectometry is to retrieve the wavefront from objects that present discontinuities or non-differentiable gradient fields. Here, we propose the integration of such gradient fields based on an L p-norm minimization problem. The solution of this problem results in a nonlinear partial differential equation, which can be solved with a fast and well-known numerical method and does not depend on external parameters. Numerical reconstructions on both synthetic and experimental data are presented that demonstrate the capability of the proposed method.

Wavelength-stepping algorithm for testing the thickness and 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 thicknesses 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 and the highest harmonic 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.

Simultaneous phase and amplitude modulation for dual-sensitivity profilometry of discontinuous objects.

Fringe projection profilometry (FPP) is a well-known technique for digitizing solids. In FPP, straight fringes are projected over a digitizing solid, and a digital camera grabs the projected fringes. The sensitivity of FPP depends on the spatial frequency of the projected fringes. The projected fringes as seen by the camera are phase modulated by the surface of the digitizing object; the demodulated phase is usually wrapped. If the digitizing object has discontinuities larger than the fringe period, the phase jumps are lost. To preserve large phase discontinuities, one must use very low spatial frequency (low-sensitivity) fringes. The drawback of low-sensitivity FPP is that the demodulated phase has low signal-to-noise ratio (SNR). Much higher SNR is obtained by projecting shorter wavelength, at the cost of obtaining wrapped phase. A way out of this problem is to use dual-wavelength FPP (DW-FPP). In DW-FPP, two sets of projected fringes are used, one with long wavelength and another with shorter wavelength. Due to harmonics and gamma distortion, in DW-FPP, one usually needs four phase-shifted fringes for each sensitivity. Here we are proposing to combine the two sensitivities simultaneously, one coded in phase (PM) and the other coded in amplitude (AM), in order to obtain phase and amplitude modulated (DW-PAM) fringes. The low-sensitivity phase is coded as AM of the DW-PAM fringes. The main advantage of DW-PAM fringes is that one reduces the number of phase-shifted fringes by half: instead of using eight phase-shifted fringes (four for low and four for high sensitivities), one would need only four DW-PAM fringes. Of course, if one wants to increase the harmonic rejection of the recovered phase, one may use a higher order phase-shifting algorithm (PSA).

Moire profilometry through simultaneous dual fringe projection for accurate phase demodulation: a comparative study.

In optical metrology, fringe projection and moire techniques have been widely used to measure the topography of objects. We can combine the advantages of the two techniques by applying a configuration of simultaneous dual projection in the fringe projection technique, which generates a superimposed fringe pattern containing a moire pattern that is phase modulated according to the topography. In this work, we present an analytic and comparative study of three methods to demodulate the phase of the moire pattern: the spatial, spatial-temporal, and temporal methods. Those methods consist of two steps: first, the moire pattern is extracted from the superimposed fringe pattern; next, the phase of the moire pattern is demodulated. The analytical results show that the resulting phase map has double phase sensitivity compared to that of the classical fringe projection technique. Experimental and numeric results prove the feasibility of this technique.

Optical-image communication through random-phase propagation channels using phase-shifting coding.

We propose an optical-image communication system robust to random-phase propagation using phase-shifting (PS) image coding. That is, this optical-image communication system is based on digital PS interferometry principles. Each pixel of the parallel transmitted image is coded as the phase of a sequence of N phase-shifted fringe patterns. The temporal fringe patterns may be displayed on a TV screen (or a multimedia projector) for transmission through the random-phase channel. At the receiver, the PS fringe patterns are digitized with a telescopic digital camera. The received fringes are phase-demodulated using an N-steps least-squares PS algorithm (LS-PSA). We show that the received, phase-demodulated images are less blurred and have better contrast than any received image without PS coding. We propose and analyze a mathematical model for the received PS fringes degraded by random-phase propagation. This PS communication system can also be used for robust optical communications through random refractive media such as underwater, air-water, or random-thickness textured glass. In particular, we show experiments for LS-PSA imaging through textured-glass, obtaining sharper images. As far as we know, this is the first time that PS interferometry has been used for parallel optical-image communication through random-phase channels.

Least-squares gamma estimation in fringe projection profilometry.

This paper introduces a novel, to the best of our knowledge, method to estimate and compensate the nonlinear gamma factor introduced by the optical system in fringe projection profilometry. We propose to determine this factor indirectly by adjusting the least-squares plane to the estimated phase coming from the reference plane. We only require a minimal set of three fringe sinusoidal images to estimate the gamma factor. This value can be used to rectify computational legacy data and also to generate and project the new set of fringe patterns for which we perform the inverse gamma compensation. Experimental results demonstrate the feasibility of the proposed method to estimate and correct the gamma distortion.

Tukey's robust M-estimator for phase demodulation of interferograms with nonuniform shifts.

In this paper, we introduce an iterative scheme for phase demodulation of interferograms with nonuniformly spaced phase shifts. Our proposal consists of two stages: first, the phase map is obtained through a least squares fitting; second, the phase steps are retrieved using a statistical robust estimator. In particular, we use Tukey's biweighted M-estimator because it can cope with both noisy data and outliers in comparison with the ordinary least squares estimator. Furthermore, we provide the frequency description of the algorithm and the phase demodulation allowing us to analyze the procedure and estimation according to the frequency transfer function (FTF) formalism for phase-shifting algorithms. Results show that our method can accurately retrieve the phase map and phase shifts, and it converges by the 10th iteration.

Suppressing ripple distortions and spurious pistons in phase-shifting interferometry.

In this paper, we present new phase-shifting algorithms (PSAs) that suppress the ripple distortions and spurious pistons in phase-shifting interferometry. These phase errors arise when non-uniform phase-shifting interferograms are processed with PSAs that assume uniform phase shifts. By modeling the non-uniform phase shifts as a polynomial of the unperturbed phase-shift value $\omega_0$ω0, we show that the conditions for eliminating the ripple distortion and the spurious piston are associated with the $m$mth 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 four ready-to-apply PSAs formulas. Finally, our conclusions are supported by computer simulations.

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 (RH), 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 its 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.

Phase measurement of nonuniform phase-shifted interferograms using the frequency transfer function.

In this paper, we propose a phase measurement method for interferograms with nonuniform phase shifts. First, we measure the phase shifts between consecutive interferograms. Second, we use these values to modify the spectrum of the interferogram data. Then, by analyzing this spectrum, we design a suitable phase-shifting algorithm (PSA) using the frequency transfer function formalism. Finally, we test our PSA with experimental data to estimate the surface of an aluminum thin film. Our result is better than those obtained using the Fourier transform method, the principal component analysis method, and the least-squares PSA.

Spectral analysis for the generalized least squares phase-shifting algorithms with harmonic robustness.

We introduce the frequency transfer function (FTF) formalism for generalized least squares phase-shifting algorithms (GLS-PSAs), whose phase shifts are nonuniformly spaced. The GLS-PSA's impulsive response is found by computing the Moore-Penrose pseudoinverse. FTF theory allows analyzing these GLS-PSAs spectrally, as well as easily finding figures of merit such as signal-to-noise ratio (SNR) and harmonic rejection capabilities. We show simulations depicting that the SNR slightly decreases as the harmonic rejection robustness improves.

Design of nonlinearly spaced phase-shifting algorithms using their frequency transfer function.

Here we show how to design phase-shifting algorithms (PSAs) for nonuniform/nonlinear (NL) phase-shifted fringe patterns using their frequency transfer function (FTF). Assuming that the 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 for interferograms corrupted by additive white Gaussian noise. Finally, for non-distorted noiseless fringes, the proposed NL-PSA retrieves the modulating phase error free, just as standard/linear PSAs do.

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

In optical metrology synchronous phase-stepping algorithms (PSAs) estimate the measured phase of temporal linear-carrier fringes with respect to a linear-reference. Linear-carrier fringes are normally obtained using closed-loop, feedback, optical phase-stepped devices. On the other hand, open-loop phase-stepping devices usually give fringe patterns with nonlinear phase steps. The Fourier spectrum of linear-carrier fringes is composed of 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 stepped fringes, one obtains a spurious-piston. The problem with this spurious-piston is that it may wrongly be interpreted as a real thickness in any absolute phase measurement. Here we mathematically find the origin of this spurious-piston and design nonlinear phase-stepping PSAs to cope with nonlinear phase-stepping interferometric fringes. We give a general theory to tailor nonlinear phase-stepping PSAs to synchronously demodulate nonlinear phase-stepped wideband fringes.

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 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 2-step phase shifting; this robustness against harmonics is similar to that of the popular 4-step least-squares phase-shifting algorithm. The proposed phase-demodulation method is suitable for the digitization of piecewise continuous surfaces because it does not require spatial low-pass filtering. Gamma calibration is also unnecessary because the projected fringes are binary, and the harmonics produced by the binary profile can be 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.

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

In this paper, we apply the frequency transfer function formalism to analyze the red, green and blue (RGB) phase-shifting fringe-projection profilometry technique. The phase-shifted patterns in RGB fringe projection are typically corrupted by crosstalk because the sensitivity curves of most projection-recording systems overlap. This 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 distortion, we can estimate the searched phase-map using Bruning's 3-step phase-shifting algorithm (PSA). For moderate crosstalk, the recorded data is usually preprocessed before feeding it into the PSA; alternatively, in this paper we propose a computationally more efficient approach, which combines linear crosstalk compensation with the phase-demodulation algorithm. For severe RGB crosstalk, we expect non-sinusoidal fringes' profiles (distorting harmonics) and a significant uncertainty on the linear 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. Finally, we support our conclusions with numerical simulations and experimental results.

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 (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 ad 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.

360-degrees profilometry using strip-light projection coupled to Fourier phase-demodulation.

360 degrees (360°) digitalization of three dimensional (3D) solids using a projected light-strip is a well-established technique in academic and commercial profilometers. These profilometers project a light-strip over the digitizing solid while the solid is rotated a full revolution or 360-degrees. 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 light-strip centroid estimation, we propose to use Fourier phase-demodulation for 360° solid digitalization. The advantage of Fourier demodulation over strip-centroid estimation is that the accuracy of phase-demodulation linearly-increases with the fringe density, while in strip-light the centroid-estimation errors are independent. Here we proposed first to construct a carrier-frequency fringe-pattern by closely adding the individual light-strip images recorded while the solid is being rotated. Next, this high-density fringe-pattern is phase-demodulated using the standard Fourier technique. To test the feasibility of this Fourier demodulation approach, we have digitized two solids with increasing topographic complexity: a Rubik's cube and a plastic model of a human-skull. According to our results, phase demodulation based on the Fourier technique is less noisy than triangulation based on centroid light-strip estimation. Moreover, Fourier demodulation also provides the amplitude of the analytic signal which is a valuable information for the visualization of surface details.

Temporal phase-unwrapping of static surfaces with 2-sensitivity fringe-patterns.

Here we describe a 2-step temporal phase unwrapping formula that uses 2-sensitivity demodulated phases for measuring static surfaces. The first phase demodulation has at most 1-wavelength sensitivity and the second one is G-times (G>>1.0) more sensitive. Measuring static surfaces with 2-sensitivity fringe patterns is well known and recent published methods combine 2-sensitivities measurements mostly by triangulation. Two important applications for our 2-step unwrapping algorithm is profilometry and synthetic aperture radar (SAR) interferometry. In these two applications the object or surface being analyzed is static and highly discontinuous; so temporal unwrapping is the best strategy to follow. Phase-demodulation in profilometry and SAR interferometry is very similar because both share similar mathematical models.

High-resolution low-noise 360-degree digital solid reconstruction using phase-stepping profilometry.

In this paper we describe a high-resolution, low-noise phase-shifting algorithm applied to 360 degree digitizing of solids with diffuse light scattering surface. A 360 degree profilometer needs to rotate the object a full revolution to digitize a three-dimensional (3D) solid. Although 360 degree profilometry is not new, we are proposing however a new experimental set-up which permits full phase-bandwidth phase-measuring algorithms. The first advantage of our solid profilometer is: it uses base-band, phase-stepping algorithms providing full data phase-bandwidth. This contrasts with band-pass, spatial-carrier Fourier profilometry which typically uses 1/3 of the fringe data-bandwidth. In addition phase-measuring is generally more accurate than single line-projection, non-coherent, intensity-based line detection algorithms. Second advantage: new fringe-projection set-up which avoids self-occluding fringe-shadows for convex solids. Previous 360 degree fringe-projection profilometers generate self-occluding shadows because of the elevation illumination angles. Third advantage: trivial line-by-line fringe-data assembling based on a single cylindrical coordinate system shared by all 360-degree perspectives. This contrasts with multi-view overlapping fringe-projection systems which use iterative closest point (ICP) algorithms to fusion the 3D-data cloud within a single coordinate system (e.g. Geomagic). Finally we used a 400 steps/rotation turntable, and a 640x480 pixels CCD camera. Higher 3D digitized surface resolutions and less-noisy phase measurements are trivial by increasing the angular-spatial resolution and phase-steps number without any substantial change on our 360 degree profilometer.

Role of the filter phase in phase sampling interferometry.

Any linear phase sampling algorithm can be described as a linear filter characterized by its frequency response. In traditional phase sampling interferometry the phase of the frequency response has been ignored because the impulse responses can be made real selecting the correct sample offset. However least squares methods and recursive filters can have a complex frequency response. In this paper, we derive the quadrature equations for a general phase sampling algorithm and describe the role of the filter phase.