Moises Padilla

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 Rubiks 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.

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

From 1971 to 2012 dual-wavelength optical-metrology used only the demodulated low-sensitivity phase-difference of two close-sensitive fringes. Dual-wavelength phase-metrology that additionally uses the phase-sum was first reported by Di et al. in 2013 [28]; this was an important step to increase the phase-accuracy in optical metrology. This and its derived papers however do not offer mathematical analysis for signal-to-noise ratio (SNR) for the phase-difference and phase-sum. Neither provide the mathematical analysis for unwrapping the phase-sum without errors. Here a general theory for super-sensitive two-wavelength phase-metrology is given. In particular mathematical analysis and formulas for SNR and error-free phase-unwrapping for two-wavelength metrology is provided. We start by phase-demodulating two close-sensitivity fringes by phase-shifting algorithms (PSAs). We then calculate their phase-difference and their phase-sum; the phase-difference is assumed non-wrapped. However the phase-sum is highly wrapped, super-sensitive and has much higher SNR. Spatial phase unwrapping for a highly discontinuous phase-sum is precluded. However as we show, it is possible to unwrap the noisy phase-sum from the noisier phase-difference without errors. We apply this super-sensitive phase-metrology theory to profilometry allowing us to obtain super-sensitive height measurements. To the best of our knowledge the mathematical analysis and formulas herein presented for the SNR and error-free unwrapping have not been reported before.A far-ultraviolet (FUV) equivalent-wavelength super-resolution interferometric technique is proposed. This FUV equivalent-wavelength interferometric method combines four demodulated phases from four temporal-sets of visible interferograms. Here FUV super-resolution interferometry is defined as the estimation of a modulating phase coming from an FUV equivalent-wavelength illumination laser. To this end we need to combine the demodulated phase of four visible-wavelength interferograms. FUV equivalent-wavelength phase-sensitivity is of course beyond the phase-information capacity of a single visible-wavelength interferogram. To break this visible-wavelength barrier we use the phase-information provided by four or more interferograms in the visible range. Having the four demodulated phases we calculate a phase-difference and the sum of the four phases which is the FUV-equivalent super-resolution phase. The phase-difference is in the infrared phase-sensitivity range and it is assumed non-wrapped. On the other hand the phase-sum is in the FUV phase-sensitivity range and it is highly-wrapped. As shown herein it is possible to unwrap the phase-sum in the temporal domain using the phase-difference and our previously reported extended-range 2-step temporal phase-unwrapper. Of course higher than FUV equivalent phase-sensitivity interferometry may be obtained by increasing the number of independent estimated phases from visible-wavelength interferograms. As far as we know, this FUV equivalent-wavelength super-resolution interferometric technique has the highest phase-sensitivity and highest signal-to-noise ratio ever reported to this date.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.

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.

Super-sensitive two-wavelength fringe projection profilometry with 2-sensitivities temporal unwrapping

From 1971 to 2012 dual-wavelength optical-metrology used only the demodulated low-sensitivity phase-difference of two close-sensitive fringes. Dual-wavelength phase-metrology that additionally uses the phase-sum was first reported by Di et al. in 2013 [28]; this was an important step to increase the phase-accuracy in optical metrology. This and its derived papers however do not offer mathematical analysis for signal-to-noise ratio (SNR) for the phase-difference and phase-sum. Neither provide the mathematical analysis for unwrapping the phase-sum without errors. Here a general theory for super-sensitive two-wavelength phase-metrology is given. In particular mathematical analysis and formulas for SNR and error-free phase-unwrapping for two-wavelength metrology is provided. We start by phase-demodulating two close-sensitivity fringes by phase-shifting algorithms (PSAs). We then calculate their phase-difference and their phase-sum; the phase-difference is assumed non-wrapped. However the phase-sum is highly wrapped, super-sensitive and has much higher SNR. Spatial phase unwrapping for a highly discontinuous phase-sum is precluded. However as we show, it is possible to unwrap the noisy phase-sum from the noisier phase-difference without errors. We apply this super-sensitive phase-metrology theory to profilometry allowing us to obtain super-sensitive height measurements. To the best of our knowledge the mathematical analysis and formulas herein presented for the SNR and error-free unwrapping have not been reported before.A far-ultraviolet (FUV) equivalent-wavelength super-resolution interferometric technique is proposed. This FUV equivalent-wavelength interferometric method combines four demodulated phases from four temporal-sets of visible interferograms. Here FUV super-resolution interferometry is defined as the estimation of a modulating phase coming from an FUV equivalent-wavelength illumination laser. To this end we need to combine the demodulated phase of four visible-wavelength interferograms. FUV equivalent-wavelength phase-sensitivity is of course beyond the phase-information capacity of a single visible-wavelength interferogram. To break this visible-wavelength barrier we use the phase-information provided by four or more interferograms in the visible range. Having the four demodulated phases we calculate a phase-difference and the sum of the four phases which is the FUV-equivalent super-resolution phase. The phase-difference is in the infrared phase-sensitivity range and it is assumed non-wrapped. On the other hand the phase-sum is in the FUV phase-sensitivity range and it is highly-wrapped. As shown herein it is possible to unwrap the phase-sum in the temporal domain using the phase-difference and our previously reported extended-range 2-step temporal phase-unwrapper. Of course higher than FUV equivalent phase-sensitivity interferometry may be obtained by increasing the number of independent estimated phases from visible-wavelength interferograms. As far as we know, this FUV equivalent-wavelength super-resolution interferometric technique has the highest phase-sensitivity and highest signal-to-noise ratio ever reported to this date.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.

Robust phase-shifting algorithms designed for high-dynamic range in fringe-projection profilometry

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.

General theory for super-sensitive dual-wavelength phase metrology: error-free unwrapping and signal-to-noise ratio

From 1971 to 2012 dual-wavelength optical-metrology used only the demodulated low-sensitivity phase-difference of two close-sensitive fringes. Dual-wavelength phase-metrology that additionally uses the phase-sum was first reported by Di et al. in 2013 [28]; this was an important step to increase the phase-accuracy in optical metrology. This and its derived papers however do not offer mathematical analysis for signal-to-noise ratio (SNR) for the phase-difference and phase-sum. Neither provide the mathematical analysis for unwrapping the phase-sum without errors. Here a general theory for super-sensitive two-wavelength phase-metrology is given. In particular mathematical analysis and formulas for SNR and error-free phase-unwrapping for two-wavelength metrology is provided. We start by phase-demodulating two close-sensitivity fringes by phase-shifting algorithms (PSAs). We then calculate their phase-difference and their phase-sum; the phase-difference is assumed non-wrapped. However the phase-sum is highly wrapped, super-sensitive and has much higher SNR. Spatial phase unwrapping for a highly discontinuous phase-sum is precluded. However as we show, it is possible to unwrap the noisy phase-sum from the noisier phase-difference without errors. We apply this super-sensitive phase-metrology theory to profilometry allowing us to obtain super-sensitive height measurements. To the best of our knowledge the mathematical analysis and formulas herein presented for the SNR and error-free unwrapping have not been reported before.A far-ultraviolet (FUV) equivalent-wavelength super-resolution interferometric technique is proposed. This FUV equivalent-wavelength interferometric method combines four demodulated phases from four temporal-sets of visible interferograms. Here FUV super-resolution interferometry is defined as the estimation of a modulating phase coming from an FUV equivalent-wavelength illumination laser. To this end we need to combine the demodulated phase of four visible-wavelength interferograms. FUV equivalent-wavelength phase-sensitivity is of course beyond the phase-information capacity of a single visible-wavelength interferogram. To break this visible-wavelength barrier we use the phase-information provided by four or more interferograms in the visible range. Having the four demodulated phases we calculate a phase-difference and the sum of the four phases which is the FUV-equivalent super-resolution phase. The phase-difference is in the infrared phase-sensitivity range and it is assumed non-wrapped. On the other hand the phase-sum is in the FUV phase-sensitivity range and it is highly-wrapped. As shown herein it is possible to unwrap the phase-sum in the temporal domain using the phase-difference and our previously reported extended-range 2-step temporal phase-unwrapper. Of course higher than FUV equivalent phase-sensitivity interferometry may be obtained by increasing the number of independent estimated phases from visible-wavelength interferograms. As far as we know, this FUV equivalent-wavelength super-resolution interferometric technique has the highest phase-sensitivity and highest signal-to-noise ratio ever reported to this date.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.

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.