Guillermo Garnica

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.

Coherent digital demodulation of single-camera N-projections for 3D-object shape measurement: co-phased profilometry.

Fringe projection profilometry is a well-known technique to digitize 3-dimensional (3D) objects and it is widely used in robotic vision and industrial inspection. Probably the single most important problem in single-camera, single-projection profilometry are the shadows and specular reflections generated by the 3D object under analysis. Here a single-camera along with N-fringe-projections is (digital) coherent demodulated in a single-step, solving the shadows and specular reflections problem. Co-phased profilometry coherently phase-demodulates a whole set of N-fringe-pattern perspectives in a single demodulation and unwrapping process. The mathematical theory behind digital co-phasing N-fringe-patterns is mathematically similar to co-phasing a segmented N-mirror telescope.

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.

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.

On Axis Fringe Projection

The standard fringe projection technique requires a non-zero angle between projection and observation directions to have sensitivity in the z direction. In this work, a new method is presented where the angle between projection and observation directions is zero, but the system presents sensitivity due to divergent projection which changes the fringes frequency in each one of the normal planes to z-axis. The experimental results compared with the standard fringe projection technique are presented in this work to show the accuracy of the method proposed.