Manuel Servin

Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms

An algorithm for phase demodulation of a single interferogram that may contain closed fringes is presented. This algorithm uses the regularized phase-tracker system as a robust phase estimator, together with a new scanning technique that estimates the phase that initially follows the bright zones of the interferogram. The combination of these two elements constitutes a powerful new method, the fringe-follower regularized phase tracker, that makes it possible to correctly demodulate complex, single-image interferograms for which traditional methods fail.

The general theory of phase shifting algorithms.

We have been reporting several new techniques of analysis and synthesis applied to Phase Shifting Interferometry (PSI). These works are based upon the Frequency Transfer Function (FTF) and how this new tool of analysis and synthesis in PSI may be applied to obtain very general results, among them; rotational invariant spectrum; complex PSI algorithms synthesis based on simpler first and second order quadrature filters; more accurate formulae for estimating the detuning error; output-power phase noise estimation. We have made our cases exposing these aspects of PSI separately. Now in the light of a better understanding provided by our past works we present and expand in a more coherent and holistic way the general theory of PSI algorithms. We are also providing herein new material not reported before. These new results are on; a well defined way to combine PSI algorithms and recursive linear PSI algorithms to obtain resonant quadrature filters.

General n-dimensional quadrature transform and its application to interferogram demodulation

Quadrature operators are useful for obtaining the modulating phase phi in interferometry and temporal signals in electrical communications. In carrier-frequency interferometry and electrical communications, one uses the Hilbert transform to obtain the quadrature of the signal. In these cases the Hilbert transform gives the desired quadrature because the modulating phase is monotonically increasing. We propose an n-dimensional quadrature operator that transforms cos(phi) into -sin(phi) regardless of the frequency spectrum of the signal. With the quadrature of the phase-modulated signal, one can easily calculate the value of phi over all the domain of interest. Our quadrature operator is composed of two n-dimensional vector fields: One is related to the gradient of the image normalized with respect to local frequency magnitude, and the other is related to the sign of the local frequency of the signal. The inner product of these two vector fields gives us the desired quadrature signal. This quadrature operator is derived in the image space by use of differential vector calculus and in the frequency domain by use of a n-dimensional generalization of the Hilbert transform. A robust numerical algorithm is given to find the modulating phase of two-dimensional single-image closed-fringe interferograms by use of the ideas put forward.

Two-dimensional Phase Locked Loop Demodulation of Interferograms

Abstract A new technique for continuous phase determination of an interferogram based on a digital phase locked loop is presented. The main advantage of this method, with respect to well established techniques such as Fourier or phase stepping demodulation, is that the traditional approach to phase unwrapping processes by removal of discontinuities is not required. The phase is determined continuously as the phase locked loop scans the two-dimensional interferogram. Because of the sequential nature of the algorithm proposed, this can be implemented using a special purpose video processor for phase determination at video rates. The above mentioned properties makes the presented technique a fast algorithm for phase determination of carrier frequency interferograms modulated by a two-dimensional smooth phase function.

Depth object recovery using radial basis functions

Abstract The main task of robot vision systems is to recover the shape of the objects in a scene and use this information for recognition and classification. The fringe projection technique may be used to recover depth range data. To use this technique, we need to estimate the phase of the projected fringes (related to the object height) by using conventional phase demodulation techniques (phase shifting, phase locked loop, or spatial synchronous detection among others). The final step of the process is to recover the object surface using a calibration procedure. This calibration procedure uses a phase-to-depth conversion obtained from the experimental optical geometry. Set-up parameters to consider are the optical characteristics of the lenses used to project and acquire the fringe patterns, which are usually unknown. Some experimental factors arise when the object under analysis is located near the projector–camera system, such as the diverging fringe projection, the video camera perspective distortion and the use of crossed-optical-axes geometry. In this work, we present a neural network based on radial basis functions (RBF) to estimate actual depth data of the object from the recovered phase using the projected fringe pattern technique. The training set of the neural network is the phase recovered of several calibration planes whose locations in space are known. An advantage of our method is that knowledge of the optical parameters of the experiment is not explicitly required.

WAVE-FRONT RECOVERY FROM TWO ORTHOGONAL SHEARED INTERFEROGRAMS

We present a new technique for using the information of two orthogonal lateral-shear interferograms to estimate an aspheric wave front. The wave-front estimation from sheared inteferometric data may be considered an ill-posed problem in the sense of Hadamard. We apply Thikonov regularization theory to estimate the wave front that has produced the lateral sheared interferograms as the minimizer of a positive definite-quadratic cost functional. The introduction of the regularization term permits one to find a well-defined and stable solution to the inverse shearing problem over the wave-front aperture as well as to reduce wave-front noise as desired.

Adaptive quadrature filters and the recovery of phase from fringe pattern images

A principled approach, based on Bayesian estimation theory and complex-valued Markov random-field prior models, is introduced for the design of a new class of adaptive quadrature filters. These filters are capable of adapting their tuning frequency to the local dominant spatial frequency of the input image while maintaining an arbitrarily narrow local frequency response; therefore they may be effectively used for the accurate recovery of the phase of broadband spatial-carrier fringe patterns, even when they are corrupted by a significant amount of noise. Also, by constraining the spatial variation of the adaptive frequency to be smooth, they permit the completely automatic recovery of local phase from single closed fringe pattern images, since the spurious discontinuities and sign reversals that one obtains from the classical Fourier-based methods are avoided in this case. Although the applications discussed here come from fringe pattern analysis in optics, these filters may also be useful in the solution of other problems, such as texture characterization and segmentation and the recovery of depth from stereoscopic pairs of images.

A Novel Technique for Spatial Phase-shifting Interferometry

A novel technique for phase detection using three-step spatial phase-shifting interferometry is presented. The presented technique overcomes and studies the two main problems presented in the commonly used three-step phase-stepping technique. These problems deal with the leak of carrier frequency in the detected phase and the optimal carrier frequency to obtain the highest phase noise robustness.

Local phase from local orientation by solution of a sequence of linear systems

A technique for recovering the phase from single fringe-pattern images is presented. It is based on the estimation of the local frequency of the pattern by successive decoupled estimation of the local orientation, direction, and magnitude of the frequency field. Once this field is known, the local phase is recovered from the complex output of an adaptive quadrature filter. It is shown that by the use of Gauss–Markov measure field models all these estimation steps may be implemented by solving linear systems of equations (i.e., minimizing quadratic functions), which makes the procedure robust and computationally efficient. Examples are presented of the application of this technique to the recovery of phase from single electronic speckle-pattern interferograms.

Multi-layer neural network applied to phase and depth recovery from fringe patterns

A multi-layer neural network (MLNN) is used to carry out calibration processes in fringe projection profilometry in which the explicit knowledge of the experimental set-up parameters is not required. The MLNN is trained by using the fringe pattern irradiance and the height directional gradients provided from a calibration object. After the MLNN has been trained, profilometric height data are estimated from the projected fringe patterns onto the test object. The MLNN method works adequately on an open fringe pattern, but it can be extended to closed fringe patterns. In the proposed technique, edge effects do not appear when the field view is limited in the fringe pattern. In order to show the application of the MLNN method, three different experiments are presented: (a) shape determination of a spherical optical surface; (b) optical phase calculation from a computer-simulated closed fringe pattern; and (c) height determination of a real surface target. An analysis is also made of how noise, spatial carrier frequencies, and different training sets affect the MLNN performance.