A fast-Fourier-transform method of topography and interferometry is proposed. By computer processing of a noncontour type of fringe pattern, automatic discrimination is achieved between elevation and depression of the object or wave-front form, which has not been possible by the fringe-contour-generation techniques. The method has advantages over moire topography and conventional fringe-contour interferometry in both accuracy and sensitivity. Unlike fringe-scanning techniques, the method is easy to apply because it uses no moving components.

Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry

A self-scanned 1024 element photodiode array and minicomputer are used to measure the phase (wavefront) in the interference pattern of an interferometer to lambda/100. The photodiode array samples intensities over a 32 x 32 matrix in the interference pattern as the length of the reference arm is varied piezoelectrically. Using these data the minicomputer synchronously detects the phase at each of the 1024 points by a Fourier series method and displays the wavefront in contour and perspective plot on a storage oscilloscope in less than 1 min (Bruning et al. Paper WE16, OSA Annual Meeting, Oct. 1972). The array of intensities is sampled and averaged many times in a random fashion so that the effects of air turbulence, vibrations, and thermal drifts are minimized. Very significant is the fact that wavefront errors in the interferometer are easily determined and may be automatically subtracted from current or subsequent wavefrots. Various programs supporting the measurement system include software for determining the aperture boundary, sum and difference of wavefronts, removal or insertion of tilt and focus errors, and routines for spatial manipulation of wavefronts. FFT programs transform wavefront data into point spread function and modulus and phase of the optical transfer function of lenses. Display programs plot these functions in contour and perspective. The system has been designed to optimize the collection of data to give higher than usual accuracy in measuring the individual elements and final performance of assembled diffraction limited optical systems, and furthermore, the short loop time of a few minutes makes the system an attractive alternative to constraints imposed by test glasses in the optical shop.

Digital wavefront measuring interferometer for testing optical surfaces and lenses

Fringe projection techniques: Whither we are?

To improve the performance of particle image velocimetry in measuring instantaneous velocity fields, direct cross-correlation of image fields can be used in place of auto-correlation methods of interrogation of double- or multiple-exposure recordings. With improved speed of photographic recording and increased resolution of video array detectors, cross-correlation methods of interrogation of successive single-exposure frames can be used to measure the separation of pairs of particle images between successive frames. By knowing the extent of image shifting used in a multiple-exposure and by a priori knowledge of the mean flow-field, the cross-correlation of different sized interrogation spots with known separation can be optimized in terms of spatial resolution, detection rate, accuracy and reliability.

Theory of cross-correlation analysis of PIV images : Image analysis as measuring technique in flows

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Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations

Fringe patterns in optical metrology systems need to be demodulated to yield the desired parameters. Time-frequency analysis is a useful concept for fringe demodulation, and a windowed Fourier transform is chosen for the determination of phase and phase derivative. Two approaches are developed: the first is based on the concept of filtering the fringe patterns, and the second is based on the best match between the fringe pattern and computer-generated windowed exponential elements. I focus on the extraction of phase and phase derivatives from either phase-shifted fringe patterns or a single carrier fringe pattern. Principles as well as examples are given to show the effectiveness of the proposed methods.

Windowed Fourier transform for fringe pattern analysis

The nonlinear intensity response of a digital fringe projection profilometry (FPP) system causes the captured fringe patterns to be nonsinusoidal waveforms and leads to an additional phase measurement error for commonly used three- and four-step phase-shifting algorithms. We perform theoretical analysis of the phase error owing to the nonsinusoidal waveforms. Based on the derived theoretical model, a novel and simple iterative phase compensation algorithm is proposed to compensate the nonsinusoidal phase error. Experiments show that the proposed algorithm can be used for effective phase error compensation in practical phase-shifting FPP.

Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry

Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry

Quality-guided phase unwrapping is a widely used technique with different quality definitions and guiding strategies reported. It is thus necessary to do a detailed comparison of these approaches to choose the optimal quality map and guiding strategy. For quality maps, in the presence of noise, transform-based methods are found to be the best choice. However in the presence of discontinuities, phase unwrapping is itself unresolved and hence quality-guided phase unwrapping is not sufficient. For guiding strategies, classical, two-section, and stack-chain guiding strategies are chosen for comparison. If accuracy is the foremost criterion then the classical guiding strategy with a data structure of indexed interwoven linked list is best. If speed is of essence then the stack-chain guiding strategy is the one to use.

Qian Kemao

Papers

Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations

Windowed Fourier transform for fringe pattern analysis

Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry

Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry

Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies