Tilt-compensating algorithm for phase-shift interferometry

A self-calibrating algorithm for phase-shift interferometry is described that is able to cancel the effect of accidental relative tilts that may occur during phase stepping. The algorithm is able to retrieve both the phase steps and the tilts that accompany them. Only three phase-shifted interferograms are needed, and no other information about the intentional phase shifts or possible tilts has to be supplied. This purpose is achieved by division of the interferogram space into blocks on which a previously reported self-calibrating algorithm is applied and the actual values of the local phase shifts are calculated. The information thus obtained is used for extracting the global shift and tilt values. Further improvement in the results is achieved by means of a fitting routine.     

Interframe intensity correlation matrix for self-calibration in phase-shifting interferometry

A new method of estimating reference phase shifts in phase-shifting interferometry is proposed. The reference phase shifts are determined from a matrix that represents the interframe intensity correlation (IIC) of phase-shifted interferograms. The root-mean-square error of intensity measurement is automatically obtained from the smallest eigenvalue of the IIC matrix. The proposed method requires only four interferograms, unlike others, and can extract phase shifts reliably even from interferograms without well-defined fringes, such as speckle patterns. In typical conditions, reference phase shifts and wave-front phases can be determined with an accuracy of lambda/6310 and lambda/150, respectively. The validity of the method is tested by comparing it with other methods in experiments and simulations.     

Disk-growing algorithm for phase-map unwrapping: application to speckle interferograms

Interferometric techniques combined with phase shifting allow computation of the phase that is linked to the displacement of the object under study. The phases before and after displacement are computed from three or more interferograms (called specklegrams when speckle is used as the information carrier). Subtraction of these two phase patterns leads to a raw phase map. Phase unwrapping restores the 2π discontinuities and gives a continuous phase map. The disk-growing algorithm presented allows the inner and the outer propagation of the unwrapping from a growing disk and so avoids the main problem of anisotropic error propagation for noisy phase maps. It works successfully in speckle interferometry.     

Multi-spatial-frequency and phase-shifting profilometry using a liquid crystal phase modulator

Optical profilometry is widely applied for measuring the morphology of objects by projecting predetermined patterns on them. In this technique, the compact size is one of the interesting issues for practical applications. The generation of pattern by the interference of coherent light sources has a potential to reduce the dimension of the illumination part. Moreover, this method can make fine patterns without projection optics, and the illumination part is free of restriction from the numerical aperture of the projection optics. In this paper, a phase-shifting profilometry is implemented by using a single liquid crystal (LC) cell. The LC phase modulator is designed to generate the interference patterns with several different spatial frequencies by changing selection of the spacing between the micro-pinholes. We manufactured the LC phase modulator and calibrated it by measuring the phase modulation amount depending on an applied voltage. Our optical profilometry using the single LC cell can generate multi-spatial frequency patterns as well as four steps of the phase-shifted patterns. This method can be implemented compactly, and the reconstructed depth profile is obtained without a phase-unwrapping algorithm.     

High-resolution 3D profilometry with binary phase-shifting methods

This paper presents a novel pixel-level resolution 3D profilometry technique that only needs binary phase-shifted structured patterns. This technique uses four sets of three phase-shifted binary patterns to achieve the phase error of less than 0.2%, and only requires two sets to reach similar quality if the projector is slightly defocused. Theoretical analysis, simulations, and experiments will be presented to verify the performance of the proposed technique.     

Phase-shifting algorithms for electronic speckle pattern interferometry

A set of innovative phase-shifting algorithms developed to facilitate metrology based on electronic speckle pattern interferometry (ESPI) are presented. The theory of a phase-shifting algorithm, called a (5,1) algorithm, that takes five phase-shifted intensity maps before a specimen is deformed and one intensity map after a specimen is deformed is presented first. Because a high-speed camera can be used to record the dynamic image of the specimen, this newly developed algorithm has the potential to retain the phase-shifting capability for ESPI in dynamic measurements. Also shown is an algorithm called a (1,5) algorithm that takes five phase-shifted intensity maps after the specimen is deformed. In addition, a direct-correlation algorithm was integrated with these newly developed (5,1) or (1,5) algorithms to form DC-(5,1) and DC-(1,5) algorithms, which are shown to improve significantly the quality of the phase maps. The theoretical and experimental aspects of these two newly developed techniques, which can extend ESPI to areas such as high-speed dynamic measurements, are examined in detail.     

Spatial carrier phase-shifting algorithm based on least-squares iteration

An advanced spatial carrier phase-shifting (SCPS) algorithm based on least-squares iteration is proposed to extract the phase distribution from a single spatial carrier interferogram. The proposed algorithm divides the spatial carrier interferogram into four phase-shifted interferograms. By compensating for the effects of the variations of phase shifts between pixels and the variations of background and contrast, the proposed algorithm determines the local phase shifts and phase distribution simultaneously and accurately. Numerical simulations show that the accuracy of the proposed algorithm is obviously improved by compensating for the effects of background and contrast variations. The peak to valley of the residual phase error remains less than 0.002 rad when the magnitude of spatial carrier is in the range from pi/5 to pi/2 and the direction of the spatial carrier is in the range from 25 degrees to 65 degrees. Numerical simulations and experiments demonstrate that the proposed algorithm exhibits higher precision than the existing SCPS algorithms. The proposed algorithm is sensitive to random noise, but the error can be reduced by N times if N measurements are taken and averaged.     

An online triple-frequency color-encoded fringe projection profilometry for discontinuous object

Abstract

An online triple-frequency color-encoded fringe projection profilometry is proposed to measure the complex and discontinuous object at straight-line movement. N frames of color fringe patterns are specially designed. Three grayscale sinusoidal grating patterns with geometric progression frequency growth are encoded into red (R), green (G), and blue (B) channels separately to compose a color-encoded fringe pattern. If these three grayscale sinusoidal grating patterns are phase-shifted N steps with an equivalent shift phase of 2π/N, they can compose the corresponding N frames of color-encoded fringe patterns as above respectively. In order to avoid the movement’s interference to the phase shifting, position adjustment should be done to guarantee the phase-shifting direction to be perpendicular to the moving direction. While these N frames of specially designed color-encoded fringe patterns are projected onto the moving object one by one, the corresponding deformed color patterns are captured by a CCD camera in real time. By color separating, color crosstalk compensation, pixel matching, and phase calculation, three wrapped phase at different frequencies can be extracted. The unwrapped phase can be solved by a simplified algorithm based on temporal phase unwrapping method from the relationship of the three wrapped phase at the same pixel. Thus, it is very suitable to measure the online complex and discontinuous objects at straight-line movement. The experimental results show the feasibility and the validity of the proposed method.     

Vibration Phase‐Based Ordering of Vibration Patterns Acquired With a Shearing Speckle Interferometer and Pulsed Illumination

Abstract: A method for both temporal and spatial characterisation of harmonic vibrations is presented. The method is based on simultaneous acquisition of phase‐stepped speckle interference patterns using a shearing speckle interferometer and the vibration phase for a series of vibration states within the vibration period. An unsynchronised free‐running pulse laser is used for illuminating a vibrating object yielding speckle interference patterns in random vibration phase order. Two π/2 phase‐stepped speckle interference patterns are acquired simultaneously for each recorded vibration state. The data set is sorted using vibration phase as the sorting key. The sorted speckle interference patterns are processed using a two‐bucket algorithm for the calculation of phase difference and by applying temporal phase unwrapping to finally obtain unwrapped phase distributions for any vibration state of the vibration cycle.     

Video-rate fringe analyzer based on phase-shifting electronic moiré patterns

We have developed a fringe analyzer that delivers the phase distribution at a video rate from a fringe pattern containing a spatial carrier. It is based on parallel generations of three phase-shifted moiré patterns through electronic multiplication with computer-generated reference gratings and low-pass filtering. The phase distribution is derived by the subsequent parallel processing of these patterns on the basis of a three-step phase-shifting algorithm. By modification of the bias phase distribution of the reference gratings, several functions, such as correction of an initial wave-front error, are realized in real time. The usefulness of this analyzer is demonstrated experimentally for phase measurements by grating-projection surface topography and interferometry.     

Description and simulation of an active imaging technique utilizing two speckle fields: iterative reconstructors

Quasi-monochromatic light will form laser speckle upon reflection from a rough object. This laser speckle provides information about the shape of the illuminated object. In a prior paper [J. Opt. Soc. Am. A 19, 444 (2002)], it was shown that two intensities of two speckle patterns and their interference are sufficient to produce an unambiguous (except for object translation) band-limited image of the object, based on a root-matching technique described therein, in the absence of measurement error and in the case of distinct roots of the field polynomials and their complex conjugates. On the other hand, algorithms based on the root-matching technique are found to be slow and sensitive to noise. So motivated, several other techniques are applied to the problem, including phase retrieval, expectation maximization, and statistical maximization. The phase-retrieval and expectation-maximization techniques proved to be most effective for reconstructions of complex objects larger than 10 pixels across, and high-quality images were formed by using three independent sets of two-field data (three frames of two-wavelength data), each comprising two speckle intensity patterns and their interference. Two additional results of note are reported. First, the expectation-maximization algorithm produced relatively good images when three or more frames each of only one speckle intensity pattern (data at just one wavelength) were used and second, the phase-retrieval algorithm when only the object autocorrelation was used also produced relatively good images for the chosen test object.     

A new phase retrieval algorithm with pure generalized three-step phase-shifting under matrix norm processing

Phase-shifting interferometry (PSI) is one of the most effective techniques in imaging a phase specimen, in which the phase retrieval is a basic and significant process. A new phase retrieval method based on the matrix norm algorithm in PSI is proposed in this paper. In this algorithm, the value of phase shift can be determined by three different matrix norms of the intensity difference between two phase-shifted interferograms, and then the phase can be retrieved. Neither the iterative calculation nor the extra measurements of other parameters are necessary on account of this algorithm which only requires three phase-shifted interferograms. The feasibility and accuracy of this algorithm are demonstrated by the simulated results. Experimentally, the generalized phase shift can be realized by a simple device which adjusting the angle of glass accurately. It is found that this algorithm has a good application prospect in the field of dynamic imaging.     

Phase-shifting interferometer based on changing the direction of linear polarization orthogonally

We present a new type of phase-shifting interferometer, which utilizes a polarizing prism to form quadrature phase-shifted fringe patterns onto a single imaging sensor. By changing the direction of linear polarization of the incident light orthogonally, four phase-shifted fringe patterns in quadrature are obtained by taking images twice; thus it is possible to reduce phase errors caused by mechanical vibrations and air turbulence that occur in temporal phase-shifting interferometers. We present the principle of this interferometer with its theoretical analysis, using the Jones matrix, along with experimental results.     

Least-squares phase estimation with multiple parameters in phase-shifting electronic speckle pattern interferometry

We have developed an accurate and robust phase-estimation method in phase-shifting electronic speckle pattern interferometry. Unlike other methods that assume a constant phase within a fitting window, our method treats the phase variation with a gradient. A cost function that can utilize the information of pixel positions is formulated on the basis of a least-squares criterion. Powell's iteration method is applied to it to derive the phase and its gradient. An automatic consistency-checking routine and an algorithm that improves the initial guess of the iteration are developed for severe situations with large noise and steep phase variations.     

Double-exposure phase calculation method in electronic speckle pattern interferometry based on holographic object illumination

Multiple-exposure phase calculation procedures are widely used in electronic speckle pattern interferometry to calculate phase maps of displacements. We developed a double-exposure process based on holographic illumination of the object and the idea of the spatial carrier phase-shifting method to examine transient displacements. In our work, computer-generated holograms and a spatial light modulator were used to generate proper coherent illuminating masks. In this adjustment all phase-shifted states were at our disposal from one recorded speckle image for phase calculation. This technique can be used in the large scale of transient measurements. In this paper we illustrate the principle through several examples.     

Statistical phase-shifting step estimation algorithm based on the continuous wavelet transform for high-resolution interferometry metrology

We propose a statistical phase-shifting estimation algorithm for temporal phase-shifting interferometry (PSI) based on the continuous wavelet transform (CWT). The proposed algorithm explores spatial information redundancy in the intraframe interferogram dataset using the phase recovery property on the power ridge of the CWT. Despite the errors introduced by the noise of the interferogram, the statistical part of the algorithm is utilized to give a sound estimation of the phase-shifting step. It also introduces the usage of directional statistics as the statistical model, which was validated, so as to offer a better estimation compared with other statistical models. The algorithm is implemented in computer codes, and the validations of the algorithm were performed on numerical simulated signals and actual phase-shifted moiré interferograms. The major advantage of the proposed algorithm is that it imposes weaker conditions on the presumptions in the temporal PSI, which, under most circumstances, requires uniform and precalibrated phase-shifting steps. Compared with other existing deterministic estimation algorithms, the proposed algorithm estimates the phase-shifting step statistically. The proposed algorithm allows the temporal PSI to operate under dynamic loading conditions and arbitrary phase steps and also without precalibration of the phase shifter. The proposed method can serve as a benchmark method for comparing the accuracy of the different phase-step estimation methods.     

Control of phase of fringes in speckle interferometry for application of fringe scanning method

The speckle interferometry is an effective technique in the displacement measurement of a structure with a rough surface. However, when the fringe scanning technique is introduced to speckle interferometry for improving the measurement resolution, generally two speckle patterns before and after the deformation of the measurement object and another speckle pattern obtained under different conditions from these two speckle patterns are required at least. So, three speckle patterns are generally required for precise fringe analysis as a minimum condition. In this paper, a method for introducing the fringe scanning method is proposed by controlling the phase of the specklegram as a fringe image using filtering techniques. Then, the temporal fringe analysis method that uses only two speckle patterns are proposed for speckle interferometry. As the result of experiments, it is shown that high precise fringe analysis can be realized by the fringe scanning methods using only two speckle patterns for the displacement measurement with a large deformation.     

Windowed Fourier transform for fringe pattern analysis: theoretical analyses

A windowed Fourier ridges (WFR) algorithm and a windowed Fourier filtering (WFF) algorithm have been proposed for fringe pattern analysis and have been demonstrated to be versatile and effective. Theoretical analyses of their performances are of interest. Local frequency and phase extraction errors by the WFR and WFF algorithms are analyzed in this paper. Effectiveness of the WFR and WFF algorithms will thus be theoretically proven. Consider four phase-shifted fringe patterns with local quadric phase [c(20)=c(02)=0.005 rad/(pixel)(2)], and assume that the noise in these fringe patterns have mean values of zero and standard deviations the same as the fringe amplitude. If the phase is directly obtained using the four-step phase-shifting algorithm, the phase error has a mean of zero and a standard deviation of 0.7 rad. However, when using the WFR algorithm with a window size of sigma(x)=sigma(y)=10 pixels, the local frequency extraction error has a mean of zero and a standard deviation of less than 0.01 rad/pixel and the phase extraction error in the WFR algorithm has a mean of zero and a standard deviation of about 0.02 rad. When using the WFF algorithm with the same window size, the phase extraction error has a mean of zero and a standard deviation of less than 0.04 rad and the local frequency extraction error also has a mean of zero and a standard deviation of less than 0.01 rad/pixel. Thus, an unbiased estimation with very low standard deviation is achievable for local frequencies and phase distributions through windowed Fourier transforms. Algorithms applied to different fringe patterns, different noise models, and different dimensions are discussed. The theoretical analyses are verified by numerical simulations.     

Iteration of phase window correlation and least-squares fit for Young's fringe pattern processing

Correlation by means of phase windows that artificially create phase shifts is combined with the least-squares fit to provide a processing algorithm. A Young's pattern is correlated by means of the phase windows to generate four patterns with phase shifts of 0 degrees , 90 degrees , 180 degrees , and 270 degrees . The phase of the Young's pattern is derived from the four patterns. Then the derived phase is least-squares fitted to a phase plane. The above steps are repeated once to improve accuracy. Speckle noise is suppressed by the correlation operations, as well as a loop including phase calculation, fringe reconstruction, and smoothing. The magnitude and direction of the displacement associated with Young's pattern are determined from the phase plane. The algorithm takes less calculation time than the fast Fourier transform method and does so with improved accuracy. Software has been developed and is used in the experiment.