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Industrial production constraints often require technical tests and controls. Optical metrology methods allow a non destructive test of wide range of parameters, such as defects and displacements, with very good accuracy. The phase retrieval is an effective way that allows three-dimensional profile reconstruction from intensity shearograms. This research work focuses on the extraction of the phase from one uncarrier shearogram using the Hilbert–Huang transform. An algorithm for the phase calculation based on the bidimensional empirical mode decomposition, Hilbert transform (HT), and Fourier transform (FT) is presented. A spatial digital carrier has been superimposed before the application of the FT or HT which uses two π2 shifted shearograms, to get access to the phase map via a global analysis of intensity images. An evaluation was made through a numerical simulation to validate and confirm the performance of the proposed algorithm. The main advantage of this technique is its ability to provide a metrological solution for fast dynamic analysis.

Lateral shear interferometry operating in the convergent beam mode has been used for testing optical components. This method is simple and phase information of the wavefront has conventionally been extracted using phase stepping techniques. We propose to use defocus, which introduces uniform tilt as a means of extracting phase information via two procedures, namely spatial phase stepping and spatial frequency carrier method. Experimental results are presented that show the wavefront phase extracted with defocus before and after the focal point of the lens.

An algorithm for analyzing a vibrating target by a dominant harmonic order determination based on Fourier transform in a self-mixing system over wide optical feedback regimes (very weak, weak, and moderate regimes) is presented. The basic frequency of a self-mixing interference signal is used to determine the frequency of vibration, and the dominant harmonic order is proportional to the amplitude of vibration. The algorithm also allows calculation of the correct target movement at extremely low signal-to-noise ratios. The validity of the proposed algorithm was demonstrated by means of simulated signals and confirmed by several experimental measurements. The maximum error of the amplitude is about 0.16 μm and the maximum error of the frequency is about 2.5%. It can be used to demodulate the microscopic vibration of a remote target.

A generic reconstruction algorithm is proposed, which is obtained by using a phase-shifting technique with an arbitrary number of phase shifts between intensity measurements. The generic algorithm entirely describes the structure of known phase-shifting algorithms and permits the construction of new ones with an arbitrary number of phase shifts.

An optimization scheme based on a genetic algorithm (GA) is proposed for kinoform synthesis. Unlike conventional optimization schemes, the initial kinoforms here are obtained by Fourier transform of the original image with random phase masks. The phase masks are then optimized by GA in order to reduce the reconstruction noise caused by amplitude negligence and phase quantization. Compared with the conventional methods of the genetic algorithm, in which optimization is directly performed to the kinoforms, the scheme can significantly improve the convergence and reduce the computation cost.

We propose an image encryption algorithm by using random position scrambling of the amplitude and phase functions in the frequency domain of optical transform. The positions of amplitude and phase data are scrambled randomly in the horizontal or vertical direction. The random position orders can be regarded as the key of the algorithm. Moreover, random phase encoding is not used in the proposed algorithm. A feasible optical implementation of the encryption algorithm is given. Some numerical simulations have demonstrated the capability of the algorithm.

We propose an evaluation method for the reliability of the analyzed phase in a phase-shifting method using a Fourier transform (PSM/FT) for shape measurement by fringe projection method. In PSM/FT, since the initial phase information is determined only from the first frequency of the Fourier spectrum of the phase-shifted intensity values at each point of an object, and the frequency components higher than the first frequency almost depend on noise, experimental noise can almost be eliminated. The phase reliability evaluation value using a Fourier transform (PREV/FT) is, therefore, defined as the ratio of the first frequency component of the Fourier spectrum to the average of the frequency components higher than the first frequency of the Fourier spectrum. PREV/FT is useful to merge data when we change the measurement condition for shape measurement. We apply the PREV/FT to eliminate the halation effect on a metallic object. Shape measurement of an aluminum specimen that is a rotational body produced by a lathe is performed as an experiment. The results show that the halation effect is eliminated successfully by selecting the highest PREV/FT.

A multispectra method is proposed for the error reduction of the quantized Fourier modulus in the iterative Fourier transform algorithms used for phase retrieval. The Fourier modulus is usually obtained by measuring the Fourier intensity with a digital device, thus the quantization error caused in the measurements of Fourier intensity leads to the error of the reconstructed object when iterative Fourier transform algorithms are used. The idea of the multispectra method uses several Fourier intensity distributions for various measuring ranges to generate a Fourier intensity distribution with low-quantization error. The error reduction of Fourier intensity with the multispectra method connects to the improvement of the phase retrieval with the iterative Fourier transform algorithms. Simulations are performed to demonstrate the effect of the multispectra method for phase retrieval with the iterative hybrid input-output algorithm.

Determination of phase difference is important in many applications of optical metrology. Algorithms for temporal phase difference measurement using the Fourier transform and phase shifting methods are available. These algorithms are first shown to be equivalent and then are generalized for spatial phase difference measurements and a spatial carrier phase shifting method. The efficiency of the algorithm is verified by two examples for slope and curvature determination.

We implement a fully phase encryption system, using fractional Fourier transform to encrypt and decrypt a 2-D phase image obtained from an amplitude image. The encrypted image is holographically recorded in a barium titanate crystal and is then decrypted by generating through phase conjugation, a conjugate of the encrypted image. The decrypted phase image is converted into an amplitude image by the phase contrast technique using an electrically addressed spatial light modulator. Experimental results in support of the proposed idea are presented.

A method to calibrate a nonlinear phase modulator is de- scribed that is based on the minimization of a function obtained from a sequence of interferometric images. The method is equally applicable to smooth wavefront and speckled wavefront interferometry, allowing con- venient in situ calibration for a wide range of interferometer configura- tions. The main advantage over previous calibration methods is that both the linear and quadratic phase variation terms are measured. The per- formance of the proposed method is verified by experiments in which the phase shift is determined independently by a Fourier transform method. © 1998 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(98)02709-3) Subject terms: Calibration; piezoelectric translator; phase-shifting interferometry; phase modulator.

We consider the reconstruction of a complex-valued object that is illuminated with random light (a sequence of speckle distributions) and viewed through a random phase screen. The reconstruction involves a phase retrieval based on the measurement of a series of two shortexposure Fourier intensities: the intensity of the Fourier transform of the image and the intensity of the Fourier transform of the image after transmission through an exponential filter. The method of reconstructing the object consists of the two steps: the reconstruction of the object intensity distribution from the average correlation functions of the short-exposure Fourier intensities of the unfiltered and filtered image fields and the retrieval of the object phase from object intensity distributions reconstructed from the series of the two short-exposure Fourier intensities multiplied by exponential functions. Computer-simulated examples of the reconstruction of 1-D complex objects demonstrate that the reconstruction is robust.

The modification of an iterative algorithm of image recon- struction from only the phase of its Fourier transform is described. This task is important for images distorted by the Gaussian transfer function. The results of reconstructing images of the Moon and Saturn are shown. © 1996 Society of Photo-Optical Instrumentation Engineers. Subject terms: image reconstruction; Fourier transform; Gaussian transfer func- tion; iterative algorithm.

The statistical characteristics of the amplitude and phase of an optical wave propagated through a transparent randomly varying medium are examined analytically. An optical-digital method is proposed for the detection of amplitude and phase of the randomly distorted wave. With a reference point source input, this method reveals that only a single output of the detected signal is needed to obtain both the phase and amplitude variations. The relationship between the randomly varying phase of the optical field and the detected amplitude and demodulated phase is also investigated.

A new technique is proposed to reconstruct an astronomical image using phase retrieval from exponential filtered and unfiltered images based on the Fourier series expansion method. The amplitude of the Fourier transform of the filtered image is a mixture of the information of the amplitude and phase in the Fourier space corresponding to the unfiltered image. Therefore, the phase information can be derived from the Fourier modulus ofthe filtered image based on the Fourier decomposition method described. This method is applied to stellar speckle interferometry and the successful reconstruction of a double star image is obtained.

Recently, a set of conditions has been developed under which a sequence is uniquely specified by the phase or samples of the phase of its Fourier transform. These conditions are distinctly different from the minimum or maximum phase requirement and are applicable to both one-dimensional and multi-dimensional sequences. Under the specified conditions, several numerical algorithms have been developed to reconstruct a sequence from its phase. In this paper, we review the recent theoretical results pertaining to the phase-only reconstruction problem, and we discuss in detail two iterative numerical algorithms for performing the reconstrucction.