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.     

Blind phase shift estimation in phase-shifting interferometry

A blind phase shift estimation algorithm that allows simultaneous calculation of phases and phase shifts from three or more interferograms is presented. In phase-shifting interferometry, the phase shift errors introduce specific correlations between the calculated background intensity distribution and the fringe component. These correlations can be measured with a cross-power spectrum. By minimization of an objective function based on this cross-power spectrum, the actual phase shifts are estimated and used for phase recovery. The validity of this algorithm is verified by both the numerical simulation and the experiment results.     

Phase shift selection for two-step generalized phase-shifting interferometry

A phase shift selection method is proposed to design algorithms immune against phase shift errors in two-step generalized phase-shifting interferometry. A general formula for wavefront reconstruction error is derived, and its specific expressions for two common errors are also given. Calculation results suggest that the proper range of phase shift for general application is about from 0.5 to 2.0 rad for both the fixed and linear phase shift errors. Computer simulations have demonstrated the effectiveness of this phase shift selection method by decreasing the wave reconstruction errors to one-fifth.     

High-resolution scanning phase retrieval with a probe beam of unknown modulus

A high-resolution phase-retrieval method for an imaging system with scanning illumination that is capable of reconstructing the modulus and phase of an object without a holographic reference wave is proposed. Reconstruction involves the synthesis of the reconstructed objects from the data of zero- and higher-frequency components of two Fourier intensity measurements: the Fourier intensity of the product of the object and a probe beam that is scanned across the object and the Fourier intensity of the product of the object and a probe beam that is modulated with an exponential filter. This method improves on the resolution of a reconstructed object by previous methods that make use of the data of only the zero-frequency components of the two Fourier intensities. In addition, phase retrieval in the scanning-illumination system with a probe of unknown modulus can be treated by use of the synthetic procedure, provided the phase of the probe function is a constant or a known distribution. Computer-simulated examples of the reconstruction of one- and two-dimensional complex objects demonstrate that reconstruction is robust.     

Adaptive phase-shifting algorithm for temporal phase evaluation

Most standard temporal-phase-shifting (TPS) algorithms evaluate the phase by computing a windowed Fourier transform (WFT) of the intensity signal at the carrier frequency of the system. However, displacement of the specimen during image acquisition may cause the peak of the transform to shift away from the carrier frequency, leading to phase errors and even unwrapping failure. We present a novel TPS method that searches for the peak of the WFT and evaluates the phase at that frequency instead of at the carrier frequency. The performance of this method is compared with that of standard algorithms by using numerical simulations. Experimental results from high-speed speckle interferometry studies of carbon fiber panels are also presented.     

Statistical estimation method for phase shift in generalized two-step phase-shifting interferometry

Based on the statistical property of the phase and the spatial frequency relationship between the object and reference waves, a simple estimation method for phase shift extraction is proposed for generalized two-step phase-shifting interferometry. In this method, a phase shift constrained within 0–π is firstly worked out with statistics, and then it is corrected as its principal value within the range (0, 2π) except the singular value of π, according to the sign of tangent form of one half of the relative phase shift. Thus, this method provides a possible method to solve the uncertainty of the phase shift existing in some common algorithms. After the determination of the phase shift, a phase can be retrieved easily. The feasibility and the accuracy of the method were verified by numerical simulations and an optical experiment. In addition, this method is applicable to two or more frames off-axis generalized PSI, including slightly off-axis situation.     

Surface shape measurement by phase-shifting digital holography with a wavelength shift

Surface contouring by phase-shifting digital holography is proposed and verified by experiments and numerical simulations. Digital holograms are recorded before and after mode hopping of a laser diode subject to current tuning, and the difference of the reconstructed phases at each wavelength is computed to deliver surface contours of a diffusely reflecting surface. Since normal incidence on the object is employed, the method does not need the removal of the tilt component and is free from the shadowing effect as advantages over the dual-incidence method proposed before by the first author.     

Phase recovering without phase unwrapping in phase-shifting interferometry by cubic and average interpolation

A simple phase estimation employing cubic and average interpolations to solve the oversampling problem in smooth modulated phase images is described. In the context of a general phase-shifting process, without phase-unwrapping, the modulated phase images are employed to recover wavefront shapes with high fringe density. The problem of the phase reconstruction by line integration of its gradient requires a form appropriate to the calculation of partial derivatives, especially when the phase to recover has higher-order aberration values. This is achieved by oversampling the modulated phase images, and many interpolations can be implemented. Here an oversampling procedure based on the analysis of a quadratic cost functional for phase recovery, in a particular case, is proposed.     

Least-squares algorithm for phase-stepping interferometry with an unknown relative step

A pointwise least-squares phase-stepping algorithm with an unknown relative phase step is proposed. In phase-stepping interferometry the recorded temporal intensity sequence is a discrete sinusoidal signal biased by a direct-current component. Its value at a certain time can be predicted from its three past samples by use of a recursive formula. Based on this linear prediction property, an unbiased least-squares estimator is deduced to determine the relative phase step from a sequence of intensity values, and the result is used to evaluate the phase value. The validity and performance of this algorithm are verified by computer simulations.     

Lensless coherent imaging by a deterministic phase retrieval method with an aperture-array filter

Recently we have proposed a noniterative and analytic phase retrieval method using the filter of an aperture array to reconstruct a complex-valued object from a diffraction intensity pattern [Phys. Rev. Lett.98, 223901 (2007)], but this method suffers from the restriction of the far-field condition of two distances between the object and the filter and between the filter and the detector for the intensity measurement. An improved method, which extends the adaptable condition of those distances to the region of Fresnel diffraction, is proposed here. In addition, the procedure for reducing the influence of noises on the phase retrieval is presented, in which the phase information contained in multiple groups of sampling data of a single intensity distribution is utilized. The usefulness of this method is shown in computer-simulated examples of the object reconstructions, including an object with phase vortices.     

Phase retrieval with Fourier-weighted projections

In coherent lensless imaging, the presence of image sidelobes, which arise as a natural consequence of the finite nature of the detector array, was early recognized as a convergence issue for phase retrieval algorithms that rely on an object support constraint. To mitigate the problem of truncated far-field measurement, a controlled analytic continuation by means of an iterative transform algorithm with weighted projections is proposed and tested. This approach avoids the use of sidelobe reduction windows and achieves full-resolution reconstructions.     

Convergence of the Schulz-Snyder phase retrieval algorithm to local minima

The Schulz-Snyder iterative algorithm for phase retrieval attempts to recover a nonnegative function from its autocorrelation by minimizing the I-divergence between a measured autocorrelation and the autocorrelation of the estimated image. We illustrate that the Schulz-Snyder algorithm can become trapped in a local minimum of the I-divergence surface. To show that the estimates found are indeed local minima, sufficient conditions involving the gradient and the Hessian matrix of the I-divergence are given. Then we build a brief proof showing how an estimate that satisfies these conditions is a local minimum. The conditions are used to perform numerical tests determining local minimality of estimates. Along with the tests, related numerical issues are examined, and some interesting phenomena are discussed.     

Lensless multiple-image optical encryption based on improved phase retrieval algorithm

A novel architecture of the optical multiple-image encryption based on the modified Gerchberg-Saxton algorithm (MGSA) by using cascading phase only functions (POFs) in the Fresnel transform (FrT) domain is presented. This proposed method can greatly increase the capacity of the system by avoiding the crosstalk, completely, between the encrypted target images. Each present stage encrypted target image is encoded as to a complex function by using the MGSA with constraining the encrypted target image of the previous stage. Not only the wavelength and position parameters in the FrT domain can be keys to increase system security, the created POFs are also served mutually as the encryption keys to decrypt target image from present stage into next stage in the cascaded scheme. Compared with a prior method [Appl. Opt.48, 2686-2692 (2009)], the main advantages of this proposed encryption system is that it does not need any transformative lenses and this makes it very efficient and easy to implement optically. Simulation results show that this proposed encryption system can successfully achieve the multiple-image encryption via fewer POFs, which is more advantageous in simpler implementation and efficiency than a prior method where each decryption stage requires two POFs to accomplish this task.     

Artifact removal in complex frequency domain optical coherence tomography with an iterative least-squares phase-shifting algorithm

We present what we believe to be a new computational algorithm for complex frequency domain optical coherence tomography that can effectively suppress artifacts that are caused by uncertainty in phase shift due to sample motion and errors in reference phases. The algorithm treats the phase-shifting values as additional unknowns, and we can determine their exact values by analyzing interference fringes using the numerical least-squares technique. A series of simulations and experiments prove that this algorithm can effectively remove strong mirror-image artifacts because it is unaffected by random phase fluctuation.     

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.     

Recursive algorithm for phase retrieval in the fractional fourier transform domain

We first discuss the discrete fractional Fourier transform and present some essential properties. We then propose a recursive algorithm to implement phase retrieval from two intensities in the fractional Fourier transform domain. This approach can significantly simplify computational manipulations and does not need an initial phase estimate compared with conventional iterative algorithms. Simulation results show that this approach can successfully recover the phase from two intensities.     

Double-image encryption by iterative phase retrieval algorithm in fractional Fourier domain

A novel double-image encryption algorithm is proposed, which can simultaneously encrypt two images into a single one as the amplitude of a fractional Fourier transform with two different groups of fractional orders. The two original images can be retrieved independently by fractional Fourier transforms with two different groups of fractional orders, one public phase mask, and two different private phase masks. The proposed approach can enlarge the key space, achieve faster convergence in the iterative process, and avoid cross-talk between the two images in reconstruction. Numerical simulations are presented to verify its validity and efficiency.