Dynamic electronic speckle pattern interferometry (DESPI) phase analyses with temporal Hilbert transform

Transform-based phase retrieval techniques from a single off-axis interferogram

Optical phase retrieval (OPR) methods are important because they are used to obtain the transverse phase profile information of a beam. Interference methods are extensively used to convert the phase information into an intensity pattern, which can then be processed further to retrieve the unknown phase. The most widely used interference method involves the interference of the unknown object beam and a known reference beam with an angle between them. There are several algorithms that retrieve the phase information from such a single off-axis interference pattern. For a particular application, the choice of an algorithm for OPR is very important. Therefore, it is necessary to choose between them, depending on the requirements. Three entirely different noniterative, transform-based algorithms, namely the Fourier transform (FT) method, the continuous wavelet transform (CWT) method, and the Hilbert transform (CWT) method, are explained in detail. A quantitative comparison is made using a combination of rms error and standard structural similarity measure. The advantages of using a standard unwrapping algorithm are also validated using the same combination of comparison metrics. We show that the HT method has a better response with object beam with higher spatial frequency content, but with the penalty of affected noise. The FT method and CWT method have better noise immunity, but have the limitation of the spatial frequency range of the object beam. The different constraints, advantages, and some practical limitations of the methods are discussed with the help of a quantitative phase imaging experiment of monodispersed polymethyl methacrylate beads.

Dynamic parallel phase-shifting electronic speckle pattern interferometer

Methods for measuring variations in diffuse surfaces using electronic speckle pattern interferometry (ESPI) are widely used and well known. In this research, we present an out-of-plane ESPI system coupled to a Michelson configuration to generate simultaneous parallel interferograms with different phase shifts. The system uses circular polarization states to generate parallel phase shifted interferograms. Due to the polarization states, the fringes do not experience a contrast reduction, thus avoiding measurement errors that affect spatial or temporal phase-shifting in interferometry. The basic operating principle of polarization modulation is described, and results that represent the temporal evolution of an aluminum plate are presented. The generation of two simultaneous patterns allows one to track the dynamic performance of the plate.

Exhaustive dithering algorithm for 3D shape reconstruction by fringe projection profilometry

Three-dimensional (3D) shape reconstruction by projection of defocused binary patterns overcomes the nonlinearity introduced by the projector. Current patch-based procedures that generate dithered patterns are time consuming and are affected by the harmonics introduced through the tiling process. To overcome this problem, we propose a novel idea, to the best of our knowledge, to generate dithering patterns using the composition of two-dimensional patches as a stack of one-dimensional arrays obtained through an efficient deterministic approach. This procedure is a one-dimension optimization problem in the intensity domain, employing only a quarter of the fringe pitch. Furthermore, the unwanted distorting harmonics are eliminated using a Hilbert transform method. Both numerical simulations and experimental results verify the effectiveness of the proposal.

Deformation reconstruction by means of surface optimization. Part II: time-resolved electronic speckle pattern interferometry

An analysis of time-resolved electronic speckle pattern interferograms using an optimization algorithm is shown to provide full-field measurements of transient surface deformation. The arrangement uses a continuous-wave laser and high-speed camera to capture speckle images, with the recovery of the time-resolved deformation achieved by spatiotemporal processing using an optimization algorithm. It is shown that the process allows imaging of high-speed non-monotonic out-of-plane displacements with sub-micrometer amplitude. Time-resolved amplitude and phase recovery is demonstrated by analyzing the out-of-plane deformation of harmonic and transient events in a friction membranophone.

Temporal electronic speckle pattern interferometry for real-time in-plane rotation analysis

A temporal electronic speckle pattern interferometry (ESPI) system is proposed for in-plane rotation measurement. The relationship between the rotation angle and the phase change distribution is established and the rotation direction is indicated by the sign of the partial differential of the phase change distribution. Temporal phase modulation is applied in the proposed symmetric illumination ESPI system. The phase is recovered by the temporal intensity analysis method which uses the temporal evolution history of the light intensity. The system can perform dynamic measurements and provide results in off-line real-time. Preliminary experiments were carried out with a continuously rotating target to show the feasibility and the dynamic feature of the temporal ESPI system. At present, the mean absolute error of the experiment is 0.39 arcsec.

Real-time reference-based dynamic phase retrieval algorithm for optical measurement

To study dynamic behaviors of a phenomenon, measuring the evolving field of a specimen/material/structure is required. Optical interferometry, as a full-field, non-contact, and highly sensitive optical measurement technique, has been applied, where the evolving field is represented as dynamic phase distribution. A dynamic phase retrieval algorithm, called least-squares with 3 unknowns (LS3U), which estimates the phase change between each two consecutive patterns by a least-squares fitting method and denoises the phase change by a windowed Fourier filtering (WFF) algorithm, has been shown to be a simple yet effective algorithm. However, LS3U is computationally expensive, restricting its potential application in real-time dynamic phase retrieval systems. In this paper, a real-time LS3U algorithm powered by GPU parallel computing is proposed, with which frame rates of up to 64.5 frames per second (fps) and 131.8 fps are achieved on NVIDIA's GTX 680 and GTX 1080 graphics cards, respectively.

Ultra-fast line-field low coherence holographic elastography using spatial phase shifting

Optical coherence elastography (OCE) is an emerging technique for quantifying tissue biomechanical properties. Generally, OCE relies on point-by-point scanning. However, long acquisition times make point-by-point scanning unfeasible for clinical use. Here we demonstrate a noncontact single shot line-field low coherence holography system utilizing an automatic Hilbert transform analysis based on a spatial phase shifting technique. Spatio-temporal maps of elastic wave propagation were acquired with only one air-pulse excitation and used to quantify wave velocity and sample mechanical properties at a line rate of 200 kHz. Results obtained on phantoms were correlated with data from mechanical testing. Finally, the stiffness of porcine cornea at different intraocular pressures was also quantified in situ.

Comparative analysis of nanometric inspection methods in fringeless speckle pattern interferometry

In digital speckle pattern interferometry, fringeless speckle pattern interferograms are obtained when the object field deformation is insufficient to produce local phase variations higher than 2π. Therefore, the use of the well-known phase recovery algorithms based on fringe processing is not adequate. In this work, distinct algorithms based on the application of a straightforward arccosine function to a filtered interferogram and the correlation of intensity images and implicit smoothing splines are proposed, analyzed, and compared for the fast inspection of nanometric displacement fields, avoiding the acquisition of several images. In addition, three different methods for the normalization of fringeless speckle pattern interferograms are proposed. The Structural Similarity Index is used to assess the performance of the tested methods by means of numerical simulations under different illuminations, signal-to-noise ratios, phase excursions, and mean speckle size conditions. The analysis shows that the phase recovered by the methods based on the arccosine function and correlation are appropriate for a fast inspection solution. The implicit smoothing spline outperforms other methods in almost all conditions.

Flexible phase error compensation based on Hilbert transform in phase shifting profilometry

This paper makes use of Hilbert transform to analyze and compensate the phase error caused by the nonlinear effect in phase shifting profilometry (PSP). The characteristics of the phase error distribution in Hilbert transform domain was analyzed and compared with spatial domain. A simple and flexible phase error compensation method was proposed to directly process the phase-shifting fringe images without any auxiliary conditions or complicated computation. Experimental results demonstrated that the phase error can be reduced by about 80% in three-step PSP, and more than 95% in four or more step PSP, which verified the effectiveness, flexibility, robustness and automation of the proposed phase error compensation method.

Comparison of real-time phase-reconstruction methods in temporal speckle-pattern interferometry

Three real-time methods for object-phase recovery are implemented and compared in temporal speckle-pattern interferometry. Empirical mode and intrinsic time-scale decompositions are used and compared as real-time nonstationary and nonlinear filtering techniques for the extraction of the spatio-temporal evolution of the object phase. The proposed real-time methods avoid the application of the Hilbert transform and improve the accuracy of the measurement by filtering under-modulated pixels using Delaunay triangulation. The performance of the proposed methods is evaluated by comparing phase-recovery accuracy and computation time by means of numerical simulations and experimental data obtained from common and simultaneous π/2 phase-shifting heterodyne interferometry.

Measurement of large discontinuities using single white light interferogram

White light interferometry is a widely used tool to extend the unambiguous measurement range of a monochromatic interferometer. In this work, we discuss Hilbert transformation analysis of a single white light interferogram acquired with a single-chip color CCD camera for step height measurement which lies beyond the unambiguous range of the monochromatic interferometry. The color interferogram is decomposed and phase maps for red, green, and blue components are calculated independently using Hilbert transformation. This procedure makes the measurement faster, simpler, and cost-effective. The usefulness of the technique is demonstrated on micro-sample.

Simplified phase-recovery method in temporal speckle pattern interferometry

A simplified method for object phase recovering is implemented in temporal speckle pattern interferometry when the employed interferometer admits the introduction of a temporal carrier, and the well-known two-beam interferometry equation is verified. The spatiotemporal evolution of the object phase is isolated by modulating the acquired interferometric intensity signals with a known temporal carrier in order to carry out its analysis by using a bivariate empirical mode decomposition framework that avoids the application of the Hilbert transform, which is not suitable for intensity signals with abrupt fluctuations. The advantages and limitations of this technique are analyzed and discussed by comparing computation time and phase recovery capability with well-known phase-retrieval methods by means of numerical simulations and experimental data.

From speckle pattern photography to digital holographic interferometry [Invited]

Speckles are inherently an interference phenomenon produced when an optically rough surface or a turbulent medium introduces some degree of randomness to a reflected or a transmitted electromagnetic field. Speckles are often nuisance in coherent image formation. Speckle patterns are however a useful tool for displacement and deformation as well as vibration and stress analysis. The development of speckle photography to speckle interferometry and digital holographic interferometry is described in this paper.

Dynamic phase retrieval in temporal speckle pattern interferometry using least squares method and windowed Fourier filtering

An algorithm for dynamic phase retrieval in temporal speckle pattern interferometry using least squares method and windowed Fourier filtering is proposed. The least squares method is used to evaluate the phase change between two speckle patterns provided that the phase of either one speckle pattern has been estimated. The windowed Fourier filtering is used to eliminate the noise in the phase change. Based on these two techniques, the proposed algorithm determines the phase of the initial speckle pattern by phase shifting method at first, then the phase of the rest speckle patterns are retrieved by sequentially evaluating the phase changes between every two consecutive speckle patterns. The algorithm solves the problem of speckle decorrelation by refreshing the reference image frame by frame, and also avoids the problem of error accumulation during the reference image refreshing process by the windowed Fourier filtering. Two experimental results are presented to demonstrate the effectiveness and robustness of the proposed algorithm.

Time average vibration fringe analysis using Hilbert transformation

Quantitative phase information from a single interferogram can be obtained using the Hilbert transform (HT). We have applied the HT method for quantitative evaluation of Bessel fringes obtained in time average TV holography. The method requires only one fringe pattern for the extraction of vibration amplitude and reduces the complexity in quantifying the data experienced in the time average reference bias modulation method, which uses multiple fringe frames. The technique is demonstrated for the measurement of out-of-plane vibration amplitude on a small scale specimen using a time average microscopic TV holography system.

Vibration measurement of a miniature component by high-speed image-plane digital holographic microscopy

Measuring deformation of vibrating specimens whose dimensions are in the submillimeter range introduces a number of difficulties using laser interferometry. Normal interferometry is not suitable because of a phase ambiguity problem. In addition, the noise effect is much more serious in the measurement of small objects because a high-magnification lens is used. We present a method for full-field measurement of displacement, velocity, and acceleration of a vibrating miniature object based on image-plane digital holographic microscopy. A miniature cantilever beam is excited by a piezoelectric transducer stage with a sinusoidal configuration. A sequence of digital holograms is captured using a high-speed digital holographic microscope. Windowed Fourier analysis is applied in the spatial and spatiotemporal domains to extract the displacement, velocity and acceleration. The result shows that a combination of image-plane digital holographic microscopy and windowed Fourier analyses can be used to study vibration without encountering a phase ambiguity problem, and one can obtain instantaneous kinematic parameters on each point.

The empirical mode decomposition: a must-have tool in speckle interferometry?

In a wider and wider range of research and engineering activities, there is a growing interest for full-field techniques, featuring nanometric sensitivities, and able to be addressed to dynamic behaviors characterization. Speckle interferometry (SI) techniques are acknowledged as good candidates to tackle this challenge. To get rid of the intrinsic correlation length limitation and simplify the un-wrapping step, a straightforward approach lies in the pixel history analysis. The need of increasing performances in terms of accuracy and computation speed is permanently demanding new efficient processing techniques. We propose in this paper a fast implementation of the Empirical Mode Decomposition (EMD) to put the SI pixel signal in an appropriate shape for accurate phase computation. As one of the best way to perform it, the analytic method based on the Hilbert transform (HT) of the so-transformed signal will then be reviewed. For short evaluation, a zero-crossing technique which exploits directly the extrema finding step of the EMD will be presented. We propose moreover a technique to discard the under-modulated pixels which yield wrong phase evaluation. This work is actually an attempt to elaborate a phase extraction procedure which exploits all the reliable information in 3D, - two space and one time coordinates -, to endeavor to make the most of SI raw data.

Use of dynamic electronic speckle pattern interferometry with the Hilbert transform method to investigate thermal expansion of a joint material

A dynamic electronic speckle pattern interferometry method is applied to investigate thermal expansion of a joint material (ceramic-stainless steel) as a practical industrial object. The speckle interference signal is considered in the temporal domain and the phase is analyzed by the Hilbert transform method. Errors caused by the bias and modulation variations over the phase values are first examined by numerical simulation. Two experiments are performed with in-plane and out-of-plane sensitive systems to study the 3D deformation field thoroughly. The deformation field showed clearly the difference between the thermal expansions of the stainless steel and ceramic. It was also revealed that the boundary of materials and its vicinity suffer very large thermal strain due to the significantly large difference in the linear coefficient of thermal expansions.

Fourier-transform method of data compression and temporal fringe pattern analysis

Temporal fringe pattern analysis is invaluable in studies of transient phenomena but necessitates large data storage for two essential sets of data, i.e., fringe pattern intensity and deformation phase. We describe a compression scheme based on the Fourier-transform method for temporal fringe data storage that permits retrieval of both the intensity and the deformation phase. When the scheme was used with simulated temporal wavefront interferometry intensity fringe patterns, a high compression ratio of 10.77 was achieved, with a significant useful data ratio of 0.859. The average root-mean-square error in phase value restored was a low 0.0015 rad. With simulated temporal speckle interferometry intensity fringe patterns, the important paremeters varied with the modulation cutoff value applied. For a zero modulation cutoff value, the ratio of data points and the compression ratio values obtained were roughly the same as in wavelength interferometry, albeit the average root-mean-square error in the phase value restored was far higher. By increasing the modulation cutoff value we attained significant reduction and increase in the ratio of data points and the compression ratio, respectively, whereas the average root-mean-square error in the restored phase values was reduced only slightly.

Temporal fringe pattern analysis with parallel computing

Temporal fringe pattern analysis is invaluable in transient phenomena studies but necessitates long processing times. Here we describe a parallel computing strategy based on the single-program multiple-data model and hyperthreading processor technology to reduce the execution time. In a two-node cluster workstation configuration we found that execution periods were reduced by 1.6 times when four virtual processors were used. To allow even lower execution times with an increasing number of processors, the time allocated for data transfer, data read, and waiting should be minimized. Parallel computing is found here to present a feasible approach to reduce execution times in temporal fringe pattern analysis.

Data compression for speckle correlation interferometry temporal fringe pattern analysis

Temporal fringe pattern analysis is gaining prominence in speckle correlation interferometry, in particular for transient phenomena studies. This form of analysis, nevertheless, necessitates large data storage. Current compression schemes do not facilitate efficient data retrieval and may even result in important data loss. We describe a novel compression scheme that does not result in crucial data loss and allows for the efficient retrieval of data for temporal fringe analysis. In sample tests with digital speckle interferometry on fringe patterns of a plate and of a cantilever beam subjected to temporal phase and load evolution, respectively, we achieved a compression ratio of 1.6 without filtering out any data from discontinuous and low fringe modulation spatial points. By eliminating 38% of the data from discontinuous and low fringe modulation spatial points, we attained a significant compression ratio of 2.4.