Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform

High space-time bandwidth product imaging in low coherence quantitative phase microscopy

Current low coherence quantitative phase microscopy (LC-QPM) systems suffer from either reduced field of view (FoV) or reduced temporal resolution due to the short temporal coherence (TC) length of the light source. Here, we propose a hybrid, experimental and numerical approach to address this core problem associated with LC-QPM. We demonstrate high spatial resolution and high phase sensitivity in LC-QPM at high temporal resolution. High space-time bandwidth product is achieved by employing incoherent light source for sample illumination in QPM to increase the spatial resolution and single-shot Hilbert spiral transform (HST) based phase recovery algorithm to enhance the temporal resolution without sacrificing spatial resolution during the reconstruction steps. The high spatial phase sensitivity comes by default due to the use of incoherent light source in QPM which has low temporal coherence length and does not generate speckle noise and coherent noise. The spatial resolution achieved by the HST is slightly inferior to the temporal phase-shifting (TPS) method when tested on a specimen but surpasses that of the single-shot Fourier transform (FT) based phase recovery method. Contrary to HST method, FT method requires high density fringes for lossless phase recovery, which is difficult to achieve in LC-QPM over entire FoV. Consequently, integration of HST algorithm with LC-QPM system makes an attractive route. Here, we demonstrate scalable FoV and resolution in single-shot LC-QPM and experimentally corroborate it on a test object and on both live and fixed biological specimen such as MEF, U2OS and human red blood cells (RBCs). LC-QPM system with HST reconstruction offer high-speed single-shot QPM imaging at high phase sensitivity and high spatial resolution enabling us to study sub-cellular dynamic inside U2OS for extended duration (3 h) and observe high-speed (50 fps) dynamics of human RBCs. The experimental results validate the effectiveness of the present approach and will open new avenues in the domain of biomedical imaging in the future.

DeepOrientation: convolutional neural network for fringe pattern orientation map estimation

Fringe pattern based measurement techniques are the state-of-the-art in full-field optical metrology. They are crucial both in macroscale, e.g., fringe projection profilometry, and microscale, e.g., label-free quantitative phase microscopy. Accurate estimation of the local fringe orientation map can significantly facilitate the measurement process in various ways, e.g., fringe filtering (denoising), fringe pattern boundary padding, fringe skeletoning (contouring/following/tracking), local fringe spatial frequency (fringe period) estimation, and fringe pattern phase demodulation. Considering all of that, the accurate, robust, and preferably automatic estimation of local fringe orientation map is of high importance. In this paper we propose a novel numerical solution for local fringe orientation map estimation based on convolutional neural network and deep learning called DeepOrientation. Numerical simulations and experimental results corroborate the effectiveness of the proposed DeepOrientation comparing it with a representative of the classical approach to orientation estimation called combined plane fitting/gradient method. The example proving the effectiveness of DeepOrientation in fringe pattern analysis, which we present in this paper, is the application of DeepOrientation for guiding the phase demodulation process in Hilbert spiral transform. In particular, living HeLa cells quantitative phase imaging outcomes verify the method as an important asset in label-free microscopy.

Deep learning in optical metrology: a review

With the advances in scientific foundations and technological implementations, optical metrology has become versatile problem-solving backbones in manufacturing, fundamental research, and engineering applications, such as quality control, nondestructive testing, experimental mechanics, and biomedicine. In recent years, deep learning, a subfield of machine learning, is emerging as a powerful tool to address problems by learning from data, largely driven by the availability of massive datasets, enhanced computational power, fast data storage, and novel training algorithms for the deep neural network. It is currently promoting increased interests and gaining extensive attention for its utilization in the field of optical metrology. Unlike the traditional "physics-based" approach, deep-learning-enabled optical metrology is a kind of "data-driven" approach, which has already provided numerous alternative solutions to many challenging problems in this field with better performances. In this review, we present an overview of the current status and the latest progress of deep-learning technologies in the field of optical metrology. We first briefly introduce both traditional image-processing algorithms in optical metrology and the basic concepts of deep learning, followed by a comprehensive review of its applications in various optical metrology tasks, such as fringe denoising, phase retrieval, phase unwrapping, subset correlation, and error compensation. The open challenges faced by the current deep-learning approach in optical metrology are then discussed. Finally, the directions for future research are outlined.

Adaptive carrier fringe pattern enhancement for wavelet transform profilometry through modifying intrinsic time-scale decomposition

The uneven background illumination and random noise will degrade the quality of the optical fringe pattern, resulting in reduced accuracy or errors in phase extraction of wavelet transform profilometry (WTP). An adaptive fringe pattern enhancement method is proposed in this paper, which can effectively solve the above problems and improve the robustness of WTP. First, a modified intrinsic time-scale decomposition (MITD) algorithm is used to decompose each row of the fringe pattern adaptively, which can obtain a set of reasonable and pure proper rotation components (PRCs) with a frequency ranging from high to low and a monotonic trend. The MITD algorithm can overcome the mode mixing problem while ensuring the completeness of decomposition. Then, based on the obtained pure PRCs, an innovative background-carrier signal-noise automatic grouping strategy is proposed. Specifically, weighted-permutation entropy (WPE) is adopted to handle noise removal, and fuzzy gray correlation analysis (FGCA) is used to separate the background and carrier signal. Finally, the desired phase information can be easily and accurately extracted from the enhanced carrier signal component by a direct wavelet ridge detection method. Both the simulation and experimental results demonstrate the effectiveness and functionality of the proposed method.

Despeckling Algorithm for Removing Speckle Noise from Ultrasound Images

Ultrasound (US) imaging can examine human bodies of various ages; however, in the process of obtaining a US image, speckle noise is generated. The speckle noise inhibits physicians from accurately examining lesions; thus, a speckle noise removal method is essential technology. To enhance speckle noise elimination, we propose a novel algorithm using the characteristics of speckle noise and filtering methods based on speckle reducing anisotropic diffusion (SRAD) filtering, discrete wavelet transform (DWT) using symmetry characteristics, weighted guided image filtering (WGIF), and gradient domain guided image filtering (GDGIF). The SRAD filter is exploited as a preprocessing filter because it can be directly applied to a medical US image containing speckle noise without a log-compression. The wavelet domain has the advantage of suppressing the additive noise. Therefore, a homomorphic transformation is utilized to convert the multiplicative noise into additive noise. After two-level DWT decomposition is applied, to suppress the residual noise of an SRAD filtered image, GDGIF and WGIF are exploited to reduce noise from seven high-frequency sub-band images and one low-frequency sub-band image, respectively. Finally, a noise-free image is attained through inverse DWT and an exponential transform. The proposed algorithm exhibits excellent speckle noise elimination and edge conservation as compared with conventional denoising methods.

Automatic fringe pattern enhancement using truly adaptive period-guided bidimensional empirical mode decomposition

Fringe patterns encode the information about the result of a measurement performed via widely used optical full-field testing methods, e.g., interferometry, digital holographic microscopy, moiré techniques, structured illumination etc. Affected by the optical setup, changing environment and the sample itself fringe patterns are often corrupted with substantial noise, strong and uneven background illumination and exhibit low contrast. Fringe pattern enhancement, i.e., noise minimization and background term removal, at the pre-processing stage prior to the phase map calculation (for the measurement result decoding) is therefore essential to minimize the jeopardizing effect the mentioned error sources have on the optical measurement outcome. In this contribution we propose an automatic, robust and highly effective fringe pattern enhancement method based on the novel period-guided bidimensional empirical mode decomposition algorithm (PG-BEMD). The spatial distribution of the fringe period is estimated using the novel windowed approach and then serves as an indicator for the truly adaptive decomposition with the filter size locally adjusted to the fringe pattern density. In this way the fringe term is successfully extracted in a single (first) decomposition component alleviating the cumbersome mode mixing phenomenon and greatly simplifying the automatic signal reconstruction. Hence, the fringe term is dissected without the need for modes selection nor summation. The noise removal robustness is ensured employing the block matching 3D filtering of the fringe pattern prior to its decomposition. Performance validation against previously reported modified empirical mode decomposition techniques is provided using numerical simulations and experimental data verifying the versatility and effectiveness of the proposed approach.

Full-field vibration profilometry using time-averaged interference microscopy aided by variational analysis

Full-field vibration testing is indispensable in characterization of micro-electro-mechanical components. Time-averaged interference (TAI) microscopy is a very capable and accurate vibration profilometry technique. It employs natural all-optical multiplexing of required information, i.e., recorded interferogram is amplitude-modulated by the Bessel pattern, which in turn encodes spatial distribution of vibration amplitude in its underlying phase function. We propose a complete end-to-end numerical scheme for efficient and robust vibration amplitude map demodulation based on the variational data-analysis paradigm. First, interferogram is variationally pre-filtered and complex analytic-interferogram is generated, exploiting the Hilbert spiral transform. The amplitude term of analytic-interferogram is accessed for Besselogram, i.e., TAI amplitude modulation distribution. Next, the Besselogram is variationally pre-filtered and complex analytic-Besselogram is calculated applying the Hilbert spiral transform. Finally, the phase term of the analytic-Besselogram is determined, unwrapped and post-filtered to achieve spatial distribution of vibration amplitude. Proposed approach is verified using simulated interferograms and corroborated upon experimental vibration testing. Reported method compares favorably with the reference Hilbert-Huang transform-based method. The improvement was gained by adding two new steps to the calculation path: (1) additional removal of the interferogram's residual background and noise and (2) variational based vibration amplitude map error correction method.

Speckle denoising by variant nonlocal means methods

This study aims to demonstrate the performances of nonlocal means (NLM) and their variant denoising methods, mainly focusing on NLM-shaped adaptive patches and several NLM-reprojection schemes for speckle noise reduction in amplitude and phase images of the digital coherent imaging systems. In the digital coherent imaging systems such as digital speckle pattern interferometry, digital holographic interferometry, etc., the image quality is severely degraded by additive uncorrelated speckle noise, due to the coherent nature of the light source, and therefore limits the development of several applications of these imaging systems in many fields. NLM and its variant denoising methods are employed to denoise the intensity/phase images obtained from these imaging systems, and their effectiveness is evaluated by considering various parameters. The performance comparison of these methods with other existing speckle denoising methods is also presented. The performance of these methods for speckle noise reduction is quantified on the basis of two criteria matrices, namely, the peak-to-signal noise ratio and the image quality index. Based on these criteria matrices, it is observed that these denoising methods have the ability to improve the intensity and phase images favorably in comparison to other speckle denoising techniques, and these methods are more effective and feasible in speckle-noise reduction.

Automatized fringe pattern preprocessing using unsupervised variational image decomposition

Successful single-frame fringe pattern preprocessing comprising high-frequency noise minimization and low-frequency background removal represents often the crucial step of the fringe pattern based full-field optical metrology (i.e., interferometry, moiré, structured light). It directly determines the measurement accuracy. Data-driven decomposition by means of the 2D empirical mode decomposition (EMD) serves the filtering purpose in adaptive and detail-preserving manner. The mode-mixing phenomenon resulting in troublesome automatic grouping of modes into three main fringe pattern components (background, information part and noise) is significantly limiting this process, however. In this paper we are introducing the unsupervised variational image decomposition (uVID) model especially tailored to overcome this preprocessing problem and assure successful sparse three-component fringe pattern decomposition. Comprehensive analysis and detailed studies of accomplished significant advancements ensuring automation, versatility and robustness of the proposed approach are provided. Main advancements include: (1) tailoring the VID calculation scheme to fringe pattern preprocessing purpose by focusing onto accurate fringe extraction with tolerance parameter and custom-made decomposition parameter values; (2) fringe pattern tailored BM3D denoising algorithm with fixed parameter values. Numerical and experimental investigations corroborate that the demonstrated uVID method compares favorably with the reference 2D EMD algorithm and classical VID model. Remarkable range of acceptable local variations of the fringe pattern orientation, period, noise, contrast and background terms is to be highlighted.

Impact of background and modulation on parameter estimation using fractional Fourier transform and its solutions

Analysis of closed-fringe patterns with quadratic phase that are often encountered plays an important role in optical interferometry. But because the frequency spectra of the two exponential signals that compose closed-fringe patterns overlap in the Fourier domain while one is clearly distinct from the other in the fractional Fourier domain, fractional Fourier transform (FRFT) is a useful method for analyzing the images to provide parameter estimation. However, when the fringe pattern has varying background and/or modulation due to non-uniform illumination, parameter estimation accuracy based on FRFT is affected, which lacks theoretical justification. Thus, the impact of varying background and/or modulation on the FRFT is studied with theoretic analysis and presented in this paper. Key factors that contribute to the optimal results are discussed when employing three kinds of fringe normalization methods to eliminate the impact. Here, the fringe pattern is first processed by the normalization technique. Then the cosine-only term is used to estimate parameter by use of the FRFT-based method. Physical quantities are then obtained by parameter estimation. In comparison with our previous method based on FRFT, more accurate results are achieved. The feasibility and applicability of the proposed approach are demonstrated using simulation and experimental results.

Wavelength-tuning point diffraction interferometer resisting inconsistent light intensity and environmental vibration: application to high-precision measurement of a large-aperture spherical surface

Phase-shifting point diffraction interferometers are used to measure large-aperture spherical optical elements because of their high level of measurement accuracy. As the most commonly used phase-shifting method, piezoelectric ceramic transducers are faced with nonlinear problems. Wavelength tuning is typically used to solve the phase-shifting problem; however, it is sensitive to environmental vibrations and light intensity changes during tuning. Therefore, a global optimization iterative algorithm based on interferogram normalization is proposed here. The method is insensitive to noise, defects, and light intensity changes and has a strong ability to resist environmental disturbance. Simulations and experiments are performed to verify the effectiveness of the proposed method.

Speckle noise reduction in digital speckle pattern interferometric fringes by nonlocal means and its related adaptive kernel-based methods

Digital speckle pattern interferometry (DSPI) is widely used in many scientific and industrial applications. Besides its several advantages, one of the basic problems encountered in DSPI is the undesired speckle noise existing in the fringe pattern. In this paper, we demonstrate the performance of nonlocal means (NLM) and its related adaptive kernel-based filtering methods for speckle noise reduction in DSPI fringes. The NLM filter and its related kernel-based filters such as NLM-average, NLM-local polynomial regression, and NLM-shape adaptive patches are implemented first on simulated DSPI fringes, and their performances are quantified on the basis of peak signal-to-noise ratio (PSNR), mean square error (MSE), and quality index (Q). Further, their effectiveness and abilities in reducing speckle noise are compared with other speckle denoising methods. These filtering methods are then employed on experimental DSPI fringes. The obtained results reveal that these filtering methods have the ability to improve the PSNR and Q of the DSPI fringes and provide better visual and quantitative results. It is also observed that the proposed filtering methods preserve the edge information of the DSPI fringes, which is evaluated on the basis of the edge preservation index of the resultant filtered images.

Hilbert-Huang single-shot spatially multiplexed interferometric microscopy

Hilbert-Huang single-shot spatially multiplexed interferometric microscopy (H2S2MIM) is presented as the implementation of a robust, fast, and accurate single-shot phase estimation algorithm with an extremely simple, low-cost, and highly stable way to convert a bright field microscope into a holographic one using partially coherent illumination. Altogether, H2S2MIM adds high-speed (video frame rate) quantitative phase imaging capability to a commercially available nonholographic microscope with improved phase reconstruction (coherence noise reduction). The technique has been validated using a 20×/0.46  NA objective in a regular Olympus BX-60 upright microscope for static, as well as dynamic, samples showing perfect agreement with the results retrieved from a temporal phase-shifting algorithm.

Automatic fringe enhancement with novel bidimensional sinusoids-assisted empirical mode decomposition

Fringe-based optical measurement techniques require reliable fringe analysis methods, where empirical mode decomposition (EMD) is an outstanding one due to its ability of analyzing complex signals and the merit of being data-driven. However, two challenging issues hinder the application of EMD in practical measurement. One is the tricky mode mixing problem (MMP), making the decomposed intrinsic mode functions (IMFs) have equivocal physical meaning; the other is the automatic and accurate extraction of the sinusoidal fringe from the IMFs when unpredictable and unavoidable background and noise exist in real measurements. Accordingly, in this paper, a novel bidimensional sinusoids-assisted EMD (BSEMD) is proposed to decompose a fringe pattern into mono-component bidimensional IMFs (BIMFs), with the MMP solved; properties of the resulted BIMFs are then analyzed to recognize and enhance the useful fringe component. The decomposition and the fringe recognition are integrated and the latter provides a feedback to the former, helping to automatically stop the decomposition to make the algorithm simpler and more reliable. A series of experiments show that the proposed method is accurate, efficient and robust to various fringe patterns even with poor quality, rendering it a potential tool for practical use.

Quantitative phase imaging by single-shot Hilbert-Huang phase microscopy

We propose a novel single-shot Hilbert-Huang transform-based algorithm applied to digital holographic microscopy (DHM) for robust, fast, and accurate single-shot quantitative phase imaging in on-axis and off-axis configurations. Fringe pattern with possible defects and closed fringes are adaptively filtered and accurately phase demodulated using local fringe direction estimation. Experimental validation of the proposed techniques is presented as the DHM study of microbeads and red blood cells phase samples. Obtained results compare very favorably with the Fourier approach (off-axis) and temporal phase shifting (on-axis).

Gram-Schmidt orthonormalization for retrieval of amplitude images under sinusoidal patterns of illumination

Structured illumination using sinusoidal patterns has been used for optical imaging of biological tissues in biomedical research, and of horticultural products in food quality evaluation. Implementation of structured-illumination imaging relies on retrieval of amplitude images, which is conventionally achieved by a phase-shifting technique that requires collecting a minimum of three phase-shifted images. In this study, we have proposed Gram-Schmidt orthonormalization (GSO) to retrieve amplitude component (AC) images using only two phase-shifted images. We have proposed two forms of GSO implementation, and prior to GSO processing, we eliminated the direct component (DC) background by subtracting a DC image we recovered using a spiral phase function (SPF) in the Fourier space. We demonstrated the GSO methods through numerical simulations and application examples of detection of bruise defects in apples by structured-illumination reflectance imaging (SIRI). GSO performed comparably to conventional three-phase-based demodulation. It is simple, fast and effective for amplitude retrieval and requires no prior phase information, which could facilitate fast implementation of structured-illumination imaging.

Demodulation of two-shot fringe patterns with random phase shifts by use of orthogonal polynomials and global optimization

We propose a simple and robust phase demodulation algorithm for two-shot fringe patterns with random phase shifts. Based on a smoothness assumption, the phase to be recovered is decomposed into a linear combination of finite terms of orthogonal polynomials, and the expansion coefficients and the phase shift are exhaustively searched through global optimization. The technique is insensitive to noise or defects, and is capable of retrieving phase from low fringe-number (less than one) or low-frequency interferograms. It can also cope with interferograms with very small phase shifts. The retrieved phase is continuous and no further phase unwrapping process is required. The method is expected to be promising to process interferograms with regular fringes, which are common in optical shop testing. Computer simulation and experimental results are presented to demonstrate the performance of the algorithm.

Single shot fringe pattern phase demodulation using Hilbert-Huang transform aided by the principal component analysis

Hybrid single shot algorithm for accurate phase demodulation of complex fringe patterns is proposed. It employs empirical mode decomposition based adaptive fringe pattern enhancement (i.e., denoising, background removal and amplitude normalization) and subsequent boosted phase demodulation using 2D Hilbert spiral transform aided by the Principal Component Analysis method for novel, correct and accurate local fringe direction map calculation. Robustness to fringe pattern significant noise, uneven background and amplitude modulation as well as local fringe period and shape variations is corroborated by numerical simulations and experiments. Proposed automatic, adaptive, fast and comprehensive fringe analysis solution compares favorably with other previously reported techniques.

Double-exposure optical sectioning structured illumination microscopy based on Hilbert transform reconstruction

Structured illumination microscopy (SIM) with axially optical sectioning capability has found widespread applications in three-dimensional live cell imaging in recent years, since it combines high sensitivity, short image acquisition time, and high spatial resolution. To obtain one sectioned slice, three raw images with a fixed phase-shift, normally 2π/3, are generally required. In this paper, we report a data processing algorithm based on the one-dimensional Hilbert transform, which needs only two raw images with arbitrary phase-shift for each single slice. The proposed algorithm is different from the previous two-dimensional Hilbert spiral transform algorithm in theory. The presented algorithm has the advantages of simpler data processing procedure, faster computation speed and better reconstructed image quality. The validity of the scheme is verified by imaging biological samples in our developed DMD-based LED-illumination SIM system.

Two-shot fringe pattern phase-amplitude demodulation using Gram-Schmidt orthonormalization with Hilbert-Huang pre-filtering

Gram-Schmidt orthonormalization is a very fast and efficient method for the fringe pattern phase demodulation. It requires only two arbitrarily phase-shifted frames. Images are treated as vectors and upon orthogonal projection of one fringe vector onto another the quadrature fringe pattern pair is obtained. Orthonormalization process is very susceptible, however, to noise, uneven background and amplitude modulation fluctuations. The Hilbert-Huang transform based preprocessing is proposed to enhance fringe pattern phase demodulation by filtering out the spurious noise and background illumination and performing fringe normalization. The Gram-Schmidt orthonormalization process error analysis is provided and its filtering-expanded capabilities are corroborated analyzing DSPI fringes and performing amplitude demodulation of Bessel fringes. Synthetic and experimental fringe pattern analyses presented to validate the proposed technique show that it compares favorably with other pre-filtering schemes, i.e., Gaussian filtering and continuous wavelet transform.

Principal-vector-directed fringe-tracking technique

Fringe tracking is one of the most straightforward techniques for analyzing a single fringe pattern. This work presents a principal-vector-directed fringe-tracking technique. It uses Gaussian derivatives for estimating fringe gradients and uses hysteresis thresholding for segmenting singular points, thus improving the principal component analysis method. Using it allows us to estimate the principal vectors of fringes from a pattern with high noise. The fringe-tracking procedure is directed by these principal vectors, so that erroneous results induced by noise and other error-inducing factors are avoided. At the same time, the singular point regions of the fringe pattern are identified automatically. Using them allows us to determine paths through which the "seed" point for each fringe skeleton is easy to find, thus alleviating the computational burden in processing the fringe pattern. The results of a numerical simulation and experiment demonstrate this method to be valid.

Continuous phase estimation from noisy fringe patterns based on the implicit smoothing splines

We introduce the algorithm for the direct phase estimation from the single noisy interferometric pattern. The method, named implicit smoothing spline (ISS), can be regarded as a formal generalization of the smoothing spline interpolation for the case when the interpolated data is given implicitly. We derive the necessary equations, discuss the properties of the method and address its application for the direct estimation of the continuous phase in both classical interferometry and digital speckle pattern interferometry (DSPI). The numerical illustrations of the algorithm performance are provided to corroborate the high quality of the results.

Optically-sectioned two-shot structured illumination microscopy with Hilbert-Huang processing

We introduce a fast, simple, adaptive and experimentally robust method for reconstructing background-rejected optically-sectioned images using two-shot structured illumination microscopy. Our innovative data demodulation method needs two grid-illumination images mutually phase shifted by π (half a grid period) but precise phase displacement between two frames is not required. Upon frames subtraction the input pattern with increased grid modulation is obtained. The first demodulation stage comprises two-dimensional data processing based on the empirical mode decomposition for the object spatial frequency selection (noise reduction and bias term removal). The second stage consists in calculating high contrast image using the two-dimensional spiral Hilbert transform. Our algorithm effectiveness is compared with the results calculated for the same input data using structured-illumination (SIM) and HiLo microscopy methods. The input data were collected for studying highly scattering tissue samples in reflectance mode. Results of our approach compare very favorably with SIM and HiLo techniques.

Denoising and extracting background from fringe patterns using midpoint-based bidimensional empirical mode decomposition

We propose a 2D generalization to the midpoint-based empirical mode decomposition algorithm (MBEMD). Unlike with the regular bidimensional empirical mode decomposition algorithm (BEMD), we do not interpolate the upper and lower envelopes, but rather directly find the mean envelope, utilizing well-defined points between two extrema of different kinds (midpoints). This approach has several advantages, such as improved spectral selectivity and better time performance over the regular BEMD process. The MBEMD algorithm is then applied to the task of the interferometric fringe pattern analysis, to identify its distinct components. This allows separating the oscillatory pattern component, which is of interest, from the background, noise, and possibly other spurious interferometric patterns. Such an enhancement is meant to aid further phase demodulation and reduce its errors. Flexibility of the adaptive method allows for processing correlation fringe patterns met in the digital speckle pattern interferometry as well as the regular interferometric fringe patterns without any special tuning of the algorithm.

Optical correlation algorithm for reconstructing phase skeleton of complex optical fields for solving the phase problem

We propose an optical correlation algorithm illustrating a new general method for reconstructing the phase skeleton of complex optical fields from the measured two-dimensional intensity distribution. The core of the algorithm consists in locating the saddle points of the intensity distribution and connecting such points into nets by the lines of intensity gradient that are closely associated with the equi-phase lines of the field. This algorithm provides a new partial solution to the inverse problem in optics commonly referred to as the phase problem.

Hilbert-Huang processing for single-exposure two-dimensional grating interferometry

Single-shot crossed-type fringe pattern processing and analysis method called Hilbert-Huang grating interferometry (HHGI) is proposed. It consist of three main procedures: (1) crossed pattern is resolved into two fringe families using novel orthogonal empirical mode decomposition approach, (2) separated fringe sets are filtered using modified automatic selective reconstruction aided by enhanced fast empirical mode decomposition and mutual information detrending, and (3) Hilbert spiral transform is employed for fringe phase demodulation. Numerical and experimental studies corroborate the validity, versatility and robustness of the proposed HHGI technique. It can be successfully applied to multiplicative and additive type crossed patterns with sinusoidal and binary orthogonal component structures. Efficient adaptive filtering enables successful fast processing and analysis of complex and defected patterns.

Enhancement strategy based on three-layer filtering for a single fringe pattern

The quality enhancement of a single fringe pattern is a challenging task in speckle interferometry. Fringe patterns suffer greatly from three adverse variations (nonuniform background, speckle noise, and intensity modulation). In this Letter, we propose a three-layer filtering strategy for noisy fringe patterns. The first layer is aimed at high-frequency speckle noises and low-frequency background. The second layer is for remaining speckle noises distributed in the middle-frequency band. The third layer will further implement quality enhancement by a phase-recovery technique. The proposed strategy is quantitatively evaluated by different indexes and verified to be effective through numerous comparative experiments.

Highly contrasted Bessel fringe minima visualization for time-averaged vibration profilometry using Hilbert transform two-frame processing

Time-averaged fringe patterns in vibration testing of MEMS (microelectromechanical systems) are unaffected by carrier displacements. They are additive superimposition type moirés. These features and Hilbert transform vulnerability to additive trend are utilized for visualization of centers of dark Bessel fringes. Two frames with shifted carrier are subtracted for background and noise correction. Two normalized images of this pattern are calculated with slightly different bias levels and subtracted. The method does not require precise phase shifting between two frames, cosinusoidal carrier and linear recording. It enables detecting light power variations and phase shifting nonuniformities. Synthetic and experimental results corroborate the robustness of the method.