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

Phase retrieval of two random phase-shifting interferograms using Zernike coefficient extraction network

This paper proposes a deep learning method for phase retrieval from two interferograms. The proposed method converts phase retrieval into the Zernike coefficient extraction problem, which can achieve Zernike coefficient extraction from two interferograms with random phase shifts. After knowing Zernike coefficients, the phase distribution can be retrieved using Zernike polynomials. The pre-filtering and phase unwrapping process are not required using the proposed method. The simulated data are analyzed, and the root mean square (RMS) of phase error reaches 0.01 λ. The effectiveness of the method is verified by the measured data.

Fringe analysis: single-shot or two-frames? Quantitative phase imaging answers

Conditions of the digital recording of the fringe pattern determine the phase reconstruction procedure, which in turn directly shapes the final accuracy and throughput of the full-field (non-scanning) optical measurement technique and defines the system capabilities. In this way, the fringe pattern analysis plays a crucial role in the ubiquitous optical measurements and thus is under constant development focused on high temporal/spatial resolution. It is especially valuable in the quantitative phase imaging technology, which emerged in the high-contrast label-free biomedical microscopy. In this paper, I apply recently blossomed two-frame phase-shifting techniques to the QPI and merge them with advanced adaptive interferogram pre-filtering algorithms. Next, I comprehensively test such frameworks against classical and adaptive single-shot methods applied for phase reconstruction in dynamic QPI enabling highest phase time-space-bandwidth product. The presented study systematically tackles important question: what is the gain, if any, in QPI realized by recording two phase-shifted interferograms? Counterintuitively, the results show that single-shot demodulation exhibited higher phase reconstruction accuracy than two-frame phase-shifting methods in low and medium interferogram signal-to-noise ratio regimes. Thus, the single-shot approach is promoted due to not only high temporal resolution but also larger phase-information throughput. Additionally, in the majority of scenarios, the best option is to shift the paradigm and employ two-frame pre-filtering rather than two-frame phase retrieval. Experimental fringe analysis in QPI of LSEC/RWPE cell lines successfully corroborated all novel numerical findings. Hence, the presented numerical-experimental research advances the important field of fringe analysis solutions for optical full-field measurement methods with widespread bio-engineering applications.

Robust weighted principal components analysis demodulation algorithm for phase-shifting interferometry

We present an asynchronous phase-shifting demodulation approach based on the principal component analysis demodulation method that is robust to typical problems as turbulence, vibrations, and temporal instabilities of the optical setup. The method brings together a two-step and a phase-shifting asynchronous demodulation method to share their benefits while reducing their intrinsic limitations. Thus, the proposed approach is based on a two-fold process. First, the modulating phase is estimated from a two-step demodulation approach. Second, this information is used to compute weights to each phase-shifted pattern of the interferogram sequence, which are used in a novel weighted principal component demodulation approach. The proposed technique has been tested with simulated and real interferograms affected by turbulence and vibrations providing very satisfactory results in challenging cases.

Single-shot detection of 8 unique monochrome fringe patterns representing 4 distinct directions via multispectral fringe projection profilometry

Spatial resolution in three-dimensional fringe projection profilometry is determined in large part by the number and spacing of fringes projected onto an object. Due to the intensity-based nature of fringe projection profilometry, fringe patterns must be generated in succession, which is time-consuming. As a result, the surface features of highly dynamic objects are difficult to measure. Here, we introduce multispectral fringe projection profilometry, a novel method that utilizes multispectral illumination to project a multispectral fringe pattern onto an object combined with a multispectral camera to detect the deformation of the fringe patterns due to the object. The multispectral camera enables the detection of 8 unique monochrome fringe patterns representing 4 distinct directions in a single snapshot. Furthermore, for each direction, the camera detects two π-phase shifted fringe patterns. Each pair of fringe patterns can be differenced to generate a differential fringe pattern that corrects for illumination offsets and mitigates the effects of glare from highly reflective surfaces. The new multispectral method solves many practical problems related to conventional fringe projection profilometry and doubles the effective spatial resolution. The method is suitable for high-quality fast 3D profilometry at video frame rates.

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.

Parallel phase shifting radial shear interferometry with complex fringes and unknown phase shift

We present an interferometric method to analyze transparent samples using complex fringes generated by a parallel phase shifting radial shear interferometer using two coupled interferometers. Parallel interferograms are generated using two interferometers: the first one generates the polarized base pattern, and the second system is used to generate parallel interferograms allowing the adjustment of the x-y positions of the parallel interferograms. To obtain the optical phase map, parallel phase shift is generated by collocating polarizing filters at the output of the system; the polarizers are placed at arbitrary angles since they do not require adjustment because of the phase-recovery algorithm. The optical phase was processed using a two-step algorithm based on a modified Gram-Schmidt orthogonalization method. Such an algorithm has the advantage of not being iterative and is robust to amplitude modulation. The proposed method reduces the number of captures needed in phase-shifting interferometry. We applied the developed system to examine static and dynamics phase objects.

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.

Two-frame fringe pattern phase demodulation using Gram-Schmidt orthonormalization with least squares method

Gram-Schmidt (GS) orthogonal normalization is a fast and efficient two-frame fringe phase demodulation method. However, the precision of the GS method is limited due to the residual background terms and noise, as well as several approximation operations in the GS method. To obtain a phase map with higher accuracy, we propose an algorithm combining GS orthogonal normalization and least squares iterative (LSI) phase shift algorithm (GS&LSI). In our method, the phase was first obtained using GS method, and then a refinement operation using LSI was adopted to get the final wrapped phase map. Because of the LSI process, the demodulation result is greatly improved in many cases. Simulation and experimental result are presented to validate the potential of the proposed method.

Self-adaptive terahertz spectroscopy from atmospheric vapor based on Hilbert-Huang transform

Absorption lines of atmospheric vapor commonly appear in terahertz (THz) spectra measured in a humid air environment. However, these effects are generally undesirable because they may mask critical spectroscopic information. Here, a self-adaptive method is demonstrated for effectively identifying and eliminating atmospheric vapor noise from THz spectra of an all-fiber THz system with the Hilbert-Huang transform. The THz signal was decomposed into eight components in different time scales called the intrinsic mode functions and the interference of atmospheric vapor was accurately isolated. A series of experiments confirmed the effectiveness and strong self-adaptiveness of the proposed system in vapor noise elimination.

Simultaneous 2-step phase-shifting interferometry with a full-band interferometric signal being recovered

2-step phase-shifting interferometry (PSI) is an important technology for phase retrieval due to its outstanding performance in balancing detector bandwidth, temporal resolution, and quantitative quality. The most significant difficulty in this technology should ascribe to the distortion of low spatial frequencies in the retrieved interference signals, which is caused by the imperfect background intensity estimation. In this Letter, we overcome this key difficulty by iteratively recovering the lost spectrum of interferometric signals during spatial filtering and realize truly full space-bandwidth utilization in 2-step PSI. We also use this method to reduce the necessary number of spatially matched cameras in the quadrature simultaneous phase-shifting interferometer to significantly simplify its optical setup.

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.

Fast and accurate wavefront reconstruction in two-frame phase-shifting interferometry with unknown phase step

A fast and accurate wavefront reconstruction method for two-frame phase-shifting interferometry is proposed. The unknown phase step between the two interferograms is estimated directly by solving a quartic polynomial equation, and then the phase map is readily reconstructed after obtaining the phase step. The whole phase reconstruction process is nearly analytical and thus very fast and easy to realize. Good performance of the proposed method is demonstrated by reconstructing the phase maps from simulated and real fringes along with comparisons to several existing well-established algorithms.

Self-calibration phase-shifting algorithm with interferograms containing very few fringes based on Fourier domain estimation

Based on Fourier domain estimation, a novel self-calibration phase-shifting algorithm, named Mid-Band Spatial Spectrum Matching (MSSM), is proposed to achieve phase retrieval from a small amount of phase-shifting interferograms containing very few fringes (defined as ultra-sparse fringe pattern(USFP)), which is still a difficult problem for optical interferometry. Both simulation and experimental results demonstrate that the proposed MSSM algorithm can accurately and rapidly achieve the phase distribution encoded in USFP while other current self-calibration algorithms fail, and this will supply a powerful tool to extend the application of phase-shifting interferometry.

Phase retrieval in two-shot phase-shifting interferometry based on phase shift estimation in a local mask

Fringe analysis in two-shot phase-shifting interferometry is important but meets challenges due to a limited number of images, corrupting noise, and background modulation. Here we propose an effective algorithm for phase retrieval from two interferograms with unknown phase shifts. The algorithm first evaluates the phase shift in a local mask through phase fitting and global optimization and then computes a full-field phase map using an arctangent function. Since the phase shift evaluation is performed within a local mask, the algorithm is fast compared with conventional optimization-based algorithms and typically needs tens of seconds to complete the processing. Computer simulation and experimental results show that the proposed algorithm has excellent performance compared with state-of-the-art algorithms. A complete software package of the algorithm in MATLAB is available at http://two-shot.sourceforge.io/.

Evaluation of adaptively enhanced two-shot fringe pattern phase and amplitude demodulation methods

Phase-shifting interferometry is a standard tool in optical metrology. Most frequently, it needs three or more interferograms to solve the system of fringe equations for phase or amplitude retrieval, which limits its time resolution. Recently, the topic of two-shot, arbitrary-phase-step fringe pattern phase and amplitude demodulation has been flourishing and attracting attention with several novel and interesting methods being proposed. In this work, we evaluate six up-to-date two-shot phase-shifting methods analyzing their main error sources and proposing efficient ways to minimize their influence by adaptive filtering using the Hilbert-Huang transform.

Random phase-shifting algorithm by constructing orthogonal phase-shifting fringe patterns

The intensity distribution of fringe patterns becomes nonsinusoidal in real testing environments. Thus, the performance of existing phase shift extraction algorithms, which usually compute the desired phase shift by arccosine function or arcsine function, may be affected. In the presented paper, we report an arctangent-function-based technique to solve this disturbance. First, two orthogonal fringe patterns are constructed through subtraction and addition of two background-removed images. Second, the unequal amplitude between two new fringe patterns is eliminated using a normalization process. Third, the phase shift is determined by computing the norms of the two new images. The proposed method is fast and can be implemented easily in many applications. We verify the algorithm performance and robustness using both simulated and experimental data, indicating the high accuracy of the presented method.

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.

Three-frame self-calibration phase shift algorithm using the Gram-Schmidt orthonormalization approach

Affected by the height dependent effects, the phase-shifting shadow moiré can only be implemented in an approximate way. In the technique, a fixed phase step around π/2 rad between two adjacent frames is usually introduced by a grating translation in its own plane. So the method is not flexible in some situations. Additionally, because the shadow moiré fringes have a complex intensity distribution, computing the introduced phase shift from the existing arccosine function or arcsine function-based phase shift extraction algorithm always exhibits instability. To solve it, we developed a Gram-Schmidt orthonormalization approach based on a three-frame self-calibration phase-shifting algorithm with equal but unknown phase steps. The proposed method using the arctangent function is fast and can be implemented robustly in many applications. We also do optical experiments to demonstrate the correction of the proposed method by referring to the result of the conventional five-step phase-shifting shadow moiré. The results show the correctness of the proposed method.

Two-frame phase-shifting interferometry for testing optical surfaces

Standard phase-shifting interferometry (PSI) generally requires collecting at least three phase-shifted interferograms to extract the physical quantity being measured. Here, we propose the application of a simple two-frame PSI for the testing of a range of optical surfaces, including flats, spheres, and aspheres. The two-frame PSI extracts modulated phase from two randomly phase-shifted interferograms using a Gram-Schmidt algorithm, and can work in either null testing or non-null testing modes. Since only two interferograms are used for phase demodulation and the phase shift amount can be random, requirements on environmental conditions and phase shifter calibration are greatly relaxed. Experimental results of three different mirrors suggest that the two-frame PSI can achieve comparable measurement precision with conventional multi-frame PSI, but has faster data acquisition speed and less stringent hardware requirements. The proposed two-frame PSI expands the flexibility of PSI and holds great potential in many applications.

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.

Two-frame tilt-shift error estimation and phase demodulation algorithm

We present an algorithm capable of performing fringe pattern phase demodulation from two frames with unknown, linearly nonuniform phase shift, i.e., under presence of the tilt-shift error. The method consists of intensity-based filtration of the tilt-shift component and subsequent two steps of a nonlinear error functional minimization. We verify the algorithm performance and robustness using both simulated and experimental data, indicating high accuracy of the presented method and its good numerical properties. Both small and large tilts can be treated. The Letter is complemented by numerical codes available online in Wielgus, "Two-frame tilt-shift error estimation and phase demodulation algorithm" (2015).