Two-step interferometry by a regularized optical flow algorithm

Polarizing white light interferometry for phase measurements using two simultaneous interferograms

Our research introduces a design for a polarization phase-shifting white light interferometric system (PPS-WLIS) that operates in a transmissive mode for measuring the slope phase of transparent objects. It comprises a cyclic path interferometer (lateral shear interferometer) coupled with a multiplexing Michelson interferometer. The system uses polarization to produce two parallel interferograms with polarization modulated with relative shifts simultaneously. To determine the optical phase, we used a two-step algorithm for phase demodulation that does not necessitate precise phase shifts, making the system more straightforward to operate. As a result, we could observe variations in the object associated with optical phase changes. Furthermore, our method simplifies the phase-shift interferometry process by requiring only one capture, making it an effective way to examine objects at dynamic events. As an illustration, we demonstrated the temperature measurement generated by a section of a candle flame.

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.

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.

Fast and robust two-frame random phase-shifting interferometry without pre-filtering

To obtain higher phase accuracy with less computation time in phase-shifting interferometry, a random phase-shifting algorithm based on principal component analysis and least squares iteration (PCA&LSI) is proposed. The algorithm does not require pre-filtering, and only requires two-frame phase-shifted interferograms and less computation time to obtain a relatively accurate phase distribution. This method can still extract the phase with high precision when there are few fringes in the interferogram. Moreover, it eliminates the limitation that the PCA algorithm needs more than three frames of interferograms with uniform phase shift distribution to accurately extract the phase. Numerical simulations and experiments confirm that the method is suitable for complex situations with different fluctuations in background intensity and modulation amplitude. And it can still achieve accurate phase extraction compared with other methods under different noise conditions.

Characterizations and Use of Recycled Optical Components for Polarizing Phase-Shifting Interferometry Applications

In this research, we report using optical components such as cubic beam splitters, lenses, diffraction gratings, and mirrors from broken, obsolete, or disused electronic devices to implement a simultaneous polarization-based phase-shifting interferometric system. The system is composed of a polarized Mach–Zehnder interferometer (PMZI) which generates a sample pattern coupled to a 4f imaging system with a diffraction grating placed on its Fourier plane. Such a diffractive element replicates the pattern generated by the PMZI, and each replica is centered and modulated by each diffraction order generated by the grating. The corresponding individual phase shifts are controlled by placing linear polarizers with known angles in front of each replica. Experimental results are presented using several phase samples such as an oil drop, a pseudoscorpion claw, a microarthropod, and red blood cells. In addition, a comparison of the retrieved phase was conducted by employing two different phase demodulation algorithms.

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.

Local computational methods to improve the interpretability and analysis of cryo-EM maps

Cryo-electron microscopy (cryo-EM) maps usually show heterogeneous distributions of B-factors and electron density occupancies and are typically B-factor sharpened to improve their contrast and interpretability at high-resolutions. However, 'over-sharpening' due to the application of a single global B-factor can distort processed maps causing connected densities to appear broken and disconnected. This issue limits the interpretability of cryo-EM maps, i.e. ab initio modelling. In this work, we propose 1) approaches to enhance high-resolution features of cryo-EM maps, while preventing map distortions and 2) methods to obtain local B-factors and electron density occupancy maps. These algorithms have as common link the use of the spiral phase transformation and are called LocSpiral, LocBSharpen, LocBFactor and LocOccupancy. Our results, which include improved maps of recent SARS-CoV-2 structures, show that our methods can improve the interpretability and analysis of obtained reconstructions.

Fast-iterative blind phase-shifting digital holographic microscopy using two images

Digital holographic microscopy (DHM) has consolidated as a tool for diagnosis and measuring in life sciences, thanks to its capability to perform quantitative phase imaging. The reduction of the acquisition and computation time has driven the development of diverse reconstruction methodologies using a single-shot and two-frame approach. Methods based on the Fourier transform, the Hilbert transform, and the phase derivative are counted among the most utilized. The sensitivity of those methods is highly dependent on the compensation of the phase step, which requires the accurate knowledge of the phase shift between the two recorded holograms. Here, an alternative fast-iterative method based on the demodulation of the different components of the recorded interferograms is presented. The novelties of the proposed two-frame approach are: minimum number of images, since it requires 2 recorded holograms; a minimum phase error of the order of 0.005% independently of the phase step ranging from 0 to 180 deg.; a maximum correlation coefficient equal to 1 between the phase and the retrieved phase image; and, finally, a reduced processing time compared with the previous three-frame approach. Experimental results demonstrate the goodness and feasibility of the proposed technique.

Random two-frame interferometry based on deep learning

A two-frame phase-shifting interferometric wavefront reconstruction method based on deep learning is proposed. By learning from a large number of simulation data based on a physical model, the wrapped phase can be calculated accurately from two interferograms with an unknown phase step. The phase step can be any value excluding the integral multiples of π and the size of interferograms can be flexible. This method does not need a pre-filtering to subtract the direct-current term, but only needs a simple normalization. Comparing with other two-frame methods in both simulations and experiments, the proposed method can achieve better performance.

Sub-nanometer height sensitivity by phase shifting interference microscopy under environmental fluctuations

Phase shifting interferometric (PSI) techniques are among the most sensitive phase measurement methods. Owing to its high sensitivity, any minute phase change caused due to environmental instability results into, inaccurate phase measurement. Consequently, a well calibrated piezo electric transducer (PZT) and highly-stable environment is mandatory for measuring accurate phase map using PSI implementation. Here, we present an inverse approach, which can retrieve phase maps of the samples with negligible errors under environmental fluctuations. The method is implemented by recording a video of continuous temporally phase shifted interferograms and phase shifts were calculated between all the data frames using Fourier transform algorithm with a high accuracy ≤ 5.5 × 10^-4 π rad. To demonstrate the robustness of the proposed method, a manual translation of the stage was employed to introduce continuous temporal phase shift between data frames. The developed algorithm is first verified by performing quantitative phase imaging of optical waveguide and red blood cells using uncalibrated PZT under the influence of vibrations/air turbulence and compared with the well calibrated PZT results. Furthermore, we demonstrated the potential of the proposed approach by acquiring the quantitative phase imaging of an optical waveguide with a rib height of only 2 nm and liver sinusoidal endothelial cells (LSECs). By using 12-bit CMOS camera the height of shallow rib waveguide is measured with a height sensitivity of 4 Å without using PZT and in presence of environmental fluctuations.vn.

Coordinate calculation for direct shape measurement based on optical flow

A quite novel surface profilometry is proposed, which adopts a single optical grating projection setup with a small projection angle. The height distribution of the measured surface is retrieved by calculating the coordinates of the intersection between the projecting ray and the observing sight line. While the position of the observing point in the deformed fringe pattern can be detected by fringe optical flow. The relationship between optical flow and the height distribution of the tested surface is established. Simulations and some primary experiment results are completed to prove that the proposed method is feasible to measure a complex surface. The main advantage of the proposed method is obviously that the height distribution of the measured surface can be obtained directly without phase-to-height transformation.

Random phase shifting shadow moiré using a one-dimensional minimizer

The introduction of a random phase-shifting technique into a shadow moiré system, where an equal and known (or unknown) phase step is used to demodulate the phase of interest, is beneficial for the improvement of measurement accuracy. However, in spite of recent advances in optical metrology phase-shifting techniques, simultaneously estimating unequal and unknown phase shifts from three random phase-shifting fringe patterns remains a significant challenge. This paper presents a one-dimensional minimizer-based technique to address this ill-posed problem of phase demodulation from random phase-shifting patterns. In this method, two new sets of connected fringe patterns, without background illumination, are constructed through normalizing the secondary fringe patterns. Then, a generalized phase-shifting algorithm is developed by utilizing the character of the modulation factor's standard deviation distribution. Both numerical simulations and optical experiments are performed to demonstrate the high accuracy and robustness of the proposed method.

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.

Three-step random phase retrieval approach based on difference map normalization and diamond diagonal vector normalization

To overcome the phase shift error in phase shifting interferometry, a three-step random phase retrieval approach based on difference map normalization and diamond diagonal vector normalization (DN&DDVN) is proposed. It does not need pre-filtering for the interferograms and can obtain relatively accurate phase distribution with a simple process and less computational time. This simulation and experiment verify the correctness and feasibility of DN&DDVN.

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.

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.

Field of view scanning based quantitative interferometric microscopic cytometers for cellular imaging and analysis

Microimaging is of great significance in the biological and medical fields, since it can realize observations acting as important references for cellular research and disease diagnosis. However, traditional microscopy only offers qualitative sample contours; moreover, it is difficult to reach large-amount sample observations limited by the fixed field of view (FoV). To realize massive cellular measurements quantitatively, three designed quantitative interferometric microscopic cytometers based on the FoV scanning are introduced and compared in details in this article. These devices not only retrieve the quantitative sample phase distributions in the extended FoV, but also provide the detailed information of massive cells, such as cellular volume, area, and roundness. Considering their capabilities as quantitative imaging and large-amount sampling, it is believed that these quantitative interferometric microscopic cytometers (QIMCs) can be potentially adopted in high-throughput cell imaging and statistical analysis for both the biological and medical applications.

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.

Three-dimensional displacement measurement based on the combination of digital image correlation and optical flow

This paper proposes what we believe is a novel simultaneous three-dimensional (3D) displacement measurement technique based on a combination of digital image correlation (DIC) and optical flow (OF). In this method, both the in-plane and out-of-plane displacements can be accurately extracted from only two continuous interferograms. DIC estimates the velocity field between two consecutive frames. According to the optical flow constrained equation, we can then obtain the whole-field out-of-plane displacement map by the estimations of the in-plane displacement components and the local frequency of the original image. The proposed method's operation is simple compared with other phase demodulation methods. Moreover, the new method works perfectly in areas with dense fringes. To verify its effectiveness, we applied a new algorithm to simulated and experimental interferograms. The results of our simulation and experiment show that the new method can demodulate the out-of-plane component of the deformation-phase from the visible in-plane velocity field without an unwrapping process. Further, the proposed algorithm provides a new approach to measure whole-field 3D displacement and dynamic deformation.

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.

Dynamic phase measurement based on spatial carrier-frequency phase-shifting method

Combining spatial carrier-frequency phase-shifting (SCPS) technique and Fourier transform method, from one-frame spatial carrier-frequency interferogram (SCFI), a novel phase retrieval method is proposed and applied to dynamic phase measurement. First, using the SCPS technique, four-frame phase-shifting sub-interferograms can be constructed from one-frame SCFI. Second, using Fourier transform method, the accurate phase-shifts of four sub-interferograms can be extracted rapidly, so there is no requirement of calibration for the carrier-frequency in advance compared to most existing SCPS methods. Third, the wrapped phase can be retrieved with the least-squares algorithm through using the above phase-shifts. Finally, the phase variations of a water droplet evaporation and a Jurkat cell apoptosis induced by a drug are presented with the proposed method. Both the simulation and experimental results demonstrate that in addition to maintaining high accuracy of the SCPS method, the proposed method reveals more rapid processing speed of phase retrieval, and this will greatly facilitate its application in dynamic phase measurement.

Precise phase retrieval under harsh conditions by constructing new connected interferograms

To date, no phase-shifting method can accurately retrieve the phase map from a small set of noisy interferograms with low phase-shifts. In this Letter, we develop a novel approach to resolve this limitation under such harsh conditions. The proposed new method is based on constructing a set of connected interferograms by means of simple subtraction and addition operations, in which all the subset of interferograms have the same phase-shift interval of π/2. According to this characteristic, this set of connected interferograms can be processed with conventional phase retrieval methods as PCA or AIA obtaining accurate results. The reduction in the RMS errors after using our method reaches as high as 93.7% and 89.3% respectively comparing with conventional PCA and AIA methods under harsh conditions. Both simulation and experiment results demonstrate that the new proposed method provides an effective way, with high precision and robustness against noise, for phase retrieval.

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.

Parallel-quadrature on-axis phase-shifting common-path interferometer using a polarizing beam splitter

A common-path parallel-quadrature on-axis phase-shifting interferometry using a modified Michelson configuration with a polarizing cube beam splitter is proposed for quantitative phase measurement. The frequency spectrum of the circularly polarized object beam is split into two beams using a beam splitter. One beam is converted to a 45° linearly polarized beam to act as the object beam, and the other beam is low-filtered by a pinhole mirror to act as the reference beam. Two interferograms with quadrature phase shift can be simultaneously captured by combining the 45° linearly polarized object beam with the circularly polarized reference beam through a 45° tilted polarizing cube beam splitter, and the phase of a specimen can be then retrieved through a two-step phase-shifting algorithm. Experiments are carried out to demonstrate the validity and stability of the proposed method.

Phase shifting interferometry from two normalized interferograms with random tilt phase-shift

We propose a novel phase shifting interferometry from two normalized interferograms with random tilt phase-shift. The determination of tilt phase-shift is performed by extracting the tilted phase-shift plane from the phase difference of two normalized interferograms, and with the calculated tilt phase-shift value the phase distribution can be retrieved from the two normalized frames. By analyzing the distribution of phase difference and utilizing special points fitting method, the tilted phase-shift plane is extracted in three different cases, which relate to different magnitudes of tilts. Proposed method has been applied to simulations and experiments successfully and the satisfactory results manifest that proposed method is of high accuracy and high speed compared with the three step iterative method. Additionally, both open and closed fringe can be analyzed with proposed method. What's more, it cannot only eliminate the small tilt-shift error caused by slight vibration in phase-shifting interferometry, but also detect the large tilt phase-shift in phase-tilting interferometry. Thus, it will relaxes the requirements on the accuracy of phase shifter, and the costly phase shifter may even be useless by applying proposed method in high amplitude vibrated circumstance to achieve high-precision analysis.

Visual measurement of the evaporation process of a sessile droplet by dual-channel simultaneous phase-shifting interferometry

To perform the visual measurement of the evaporation process of a sessile droplet, a dual-channel simultaneous phase-shifting interferometry (DCSPSI) method is proposed. Based on polarization components to simultaneously generate a pair of orthogonal interferograms with the phase shifts of π/2, the real-time phase of a dynamic process can be retrieved with two-step phase-shifting algorithm. Using this proposed DCSPSI system, the transient mass (TM) of the evaporation process of a sessile droplet with different initial mass were presented through measuring the real-time 3D shape of a droplet. Moreover, the mass flux density (MFD) of the evaporating droplet and its regional distribution were also calculated and analyzed. The experimental results show that the proposed DCSPSI will supply a visual, accurate, noncontact, nondestructive, global tool for the real-time multi-parameter measurement of the droplet evaporation.

Advanced principal component analysis method for phase reconstruction

Focus on the phase reconstruction from three phase-shifting interferograms with unknown phase shifts, an advanced principal component analysis method is proposed. First, use a simple subtraction operation among interferograms, two intensity difference images are obtained easily. Second, set the center region of the data of intensity difference images to zero, and then construct a covariance matrix to obtain a transformation matrix. Third, two principal components of interferograms can be determined by the Hotelling transform and then phase can be calculated from the two normalized principal components by an arctangent function. By means of the simulation calculation and the experimental research, it is proved that the phase with high precision can be obtained rapidly by the proposed algorithm.

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.

Single-wavelength phase retrieval method from simultaneous multi-wavelength in-line phase-shifting interferograms

From a sequence of simultaneous multi-wavelength phase-shifting interferograms (SMWPSIs), a novel single-wavelength phase retrieval method based on the least-squares iterative algorithm is proposed and utilized in dual-wavelength interferometry. Firstly, only one time phase-shifting procedure implements the phase shifts of all illumination wavelengths simultaneously, and then the accurate wrapped phases of each single-wavelength can be respectively retrieved from SMWPSIs by the least-squares iterative operation, so the phase of synthetic wavelength can be obtained by the subtraction easily. Using the proposed method, both the simulation and the experimental results demonstrate that the optical setup is simpler; the requirements for the displacement of the phase-shifting device and the number of the captured interferograms are smaller compared to the traditional phase-shifting multi-wavelength interferometry or off-axis multi-wavelength interferometry. Even in the case that the phase-shifts are unknown, the wrapped phases and the phase-shifts of each single-wavelength can be obtained by the proposed method.

Simultaneous phase-shifting dual-wavelength interferometry based on two-step demodulation algorithm

A simultaneous phase-shifting dual-wavelength interferometry based on two-step demodulation algorithm is proposed in this Letter. First, two lasers with different wavelengths go through the same inline phase-shifting interference system simultaneously, and a sequence of five frames of simultaneous phase-shifting dual-wavelength interferograms (SPSDWIs) with the special phase shifts are captured by a monochrome CCD. Subsequently, using the subtraction between the first SPSDWI and the other SPSDWI, each wavelength of two frames of single-wavelength interference images (SWIIs) without the background can be achieved. Finally, using two-step demodulation algorithm, the wrapped phase of each single-wavelength can be determined easily and quickly with high accuracy.

Dynamic phase imaging of microscopic measurements using parallel interferograms generated from a cyclic shear interferometer

We present a technique which allows us to generate two parallel interferograms with phase shifts of π/2 using a Cyclic Shear Interferometer (CSI) and a polarizing splitter. Because of the use of a CSI, we obtain the derivative phase data map directly, due to its configuration, it is immune to vibrations because the reference wavefront and the object wavefront have a common path; the shearing interferometer is insensitive to temperature and vibration. To obtain the optical phase data map, two interferograms are generated by collocating a polarizing device at the output of the CSI. The optical phase was processed using a Vargas-Quiroga algorithm. Related experimental results obtained for dynamic microscopic transparent samples are presented.

Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts

From three interferograms with unknown phase shifts, an innovative phase retrieval approach based on the normalized difference maps is proposed. Using the subtraction operation between interferograms, two difference maps without background can be achieved. To eliminate the amplitude inequality of difference maps, normalization process is employed so that two normalized difference maps are obtained. Finally, combining two normalized difference maps and two-step phase retrieval algorithm, the measured phase with high precision can be retrieved rapidly. Comparing with the conventional two-step phase retrieval algorithm with high-pass filtering, the accuracy and processing time of the proposed approach are greatly improved. Importantly, when the phase shift is close to π, almost all two-step algorithms become invalid, but the proposed approach still performs well. That is, the proposed normalized difference maps approach is suitable for the phase retrieval with arbitrary phase shifts.

Shack-Hartmann spot dislocation map determination using an optical flow method

We present a robust, dense, and accurate Shack-Hartmann spot dislocation map determination method based on a regularized optical flow algorithm that does not require obtaining the spot centroids. The method is capable to measure in presence of strong noise, background illumination and spot modulating signals, which are typical limiting factors of traditional centroid detection algorithms. Moreover, the proposed approach is able to face cases where some of the reference beam spots have not a corresponding one in the distorted Hartmann diagram, and it can expand the dynamic range of the Shack-Hartmann sensor unwrapping the obtained dense dislocation maps. We have tested the algorithm with both simulations and experimental data obtaining satisfactory results. A complete MATLAB package that can reproduce all the results can be downloaded from [http://goo.gl/XbZVOr].

Phase extraction from interferograms with unknown tilt phase shifts based on a regularized optical flow method

A novel method is presented to extract phase distribution from phase-shifted interferograms with unknown tilt phase shifts. The proposed method can estimate the tilt phase shift between two temporal phase-shifted interferograms with high accuracy, by extending the regularized optical flow method with the spatial image processing and frequency estimation technology. With all the estimated tilt phase shifts, the phase component encoded in the interferograms can be extracted by the least-squares method. Both simulation and experimental results have fully proved the feasibility of the proposed method. Particularly, a flat-based diffractive optical element with quasi-continuous surface is tested by the proposed method with introduction of considerably large tilt phase shift amounts (i.e., the highest estimated tilt phase shift amount between two consecutive frame reaches 6.18λ). The phase extraction result is in good agreement with that of Zygo's MetroPro software under steady-state testing conditions, and the residual difference between them is discussed. In comparison with the previous methods, the proposed method not only has relatively little restrictions on the amounts or orientations of the tilt phase shifts, but also works well with interferograms including open and closed fringes in any combination.

Calculation method for a quadrature phase-shifting interferometer and its applications

A calculation method for a quadrature phase-shifting interferometer is presented, and its applications to specular and speckle interferometers and digital holography are described. Two sets of quadrature phase-shifted interferograms are acquired, and the calculation method proposed gives the phase distribution of the interferograms. The principle of the calculation method with error analysis and experimental results for specular and speckle interferometers and digital holography are also given.

Two-step phase demodulation algorithm based on the extreme value of interference

Using the maximum and the minimum of interference, a novel two-step phase demodulation algorithm is proposed to perform the phase extraction in phase-shifting interferometry. By means of the simulation calculation and the experimental research, it is proved that both the measured phase and the phase shift with high precision can be obtained in the proposed algorithm.

Multiplicative phase-shifting interferometry using optical flow

Fringe patterns with a multiplicative phase shift among them appear in experimental techniques as photoelasticity and RGB shadow moiré, among others. These patterns cannot be processed using standard phase-shifting demodulation techniques. In this work, we propose to use a multiframe regularized optical flow algorithm to obtain the interesting modulating phase. The proposed technique has been applied to simulated and experimental interferograms obtaining satisfactory results.

Two-step demodulation based on the Gram-Schmidt orthonormalization method

This Letter presents an efficient, fast, and straightforward two-step demodulating method based on a Gram-Schmidt (GS) orthonormalization approach. The phase-shift value has not to be known and can take any value inside the range (0,2π), excluding the singular case, where it corresponds to π. The proposed method is based on determining an orthonormalized interferogram basis from the two supplied interferograms using the GS method. We have applied the proposed method to simulated and experimental interferograms, obtaining satisfactory results. A complete MATLAB software package is provided at http://goo.gl/IZKF3.