Fourier fringe analysis and its application to metrology of extreme physical phenomena: a review [Invited]

Compact Linnik-type hyperspectral quantitative phase microscope for advanced classification of cellular components

Hyperspectral quantitative phase microscopy (HS-QPM) involves the acquisition of phase images across narrow spectral bands, which enables wavelength-dependent study of different biological samples. In the present work, a compact Linnik-type HS-QPM system is developed to reduce the instability and complexity associated with conventional HS-QPM techniques. The use of a single objective lens for both reference and sample arms makes the setup compact. The capabilities of the system are demonstrated by evaluating the HS phase map of both industrial and biological specimens. Phase maps of exfoliated cheek cells at different wavelengths are stacked to form a HS phase cube, adding spectral dimensionality to spatial phase distribution. Analysis of wavelength response of different cellular components are performed using principal component analysis to identify dominant spectral features present in the HS phase dataset. Findings of the study emphasize on the efficiency and effectiveness of HS-QPM for advancing cellular characterization in biomedical research and clinical applications.

Stereo-phase rectification for metric profilometry with two calibrated cameras and one uncalibrated projector

Fringe projection profilometry requires calibrating both cameras and projectors for metric measurements. Cameras are relatively simple to calibrate, but projectors require more sophisticated procedures. In this paper, a fringe projection profilometer with two calibrated cameras and one uncalibrated projector is developed for metric measurements. A phase rectification method, which is crucial for stereo matching, is designed by minimizing the perspective distortion. Also, a simple method for point matching using stereo rectified phase maps is proposed. The principles of metric profilometry using the proposed rectification method are introduced. The developed system is evaluated experimentally by the metric measurement of three-dimensional objects. The obtained results confirm a high accuracy of metric measurement and versatility in the design of fringe projection profilometers with uncalibrated projectors.

Grating-based in-line geometric-phase-shifting incoherent digital holographic system toward 3D videography

Incoherent digital holography (IDH) with a sequential phase-shifting method enables high-definition 3D imaging under incoherent lights. However, sequential recording of multiple holograms renders IDH impractical for 3D videography. In this study, we propose grating-based in-line geometric-phase-shifting IDH. Our method divides orthogonal circularly polarized lights into four copies with a fabricated phase grating and subsequently creates self-interference holograms with geometric phases introduced by a segmented linear polarizer. This enables single-shot recording of holograms without the need for a specially designed image sensor, such as a polarization-sensitive sensor. Moreover, the achievable spatial resolution is higher than that of off-axis methods. As a proof-of-principle experiment, we demonstrated snapshot and video recording of 3D reflective objects using our IDH method. The results confirmed the feasibility of the proposed method.

Multimodal biomicroscopic system for the characterization of cells with high spatial phase sensitivity and sub-pixel accuracy

Multimodal analysis is highly advantageous for various biomedical applications including cancer and brain studies. Simultaneous measurement of quantitative phase with sub-pixel accuracy and fluorescence image is difficult to achieve in single measurement. Conventionally, off-axis interferograms are analyzed using the Fourier-transform method which limits the accuracy of the phase maps by pixel size, and usually the location of the carrier peak is in sub-pixel. We report a multimodal microscopic system consisting of high-resolution (HR) quantitative phase interferometer to retrieve sub-pixel accuracy in phase imaging and an oblique-illumination-based fluorescence imaging system which decouples the excited light from emitted signal light to avoid saturation of the camera, both integrated into a single unit. Here, highly resolved phase maps are obtained using a two-step process. First, using a speckle-free illumination which offers high spatial phase sensitivity. Second, using a hamming window for accurate estimation of original signal frequency information and HR discrete Fourier transform (DFT) which offers sub-pixel accuracy in phase measurements. HR-DFT has computational load of OABβ , where A×B is the size of the interferogram and β is the upsampling factor, making system computationally more robust and efficient compared to zero-padded FFT. The experiment is conducted on MG63 osteosarcoma and human mesenchymal stem cells (hMSCs) and their quantitative parameters are extracted with significantly improved accuracy. The average phase for MG63 cells and hMSCs, for nucleus is obtained to be 8.02 rad ± 0.80 rad and 4.29 rad ± 0.43 rad, respectively, and for cytoplasm is obtained to be 2.63 rad ± 0.96 rad and 1.73 rad ± 0.57 rad, respectively.

A super-grayscale and real-time computer-generated Moiré profilometry using video grating projection

By using the time-division multiplexing characteristics of the projector and the integral exposure characteristics of the charge coupled device (CCD) camera, a super-grayscale and real-time computer-generated Moiré profilometry based on video grating projection is proposed. The traditional digital static grating is of 256-grayscale at most. If an expected super-grayscale grating with a maximum grayscale of 766 is designed and divided into three 256-grayscale fringe patterns with balanced grayscale as far as possible, they can be synthesized into a repeated playing video grating instead of the traditional static grating. When the video grating is projected onto the measured object, as long as the exposure time is set to three times the refresh cycle of the video grating, the super-grayscale deformed patterns in the 766-grayscale can be captured with a 10-bit CCD camera, so that the deformed patterns are realistic. The digital error in computer-generated Moiré profilometry is effectively reduced. In addition, this method can expand the linear range of the deformed pattern by 20% in computer Moiré profilometry. Therefore, the proposed method has the perspectives of high accuracy and real-time measurement. Theoretical analysis and experimental results demonstrate the validity and capability of the proposed method.

High-speed 3D shape measurement using a rotary mechanical projector

In this paper, a fast rotary mechanical projector (RMP) is designed and manufactured for high-speed 3D shape measurement. Compared with the common high-speed projectors, RMP has a good performance in high-speed projection, which can obtain high quality projected fringes with shorter camera exposure time by using the error diffusion binary coding method and chrome plating technology. The magnitude, acceptability of systemic projection error is analyzed and quantified in detail. For the quantified error, the probability distribution function (PDF) algorithm is introduced to correct the error. Corrected projection error is reduced to more than one third of the original error. Subsequently, a monocular measurement system composed of the RMP and a single camera is constructed. The combination of the RMP device and PDF algorithm ensure the accuracy of a corresponding 3D shape measurement system. Experiments have demonstrated that the proposed solution has a good performance for the 3D measurement of high-speed scenes.

Depth acquisition in single-pixel imaging with multiplexed illumination

Single-pixel imaging (SPI) has drawn wide attentions due to its high signal-to-noise ratio and wide working spectrum, providing a feasible solution when array sensors are expensive or not available. In the conventional SPI, the target's depth information is lost in the acquisition process due to the 3D-to-1D projection. In this work, we report an efficient depth acquisition method that enables the existing SPI systems to obtain reflectance and depth information without any additional hardware. The technique employs a multiplexed illumination strategy that contains both random and sinusoidal codes, which simultaneously encode the target's spatial and depth information into the single measurement sequence. In the reconstruction phase, we build a convolutional neural network to decode both spatial and depth information from the 1D measurements. Compared to the conventional scene acquisition method, the end-to-end deep-learning reconstruction reduces both sampling ratio (30%) and computational complexity (two orders of magnitude). Both simulations and experiments validate the method's effectiveness and high efficiency for additional depth acquisition in single-pixel imaging without additional hardware.

Enhanced Fourier-transform method for high-density fringe analysis by iterative spectrum narrowing

We propose a model-based fringe analysis technique that enables the Fourier transform method to analyze dense fringe patterns with large phase variations. The conventional Fourier-transform method has a limited dynamic range of measurable phase because the Fourier spectra broadened by large phase variations cannot be separated by the spatial carrier frequency. Our model-based iterative technique effectively narrows the broad spectrum and reduces phase errors. Results of simulations and experiments are presented that demonstrate the validity of the proposed spectrum-narrowing technique for high-density fringe patterns.

State-of-the-art active optical techniques for three-dimensional surface metrology: a review [Invited]

This paper reviews recent developments of non-contact three-dimensional (3D) surface metrology using an active structured optical probe. We focus primarily on those active non-contact 3D surface measurement techniques that could be applicable to the manufacturing industry. We discuss principles of each technology, and its advantageous characteristics as well as limitations. Towards the end, we discuss our perspectives on the current technological challenges in designing and implementing these methods in practical applications.

Fast focus-shifter based on a unimorph deformable mirror

On-the-fly remote laser processing plays an increasingly important role in modern fabrication techniques. These processes require guiding of the focus of a laser beam along the contours of the workpiece in three dimensions. State-of-the-art galvanometer scanners already provide highly dynamic and precise transverse x-y beam steering. However, longitudinal focus shifting ("z-shifting") relying on conventional optics is restricted to a bandwidth of a few hundred Hz. We have developed and manufactured a fast piezo-based z-shifting mirror with diffraction-limited surface fidelity providing a focus shift of Δz>60mm with an actuation rate of 2 kHz.

Real-Time Dynamic 3D Shape Reconstruction with SWIR InGaAs Camera

In this paper, a real-time, dynamic three-dimensional (3D) shape reconstruction scheme based on the Fourier-transform profilometry (FTP) method is achieved with a short-wave infrared (SWIR) indium gallium arsenide (InGaAs) camera for monitoring applications in low illumination environments. A SWIR 3D shape reconstruction system is built for generating and acquiring the SWIR two-dimensional (2D) fringe pattern of the target. The depth information of the target is reconstructed by employing an improved FTP method, which has the advantages of high reconstruction accuracy and speed. The maximum error in depth for static 3D shape reconstruction is 1.15 mm for a plastic model with a maximum depth of 36 mm. Meanwhile, a real-time 3D shape reconstruction with a frame rate of 25 Hz can be realized by this system, which has great application prospects in real-time dynamic 3D shape reconstruction, such as low illumination monitoring. In addition, for real-time dynamic 3D shape reconstruction, without considering the edge areas, the maximum error in depth among all frames is 1.42 mm for a hemisphere with a depth of 35 mm, and the maximum error of the average of all frames in depth is 0.52 mm.

Enhanced phase-coding method for three-dimensional shape measurement with half-period codeword

The phase-coding method has been widely used for 3D shape measurement, which uses sinusoidal phase-shifting patterns to recover the wrapped phase and the stair phase-coding patterns to determine the fringe order. However, due to random noises and image blurring, the fringe order is always misaligned with the wrapped phase, which will lead to fringe order errors. This paper presents an enhanced phase-coding method to address this misalignment problem by using half-period codewords, in which each codeword is aligned to the half-period of the sinusoidal patterns. Then, two complementary fringe orders with half-period dislocation can be calculated, which can effectively eliminate the fringe order errors. To extend the coding range of stair phase, this paper further develops a computational scheme based on the geometric constraint method. Simulations and experiments have been carried out, and their results confirm that the enhanced method can reliably recover the 3D shape of the measured objects.

Three-dimensional measurement based on optimized circular fringe projection technique

Circular fringe projection profilometry (CFPP) is a recently proposed optical three-dimensional (3D) measurement technique. Theoretically, its optimal measurement accuracy should precede that of the conventional fringe projection profilometry. In practice, the measurement accuracy is impacted by many factors, and much research remains to be done in order to make CFPP reach its optimal precision. One of the dominant factors is the zero-phase point. For the usage of the cotangent function, error near the zero-phase point will be significantly amplified. This makes the overall measurement accuracy very low for CFPP with coaxial layout. To address this critical issue, CFPP with off-axis layout (called OCFPP for simplicity) is presented in this paper. The core theory of OCFPP is briefly introduced. The zero-phase point detection problem coming with OCFPP is explained. Then, two methods, one based on a two-dimensional ruler and the other based on plane constraint, are proposed to solve this additional problem. Simulation and experiments validate the effectiveness of the proposed zero-phase point detection methods, and convince the advantage of OCFPP. This paper contributes to distinctly improving the 3D measurement capability of CFPP, and lays an indispensible foundation for its practical application.

Modified Gray-Level Coding Method for Absolute Phase Retrieval

Fringe projection systems have been widely applied in three-dimensional (3D) shape measurements. One of the important issues is how to retrieve the absolute phase. This paper presents a modified gray-level coding method for absolute phase retrieval. Specifically, two groups of fringe patterns are projected onto the measured objects, including three phase-shift patterns for the wrapped phase, and three n-ary gray-level (nGL) patterns for the fringe order. Compared with the binary gray-level (bGL) method which just uses two intensity values, the nGL method can generate many more unique codewords with multiple intensity values. With assistance from the average intensity and modulation of phase-shift patterns, the intensities of nGL patterns are normalized to deal with ambient light and surface contrast. To reduce the codeword detection errors caused by camera/projector defocus, nGL patterns are designed as n-ary gray-code (nGC) patterns to ensure that at most, one code changes at each point. Experiments verify the robustness and effectiveness of the proposed method to measure isolated objects with complex surfaces.

Accurate efficient carrier estimation for single-shot digital holographic imaging

Off-axis digital holograms are commonly processed using the Fourier transform method. This method requires estimation of the carrier frequency peak in the two-dimensional Fourier domain. The accuracy of peak location information using the standard fast Fourier transform (FFT) routines is limited by the pixel size, thus giving rise to a residual carrier component in the reconstructed image. In this Letter, we use an efficient two-step process for cross-term peak location to sub-pixel accuracy where the standard FFT-based peak finding procedure is followed by a high-resolution discrete Fourier transform (DFT) calculation at the nominal peak location. It is shown that the residual tilt artifacts due to a small change in the magnitude of the carrier frequency or due to the rotation of the fringe pattern are uniformly removed with this method. For a digital hologram of size M×N pixels, the high-resolution DFT calculation with an up-sampling factor of α requires an additional computational effort of O(MNα) which is marginal when α≪M,N and, thus, provides a practical single-shot method for accurate demodulation of digital holograms.

High-accuracy off-axis wavefront reconstruction from noisy data: local least square with multiple adaptive windows

A variational algorithm to object wavefront reconstruction from noisy intensity observations is developed for the off-axis holography scenario with imaging in the acquisition plane. The algorithm is based on the local least square technique proposed in paper [J. Opt. Soc. Am. A21, 367 (2004)]. First, multiple reconstructions of the wavefront are produced for various size and various directional windows applied for localization of estimation. At the second stage, a special statistical rule is applied in order to select the best window size estimate for each pixel of the image and for each of the directional windows. At the third final stage the estimates of the different directions obtained for each pixel are aggregated in the final one. Simulation experiments and real data processing prove that the developed algorithm demonstrate the performance of the extraordinary quality and accuracy for both the phase and amplitude of the object wavefront.

Single-shot absolute 3D shape measurement with Fourier transform profilometry

Fourier transform profilometry (FTP) is one of the frequently adopted three-dimensional (3D) shape measurement methods due to its ability to recover single-shot 3D shapes, yet it is challenging to retrieve the absolute phase map solely from one single grayscale fringe image. This paper presents a computational framework that overcomes this limitation of FTP with digital fringe projection (DFP). By using geometric constraints, an absolute phase map can be retrieved point-by-point from one single grayscale fringe image. Experiments demonstrate the success of our proposed framework with single-shot absolute 3D shape measurement capability.

Theory and algorithms of an efficient fringe analysis technology for automatic measurement applications

Some advances in fringe analysis technology for phase computing are presented. A full scheme for phase evaluation, applicable to automatic applications, is proposed. The proposal consists of: a fringe-pattern normalization method, Fourier fringe-normalized analysis, generalized phase-shifting processing for inhomogeneous nonlinear phase shifts and spatiotemporal visibility, and a phase-unwrapping method by a rounding-least-squares approach. The theoretical principles of each algorithm are given. Numerical examples and an experimental evaluation are presented.

Birefringent phase demodulator: application to wave plate characterization

The scope of this work is to present a phase demodulator that enables the recovery of temporal phase information contained in the phase difference between two signals with different polarizations. This demodulator is a polarization interferometer that may consist only of a uniaxial crystal slab and a polarizer sheet. The phase shift between two orthogonal components of the electric field is translated into space by means of birefringent crystals, which act as demodulators or phase analyzers with great robustness. The experimental scheme utilized is based on a simple conoscopic interference setup. Each portion of the space in which the interference pattern is projected contains not only the unknown temporal phase we want to recover, but also a phase shift due to the uniaxial crystal itself. The underlying idea is developing simultaneous phase shifting with uniaxial crystals. Thus, different phase recovery techniques can be applied in order to maximize their ability to track high-speed signals. Depending on the characteristics of the fringe pattern, it will permit phase recovery via different classical procedures. In order to prove the demodulator under different experimental and signal processing schemes, we employed it for wave plate characterization. The results obtained not only allow some wave plate features such as axes determination and retardance to be characterized, but also prove the working principle and capabilities of the demodulator.

Carrier peak isolation from single interferogram using spectrum shift technique

This paper presents a new method to obtain a wrapped phase distribution from a single interferogram with a spatial carrier modulation. The Fourier transform of the interferogram has three peaks: one is a dc peak around the origin in the Fourier domain, and the other two are carrier peaks that have information of phase modulation by an object placed in the interferometer. Since the wrapped phase can be evaluated by one of the two carrier peaks, the dc peak and the adjoint peak that is the other peak of two carrier peaks should be removed by filters. The proposed filtering process consists of two stages: dc peak filtering and adjoint peak filtering. A spectrum shift filter based on symmetrical characteristics of the spectrum is applied in both stages as a basic filter that can remove most of the undesired spectrum. An additional two filters are applied to remove the remaining spectrum. The new method can automatically isolate the carrier peak, even when the boundary of peaks is not very clear. Numerical evaluations of simulation data and experimental data demonstrate that the proposed method can successfully isolate the carrier peak.

Single shot interferometric method for measuring the nonlinear refractive index

In this paper we are introducing a new single shot method for measuring the nonlinear refractive index of materials in a simple interferometric pump-probe configuration. The theoretical model proposed for extracting the nonlinear refractive index from the experimental fringe pattern and the experimental configuration are presented and discussed. The results obtained by this method are in good agreement with that obtained on the same sample using the conventional Z-scan method.