A fast-Fourier-transform method of topography and interferometry is proposed. By computer processing of a noncontour type of fringe pattern, automatic discrimination is achieved between elevation and depression of the object or wave-front form, which has not been possible by the fringe-contour-generation techniques. The method has advantages over moire topography and conventional fringe-contour interferometry in both accuracy and sensitivity. Unlike fringe-scanning techniques, the method is easy to apply because it uses no moving components.

Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry

A self-scanned 1024 element photodiode array and minicomputer are used to measure the phase (wavefront) in the interference pattern of an interferometer to lambda/100. The photodiode array samples intensities over a 32 x 32 matrix in the interference pattern as the length of the reference arm is varied piezoelectrically. Using these data the minicomputer synchronously detects the phase at each of the 1024 points by a Fourier series method and displays the wavefront in contour and perspective plot on a storage oscilloscope in less than 1 min (Bruning et al. Paper WE16, OSA Annual Meeting, Oct. 1972). The array of intensities is sampled and averaged many times in a random fashion so that the effects of air turbulence, vibrations, and thermal drifts are minimized. Very significant is the fact that wavefront errors in the interferometer are easily determined and may be automatically subtracted from current or subsequent wavefrots. Various programs supporting the measurement system include software for determining the aperture boundary, sum and difference of wavefronts, removal or insertion of tilt and focus errors, and routines for spatial manipulation of wavefronts. FFT programs transform wavefront data into point spread function and modulus and phase of the optical transfer function of lenses. Display programs plot these functions in contour and perspective. The system has been designed to optimize the collection of data to give higher than usual accuracy in measuring the individual elements and final performance of assembled diffraction limited optical systems, and furthermore, the short loop time of a few minutes makes the system an attractive alternative to constraints imposed by test glasses in the optical shop.

Digital wavefront measuring interferometer for testing optical surfaces and lenses

Abstract 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. 

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Deep learning in optical metrology: a review

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Phase unwrapping is an ill-posed classical problem in many practical applications of significance such as 3D profiling through fringe projection, synthetic aperture radar and magnetic resonance imaging. Conventional phase unwrapping techniques estimate the phase either by integrating through the confined path (referred to as path-following methods) or by minimizing the energy function between the wrapped phase and the approximated true phase (referred to as minimum-norm approaches). However, these conventional methods have some critical challenges like error accumulation and high computational time and often fail under low SNR conditions. To address these problems, this paper proposes a novel deep learning framework for unwrapping the phase and is referred to as “PhaseNet 2.0”. The phase unwrapping problem is formulated as a dense classification problem and a fully convolutional DenseNet based neural network is trained to predict the wrap-count at each pixel from the wrapped phase maps. To train this network, we simulate arbitrary shapes and propose new loss function that integrates the residues by minimizing the difference of gradients and also uses $L_{1}$ loss to overcome class imbalance problem. The proposed method, unlike our previous approach PhaseNet, does not require post-processing, highly robust to noise, accurately unwraps the phase even at the severe noise level of −5 dB, and can unwrap the phase maps even at relatively high dynamic ranges. Simulation results from the proposed framework are compared with different classes of existing phase unwrapping methods for varying SNR values and discontinuity, and these evaluations demonstrate the advantages of the proposed framework. We also demonstrate the generality of the proposed method on 3D reconstruction of synthetic CAD models that have diverse structures and finer geometric variations. Finally, the proposed method is applied to real-data for 3D profiling of objects using fringe projection technique and digital holographic interferometry. The proposed framework achieves significant improvements over existing methods while being highly efficient with interactive frame-rates on modern GPUs.

PhaseNet 2.0: Phase Unwrapping of Noisy Data Based on Deep Learning Approach

The interferometry technique is commonly used to obtain the phase information of an object in optical metrology. The obtained wrapped phase is subject to a 2π ambiguity. To remove the ambiguity and obtain the correct phase, phase unwrapping is essential. Conventional phase unwrapping approaches are time-consuming and noise sensitive. To address those issues, we propose a new approach, where we transfer the task of phase unwrapping into a multi-class classification problem and introduce an efficient segmentation network to identify classes. Moreover, a noise-to-noise denoised network is integrated to preprocess noisy wrapped phase. We have demonstrated the proposed method with simulated data and in a real interferometric system.

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Phase unwrapping in optical metrology via denoised and convolutional segmentation networks

We propose a deep learning computational ghost imaging (CGI) scheme to achieve sub-Nyquist and high-quality image reconstruction. Unlike the second-order-correlation CGI and compressive-sensing CGI, which use lots of illumination patterns and a one-dimensional (1-D) light intensity sequence (LIS) for image reconstruction, a deep neural network (DAttNet) is proposed to restore the target image only using the 1-D LIS. The DAttNet is trained with simulation data and retrieves the target image from experimental data. The experimental results indicate that the proposed scheme can provide high-quality images with a sub-Nyquist sampling ratio and performs better than the conventional and compressive-sensing CGI methods in sub-Nyquist sampling ratio conditions (e.g., 5.45%). The proposed scheme has potential practical applications in underwater, real-time and dynamic CGI.

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Sub-Nyquist computational ghost imaging with deep learning.

Deep-learning denoising computational ghost imaging

To accurately obtain the phase distribution of an optical surface under test, the accurate phase extraction algorithm is essential. To overcome the phase shift error, a random two-step phase shifting algorithm, which can be used in the fluctuating and non-uniform background intensity and modulation amplitude, Lissajous ellipse fitting, and least squares iterative phase shifting algorithm (LEFL pre-filtering interferograms are not necessary, but they can get relatively accurate phase distribution and unknown phase shift value. The simulation and experiment verify the correctness and feasibility of the LEF & LSI PSA.

Random two-step phase shifting interferometry based on Lissajous ellipse fitting and least squares technologies.

A snapshot multi-wavelength interference microscope is proposed for high-speed measurement of large vertical range discontinuous microstructures and surface roughness. A polarization CMOS camera with a linear micro-polarizer array and Bayer filter accomplishes snapshot multi-wavelength phase-shifting measurement. Four interferograms with &#x1D70B;/2 phase shift are captured at each wavelength for phase measurement, the 2&#x1D70B; ambiguities are removed by using two or three wavelengths.

Xiaobo Tian

Papers

Phase unwrapping in optical metrology via denoised and convolutional segmentation networks

Sub-Nyquist computational ghost imaging with deep learning.

Deep-learning denoising computational ghost imaging

Random two-step phase shifting interferometry based on Lissajous ellipse fitting and least squares technologies.

Snapshot multi-wavelength interference microscope.