There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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

Quantification and characterization of microporosity by image processing, geometric measurement and statistical methods: Application on SEM images of clay materials

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. 

/pdf/deep-learning-in-optical-metrology-a-review-2ggtgtrp.pdf

Deep learning in optical metrology: a review

https://eprints.lib.hokudai.ac.jp/dspace/bitstream/2115/42844/1/ACS47-1-2_65-71.pdf

Analysis of microstructural images of dry and water-saturated compacted bentonite samples observed with X-ray micro CT

In conventional least square (LS) regressions for nonlinear problems, it is not easy to obtain analytical derivatives with respect to target parameters that comprise a set of normal equations. Even if the derivatives can be obtained analytically or numerically, one must take care to choose the correct initial values for the iterative procedure of solving an equation, because some undesired, locally optimized solutions may also satisfy the normal equation. In the application of genetic algorithms (GAs) for nonlinear LS, it is not necessary to use normal equations, and a GA is also capable of avoiding localized optima. However, convergence of population and reliability of solutions depends on the initial domain of parameters, similarly to the choice of initial values in the above mentioned method using the normal equation. To overcome this disadvantage of applying GAs for nonlinear LS, we propose to use an adaptive domain method (ADM) in which the parameter domain can change dynamically by using several real-coded GAs with short lifetimes. Through an example problem, we demonstrate improvements in terms of both the convergence and the reliability by ADM. A further merit in the proposed method is that it does not require any specialized knowledge about GAs or their tuning. Therefore, the nonlinear LS by ADM with GAs are accessible to general scientists for various applications in many fields

/pdf/nonlinear-least-square-regression-by-adaptive-domain-method-32c80k3tt7.pdf

Nonlinear Least Square Regression by Adaptive Domain Method With Multiple Genetic Algorithms

Phase unwrapping for a noisy image suffers from many singular points. Singularity-spreading methods are useful for the noisy image to regularize the singularity. However, the methods have a drawback of distorting phase distribution in a regular area that contains no singular points. When the singular points are confined in some local areas, the regular region is not distorted. This paper proposes a new phase unwrapping algorithm that uses a localized compensator obtained by clustering and by solving Poisson’s equation for the localized areas. The numerical results demonstrate that the proposed method can improve the accuracy compared with other singularity-spreading methods.

/pdf/phase-unwrapping-for-noisy-phase-map-using-localized-2gfwcts5gb.pdf

Phase unwrapping for noisy phase map using localized compensator

In the process of phase unwrapping for an image obtained by an interferometer or in-line holography, noisy image data may pose difficulties. Traditional phase unwrapping algorithms used to estimate a two-dimensional phase distribution include much estimation error, due to the effect of singular points. This paper introduces an accurate phase-unwrapping algorithm based on three techniques: a rotational compensator, unconstrained singular point positioning, and virtual singular points. The new algorithm can confine the effect of singularities to the local region around each singular point. The phase-unwrapped result demonstrates that accuracy is improved, compared with past methods based on the least-squares approach.

/pdf/phase-unwrapping-for-noisy-phase-maps-using-rotational-5dde1ms75a.pdf

Phase unwrapping for noisy phase maps using rotational compensator with virtual singular points

This paper presents a gradient field representation using an analytical regularization of a hypersingular boundary integral equation for a two-dimensional time harmonic wave equation called the Helmholtz equation The regularization is based on cancelation of the hypersingularity by considering properties of hypersingular elements that are adjacent to a singular node Advantages to this regularization include applicability to evaluate corner nodes, no limitation for element size, and reduced computational cost compared to other methods To demonstrate capability and accuracy, regularization is estimated for a problem about plane wave propagation As a result, it is found that even at a corner node the most significant error in the proposed method is due to truncation error of non-singular elements in discretization, and error from hypersingular elements is negligibly small

Satoshi Tomioka

Papers

Analysis of microstructural images of dry and water-saturated compacted bentonite samples observed with X-ray micro CT

Nonlinear Least Square Regression by Adaptive Domain Method With Multiple Genetic Algorithms

Phase unwrapping for noisy phase map using localized compensator

Phase unwrapping for noisy phase maps using rotational compensator with virtual singular points

Analytical regularization of hypersingular integral for Helmholtz equation in boundary element method