The aim of this paper is to compare the Ronchi test and the Moiré deflectometry applied to a new pattern. To do so, we used two Ronchi patterns (typical Ronchigrams) produced by two identical Ronchi rulings angularly displaced and placed close to the curvature radius of a mirror. The result obtained by superposing both Ronchigrams is our new pattern, a Moiré pattern of the Ronchigram (Moiré-Ronchigram). The Zernike aberration polynomial coefficients of the tridimensional wavefront for both mirrors are obtained as a result. We also compared the Zernike coefficients obtained for each of the mentioned techniques and we found that the results with less dispersion are those where Moiré deflectometry was applied. Finally, as a confidence test for applying and testing the Moiré-Ronchigram, we compare our results with Open Fringe, FringeXP, and APEX, using the root–mean–square and standard deviation values.

In this paper, we use the irradiance transport equation and the Fourier transform-based experimental solution given by Ichikawa–Lohmann–Takeda. We analyze experimental factors such as the digital filter, the introduced error for the rotation and period of the Ronchi ruling, and a new method is demonstrated for the measurement of 3D wavefront information.

We tested different optical elements placed in three different positions by applying the irradiance transport equation (ITE), obtaining the wavefront [ W ( x , y ) ] and aberration surface [ AS ( r , θ ) ]. The existing noise in the captures I as well as in the W and A S were analyzed applying several filters: first a filter based on Legendre polynomials (LP), generating the most probable points increasing the data resolution; second, a filter based on a 50 deg 2D-LP was used as a multilineal fit (multiple linear regression); and third, an ideal bandpass filter in the Fourier space after inducing a periodicity using Ronchi simulated masks with periods in x , y , x y was used to perform data scanning (similar to the four-step phase-shifting method). Signal-to-noise ratio values were obtained for each proposed filter along with the most probable image free from noise, determined from a linear combination of the original data and the applied filters.

In this paper, we give a solution to the irradiance transport equation (ITE) using the two-dimensional (2D) Legendre polynomials (LPs) and an interpolator (i-LP) based on the LP. In the first place, we analyze the experimental data; subsequently, we proceed to fit the most probable 2D LPs’ surface to the data in order to obtain the wavefront surface ($W(x,y)$ of the elements under test) as a solution of the ITE differential equation associated with a boundary problem; and finally, we interpolate the resulting fitting. The interpolation is built from LP to increase the resolution and sharpness of the data. We apply the ITE to these results in order to obtain the wavefront as a nondeterministic solution that increases the resolution of the ITE as an optical test, and we compare our results regarding the obtained aberration surfaces ($\mathrm{AS}(x,y)$).

In this work, we propose a solution to the problem of identification of sources in the brain from measurements of the electrical potential, recorded on the scalp EEG (electroencephalogram), where boundary problems are used to model the skull, brain region, and scalp, solving the inverse problem from the EEG measurements, the so-called Electroencephalographic Inverse Problem (EIP), which is ill-posed in the Hadamard sense since the problem has numerical instability. We focus on the identification of volumetric dipolar sources of the EEG by constructing and modeling a simplification to reduce the multilayer conductive medium (two layers or regions ${\Omega }_1$ and ${\Omega }_2$) to a problem of a single layer of a homogeneous medium with a null Neumann condition on the boundary. For this simplification purpose, we consider the Cauchy problem to be solved at each time. We compare the results we obtained solving the multiple layers problem with those obtained by our simplification proposal. In both cases, we solve the direct and inverse problems for two different sources, as synthetic results for dipolar sources resembling epileptic foci, and a similar case with an external stimulus (intense light, skin stimuli, sleep problems, etc). For the inverse problem, we use the Tikhonov regularization method to handle its numerical instability. Additionally, we build an algorithm to solve both models (multiple layers problem and our simplification) in time, showing optimization of the problem when considering 128 divisions in the time interval $[0,1]$ s, solving the inverse problem at each time (interval division) and comparing the recovered source with the initial one in the algorithm. We observed a significant decrease in the computation times when simplifying the numerical calculations, resulting in a decrease up to $50\%$ in the execution times, between the EIP multilayer model and our simplification proposal, to a single layer homogeneous problem of a homogeneous medium, which translates into a numerical efficiency in this type of problem.

We introduce a technique for obtaining three-dimensional (3D) profiles of objects captured in two-dimensional (2D) flat images (FI). This technique performs a numerical approximation of the object's topography from their image gray tones, analyzing topological concepts such as algebraic bases construction, metric functions in terms of such bases, as well as modeling and development of an isomorphism to project masks (fringe patterns) in the FI, allowing us to use the optimal 3D profilometry techniques. Among these techniques, phase shifting (four steps) is applied in the pattern shifts of the projected masks, but with a 2D shift from left to right and from top to bottom, simultaneously. Moreover, the fringe patterns in the masks are binary and with superposed periods in x, y (bi-Ronchi). We show the results of the construction of the masks, as well as their projection into the FIs. We also show the 3D profilometry of the objects after the projection and phase-shifting application and a simple segmentation to observe a particular object. Subsequently, we perform the analysis of a single Mg crystal's micrography. (C) 2020 Optical Society of America