3D solid of SARS-CoV-2 viral particles applying Legendre polynomials from tomography Fourier analysis.

We show the construction of 3D solids (volumetric 3D models) of SARS-CoV-2 viral particles from the tomographic studies (videos) of SARS-CoV-2-infected tissues. To this aim, we propose a video analysis (tomographic images) by frames (medical images of the virus), which we set as our metadata. We optimize the frames by means of Fourier analysis, which induces a periodicity with simple structure patterns to minimize noise filtering and to obtain an optimal phase of the objects in the image, focusing on the SARS-CoV-2 cells to obtain a medical image under study phase (MIS) (process repeated over all frames). We build a Python algorithm based on Legendre polynomials called "2DLegendre_Fit," which generates (using multilinear interpolation) intermediate images between neighboring MIS phases. We used this code to generate m images of size M×M, resulting in a matrix with size M×M×M (3D solid). Finally, we show the 3D solid of the SARS-CoV-2 viral particle as part of our results in several videos, subsequently rotated and filtered to identify the glicoprotein spike protein, membrane protein, envelope, and the hemagglutinin esterase. We show the algorithms in our proposal along with the main MATLAB functions such as FourierM and Results as well as the data required for the program execution in order to reproduce our results.

Filter construction using Ronchi masks and Legendre polynomials to analyze the noise in aberrations by applying the irradiance transport equation.

We tested different optical elements placed in three different positions by applying the irradiance transport equation (ITE), obtaining the wavefront $[ W(x,y) ]$[W(x,y)] and aberration surface [$ \textit{AS}(r,\theta ) $AS(r,θ)]. The existing noise in the captures $ I $I as well as in the $ W $W and $ AS $AS 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,xy $x,y,xy 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.

Two-dimensional Legendre polynomials as a basis for interpolation of data to optimize the solution of the irradiance transport equation analyzed as a boundary problem on surfaces testing.

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 (AS(x,y)).

Moiré-Ronchigram analysis applied in the characterization of aberration surfaces and optical surface parameters from 3D wavefronts.

In this work, we propose the construction of the Moiré-Ronchigram as a pattern obtained from simple Ronchigrams, superposed and with a slight degree of rotation, to obtain a 3D wavefront. The aberrations in the obtained wavefront are fitted to the aberration Zernike polynomial of degree 12 in order to obtain a polynomial expression of a high degree of efficiency, which is subsequently compared with the general equation of quadrics (considering the coefficients accordingly) to obtain essential parameters (as focus, eccentricity, or F number) that allow us to understand and manipulate the optical elements under test optimally. Finally, we compare our results analyzing optical surfaces to make a weak statistic of our proposed method.

Ronchi and Moiré patterns for testing spherical and aspherical surfaces using deflectometry.

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.

Measurement of three-dimensional wavefronts using the Ichikawa-Lohmann-Takeda solution to the irradiance transport equation.

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.

Determination of early bone metastasis on Bone Scans Using the Gray Levels Histogram / Búsqueda de metástasis ósea temprana en gammagramas óseos usando el histograma de tonos de gris

ABSTRACT The aim of this paper is to show a technique to speed up the interpretation of bone scans in order to determine the presence of early bone metastasis. This is done using the gray levels histogram of the region of interest. The technique is intended to assist in the bone scans interpretation in order to provide a successful diagnosis. During the analysis, three types of histograms were observed on the regions of interest. If the histogram is narrow and shifted toward the origin, the bone scan is free of metastasis. If it is shifted to the right and slightly broadened, indicates the presence of a bone anomaly different from a metastasis. On the other hand, if the histogram is more broadened and shifted to the right, is suggests the presence of metastasis. This histogram is characterized by displaying small curls on the right side providing information about the metastatic disease stage, which could be low-amplitude peaks and have a short length, if the metastasis is in early stage, or high-amplitude peaks and a long length, if is advanced. Finally, the analyzed region is displayed in false color considering the minimum gray levels observed in the histogram.