Gaussian Weighting Function Research Articles

Slosh Dynamics of Liquid-Filled Rigid Containers: Two-Dimensional Meshless Local Petrov-Galerkin Approach

This paper presents some of the interesting effects arising from the nonlinear motion of the liquid-free surface, due to sloshing, in a partially filled rigid container subjected to forced excitation. A two-dimensional meshless local Petrov-Galerkin method is used for the numerical simulation of the problem. A local symmetric weak form (LSWF) for nonlinear sloshing of liquid is developed, and a truly meshless method, based on LSWF and moving least squares (MLS) approximation, is presented for the solution of the Laplace equation with the requisite time-dependent boundary conditions. The MLS approximation with linear basis as well as Gaussian type weight function is employed in the computation. At every instant of time, velocity potential is computed at each node and the nodal positions are updated. The choice of a suitable scaling parameter value in the MLS approximation is discussed in this study. The effectiveness of the developed algorithm is demonstrated through a few numerical examples. The accuracy and stability of the numerical method introduced are verified from the comparison with the existing reference solutions.

Classical and Quantum Models in Non-Equilibrium Statistical Mechanics: Moment Methods and Long-Time Approximations

We consider non-equilibrium open statistical systems, subject to potentials and to external “heat baths” (hb) at thermal equilibrium at temperature T (either with ab initio dissipation or without it). Boltzmann’s classical equilibrium distributions generate, as Gaussian weight functions in momenta, orthogonal polynomials in momenta (the position-independent Hermite polynomialsHn’s). The moments of non-equilibrium classical distributions, implied by the Hn’s, fulfill a hierarchy: for long times, the lowest moment dominates the evolution towards thermal equilibrium, either with dissipation or without it (but under certain approximation). We revisit that hierarchy, whose solution depends on operator continued fractions. We review our generalization of that moment method to classical closed many-particle interacting systems with neither a hb nor ab initio dissipation: with initial states describing thermal equilibrium at T at large distances but non-equilibrium at finite distances, the moment method yields, approximately, irreversible thermalization of the whole system at T, for long times. Generalizations to non-equilibrium quantum interacting systems meet additional difficulties. Three of them are: (i) equilibrium distributions (represented through Wigner functions) are neither Gaussian in momenta nor known in closed form; (ii) they may depend on dissipation; and (iii) the orthogonal polynomials in momenta generated by them depend also on positions. We generalize the moment method, dealing with (i), (ii) and (iii), to some non-equilibrium one-particle quantum interacting systems. Open problems are discussed briefly.

Element-free Galerkin modeling of neutron diffusion equation in X–Y geometry

The numerical treatment of partial differential equations with element-free discretization techniques has been attractive research area in the recent years. In this paper an Element-free Galerkin, EFG, method is applied to solve the neutron diffusion equation in X–Y geometry. The Moving Least Square (MLS), interpolation is used to construct the shape functions for the weak form of the equation. The constructed shape functions through using Gaussian and cubic weight functions, which are commonly used in Element-free Galerkin method, lack the Kronecker delta property; this causes additional numerical effort for satisfying the essential boundary conditions. In this study a new weight function which almost fulfills the essential boundary conditions with high accuracy is presented. Constructed shape functions along the new weight function provide much stable results for varying support domain size and distorted nodal arrangements. The efficiency and accuracy of the method was evaluated through a number of examples. Results are compared with the analytical and reference solutions. It is revealed that the applied EFG method provides highly accurate results and the method is attractive in some aspects such as nodes refinement and dealing with the curve boundaries.

Applications of ‘TissueQuant’– A color intensity quantification tool for medical research

This paper demonstrates the use of TissueQuant – an image analysis tool for quantification of color intensities which was developed for use in medical research where the stained biological specimen such as tissue or antigen needs to be quantified. TissueQuant provides facilities for user interaction to choose and quantify the color of interest and its shades. Gaussian weighting functions are used to provide a color score which quantifies how close the shade is to the user specified reference color. We describe two studies in medical research which use TissueQuant for quantification. The first study evaluated the effect of petroleum-ether extract of Cissus quadrangularis (CQ) on osteoporotic rats. It was found that the analysis results correlated well with the manual evaluation, p<0.001. The second study evaluated the nerve morphometry and it was found that the adipose and non adipose tissue content was maximum in radial nerve among the five nerves studied.

Inversion of magnetic measurements of the CHAMP satellite over the Pannonian Basin

The Pannonian Basin is a deep intra-continental basin that formed as part of the Alpine orogeny . In order to study the nature of the crustal basement we used the long-wavelength magnetic anomalies acquired by the CHAMP satellite. The anomalies were distributed in a spherical shell, some 107,927 data recorded between January 1 and December 31 of 2008. They covered the Pannonian Basin and its vicinity. These anomaly data were interpolated into a spherical grid of 0.5° × 0.5°, at the elevation of 324 km by the Gaussian weight function. The vertical gradient of these total magnetic anomalies was also computed and mapped to the surface of a sphere at 324 km elevation. The former spherical anomaly data at 425 km altitude continued downward to 324 km. To interpret these data at the elevation of 324 km we used an inversion method. A polygonal prism forward model was used for the inversion. The minimum problem was solved numerically by the Simplex and Simulated annealing methods; a L 2 norm in the case of Gaussian distribution parameters and a L 1 norm was used in the case of Laplace distribution parameters. We interpret that the magnetic anomaly was produced by several sources and the effect of the sable magnetization of the exsolution of hemo-ilmenite minerals in the upper crustal metamorphic rocks . ► Derivation of the total magnetic anomaly field from the CHAMP data over the Pannonian Basin and its vicinity. ► It is supposed the forward model of the source of magnetic anomalies is a triangular prismatic body which has a uniform magnetization. Its coordinates are determined by the numerical solution of an inverse problem. The applied numerical methods are the Simplex and Simulated Annealing procedures. ► Magnetic anomalies are assumed to be produced by the exsolution of the hemo-ilmenite minerals in the crust.

Detection of Left Ventricular Myocardial Contours from Ischemic Cardiac MR Images

Quantitative evaluation of cardiac function from cardiac magnetic resonance (CMR) images requires the identification of the myocardial walls. This generally requires the clinician to view the image and interactively trace the contours. The myocardial wall of the left ventricle in CMR images, obtained from subjects having serious diseases, is obscure, henceforth, its detection is a tough task. In this paper, an approach to outlining the left ventricular contour is proposed. The utilization of a random walk approach is shown in order to extract the blood pool boundary or endocardium (the inner side of the left ventricular myocardial wall). Inaccurate segmentation is resulted, while applying the approach to ischemic CMR images, because this type of images bear multilabeled blood pool and the manual seed(s) selection in these images introduces variability in the results. In view of this, the paper presents two modifications in the algorithm: (1) automatic seed(s) selection and (2) introduction of Laplacian of difference of Gaussian weighting function. Subsequently, a modified version of an active contour method is implemented to extract the epicardium (the outer side of the left ventricular myocardial wall). This outer contour is achieved by taking the blood pool boundary as its initial contour. Basically, this method is based on active contour without edges. Promising experimental results in CMR images demonstrate the potentials of our approach.

Weak gravitational lensing with deimos

We introduce a novel method for weak-lensing measurements, which is based on a mathematically exact deconvolution of the moments of the apparent brightness distribution of galaxies from the telescope's point spread function (PSF). No assumptions on the shape of the galaxy or the PSF are made. The (de)convolution equations are exact for unweighted moments only, while in practice a compact weight function needs to be applied to the noisy images to ensure that the moment measurement yields significant results. We employ a Gaussian weight function, whose centroid and ellipticity are iteratively adjusted to match the corresponding quantities of the source. The change of the moments caused by the application of the weight function can then be corrected by considering higher order weighted moments of the same source. Because of the form of the deconvolution equations, even an incomplete weighting correction leads to an excellent shear estimation if galaxies and PSF are measured with a weight function of identical size. We demonstrate the accuracy and capabilities of this new method in the context of weak gravitational lensing measurements with a set of specialized tests and show its competitive performance on the GREAT08 Challenge data. A complete c++ implementation of the method can be requested from the authors.

Phonon confinement in Ge nanocrystals in silicon oxide matrix

Spherical Ge nanocrystals well-dispersed in amorphous silicon oxide matrix have been synthesized with different sizes, and significant size-dependent Raman shift and broadening have been observed. The lattice constant of Ge nanocrystals well-bonded to silicon oxide matrix has been characterized nearly size-independent. With our proposed stress generation and relaxation mechanisms, stress effects in our samples have been analyzed to be insignificant with respect to phonon confinement effects. The phenomenological model introduced by [Richter, Wang, and Ley, Solid State Commun. 39, 625 (1981] with Gaussian weighting function and TO2 phonon dispersion function has been found to give a quite good description of the measured size-dependence of Raman shift and broadening. A 3-peak fitting method has been proposed to determine Ge nanocrystal size and film crystallinity. After physically quantizing quantum-confined one-dimensional elastic waves, we have deduced that each quantum-confined phonon possesses an instantaneous momentum of a given magnitude ℏk with an equal chance of being either positive or negative and momentum conservation is retained in an electron-phonon scattering process. Therefore, on the basis of the first-principle microscopic model and our experimental results, we deduced that Raman scattering in spherical nanocrystals is a concurrent two-phonon process, one phonon generation and one phonon transition.

Gaussian-DPSM (G-DPSM) and Element Source Method (ESM) modifications to DPSM for ultrasonic field modeling

In the last few years, Distributed Point Source Method (DPSM) a mesh-free semi-analytical technique has been developed. In spite of its many advantages, one shortcoming of the conventional DPSM method is that the field obtained by conventional DPSM method needs to be scaled to match the theoretical solutions. Two modification techniques called Gaussian-DPSM (G-DPSM) and Element Source Method (ESM) are developed here to avoid the scaling need. G-DPSM technique introduces additional fictitious point sources around every parent point source. Gaussian weight functions determine the strength of these additional fictitious point sources that are denoted as child point sources. ESM replaces discrete point sources used in the conventional DPSM by continuous sources. In the ESM formulation individual point sources are denoted as nodes. Special elements are formed on the boundary by connecting these nodes. The source strength inside the element can vary linearly or non-linearly depending on the order of the interpolation function used inside the element. Results generated by both these methods are compared with the conventional DPSM solution and analytical solution. It is shown that the ultrasonic field in front of the transducer computed by G-DPSM and ESM matches very well with the theory without using any scaling factor.

Goodness-of-fit tests for multivariate Laplace distributions

Consistent goodness-of-fit tests are proposed for symmetric and asymmetric multivariate Laplace distributions of arbitrary dimension. The test statistics are formulated following the Fourier-type approach of measuring the weighted discrepancy between the empirical and the theoretical characteristic function, and result in computationally convenient representations. For testing the symmetric Laplace distribution, and in the particular case of a Gaussian weight function, a limit value of these test statistics is obtained when this weight function approaches a Dirac delta function. Interestingly, this limit value is related to a couple of well-known measures of multivariate skewness. A Monte Carlo study is conducted in order to compare the new procedures with standard tests based on the empirical distribution function. A real data application is also included.

A Strategic Approach for Cardiac MR Left Ventricle Segmentation

Quantitative evaluation of cardiac function from cardiac magnetic resonance (CMR) images requires the identification of the myocardial walls. This generally requires the clinician to view the image and interactively trace the contours. Especially, detection of myocardial walls of left ventricle is a difficult task in CMR images that are obtained from subjects having serious diseases. An approach to automated outlining the left ventricular contour is proposed. In order to segment the left ventricle, in this paper, a combination of two approaches is suggested. Difference of Gaussian weighting function (DoG) is newly introduced in random walk approach for blood pool (inner contour) extraction. The myocardial wall (outer contour) is segmented out by a modified active contour method that takes blood pool boundary as the initial contour. Promising experimental results in CMR images demonstrate the potentials of our approach.

Spatially weighted mutual information image registration for image guided radiation therapy

To develop a new metric for image registration that incorporates the (sub)pixelwise differential importance along spatial location and to demonstrate its application for image guided radiation therapy (IGRT). It is well known that rigid-body image registration with mutual information is dependent on the size and location of the image subset on which the alignment analysis is based [the designated region of interest (ROI)]. Therefore, careful review and manual adjustments of the resulting registration are frequently necessary. Although there were some investigations of weighted mutual information (WMI), these efforts could not apply the differential importance to a particular spatial location since WMI only applies the weight to the joint histogram space. The authors developed the spatially weighted mutual information (SWMI) metric by incorporating an adaptable weight function with spatial localization into mutual information. SWMI enables the user to apply the selected transform to medically "important" areas such as tumors and critical structures, so SWMI is neither dominated by, nor neglects the neighboring structures. Since SWMI can be utilized with any weight function form, the authors presented two examples of weight functions for IGRT application: A Gaussian-shaped weight function (GW) applied to a user-defined location and a structures-of-interest (SOI) based weight function. An image registration example using a synthesized 2D image is presented to illustrate the efficacy of SWMI. The convergence and feasibility of the registration method as applied to clinical imaging is illustrated by fusing a prostate treatment planning CT with a clinical cone beam CT (CBCT) image set acquired for patient alignment. Forty-one trials are run to test the speed of convergence. The authors also applied SWMI registration using two types of weight functions to two head and neck cases and a prostate case with clinically acquired CBCT/ MVCT image sets. The SWMI registration with a Gaussian weight function (SWMI-GW) was tested between two different imaging modalities: CT and MRI image sets. SWMI-GW converges 10% faster than registration using mutual information with an ROI. SWMI-GW as well as SWMI with SOI-based weight function (SWMI-SOI) shows better compensation of the target organ's deformation and neighboring critical organs' deformation. SWMI-GW was also used to successfully fuse MRI and CT images. Rigid-body image registration using our SWMI-GW and SWMI-SOI as cost functions can achieve better registration results in (a) designated image region(s) as well as faster convergence. With the theoretical foundation established, we believe SWMI could be extended to larger clinical testing.

Un estudio de caso en el análisis de la distribución de frecuencias de tallas de Litopenaeus vannamei (Boone, 1931) mediante el uso de estimadores de densidad por Kernel

This paper, introduces the use of kernel density estimators (EDK's) as a modern tool for examining the length-frequency distribution of Litopenaeus vannamei in its estuarine stage. The data were obtained monthly at 22 sampling sites in the Carretas-Pereyra lagoon-estuarine system, from March 2004 to August 2005, covering both seasons of the year: dry and rainy. Shrimp were caught with a 4 m diameter cast net of 10 mm mesh size. The size distribution of each sample was analyzed by means of EDK's, using the Gaussian weight function and the bootstrap bandwidth. The values of the dominant modes over time were fitted to a von Bertalanffy growth function. The results suggest continuous recruitment to the fishing area, albeit with a bimodal pulse. The mean growth rate of the shrimp is the same in both climatic seasons. The estimated parameters are similar to those registered in previous research by other authors for L. vannamei in nearby systems. The study illustrates how the use of the EDK's followed by a method of growth analysis herein, we use modal progression analysis is an objective and precise way to investigate length-frequency

Fast and accurate marker-based projective registration method for uncalibrated transmission electron microscope tilt series

This paper presents a fast and accurate marker-based automatic registration technique for aligning uncalibrated projections taken from a transmission electron microscope (TEM) with different tilt angles and orientations. Most of the existing TEM image alignment methods estimate the similarity between images using the projection model with least-squares metric and guess alignment parameters by computationally expensive nonlinear optimization schemes. Approaches based on the least-squares metric which is sensitive to outliers may cause misalignment since automatic tracking methods, though reliable, can produce a few incorrect trajectories due to a large number of marker points. To decrease the influence of outliers, we propose a robust similarity measure using the projection model with a Gaussian weighting function. This function is very effective in suppressing outliers that are far from correct trajectories and thus provides a more robust metric. In addition, we suggest a fast search strategy based on the non-gradient Powell's multidimensional optimization scheme to speed up optimization as only meaningful parameters are considered during iterative projection model estimation. Experimental results show that our method brings more accurate alignment with less computational cost compared to conventional automatic alignment methods.

Non-local damage and fracture model of rock and concrete-like materials

In view of the non-local phenomena appearing in the rock and concrete-like materials, the non-local damage and fracture model of rock and concrete-like materials was established through non-local method of Gaussian weighting function. The result indicates that, the stress of one point in the material is correlated not only to its strain history, but also to the interaction of the points in its certain adjacent region of the material. Based on the established non-local model, the numerical simulation of notch containing three-point bending beam was carried out. The results show that the grid sensitivities have been avoided and the fracture direction of the material has not been influenced by the grid shape, and the model proposed can be used to better simulate the damage developing process of the rock and concrete-like materials.

Non‐linear dynamic analyses by meshless local Petrov–Galerkin formulations

AbstractIn this work, meshless methods based on the local Petrov–Galerkin approach are proposed for the solution of dynamic problems considering elastic and elastoplastic materials. Formulations adopting the Heaviside step function and the Gaussian weight function as the test functions in the local weak form are considered. The moving least‐square method is used for the approximation of physical quantities in the local integral equations. After spatial discretization is carried out, a non‐linear system of ordinary differential equations of second order is obtained. This system is solved by Newmark/Newton–Raphson techniques. At the end of the paper numerical results are presented, illustrating the potentialities of the proposed methodologies. Copyright © 2009 John Wiley &amp; Sons, Ltd.

Toward a Standard Procedure for Validation of Satellite-Derived Cloud Liquid Water Path: A Study with SEVIRI Data

Abstract Differences between satellite-derived and ground-based values of cloud liquid water path (LWPsat and LWPgr, respectively) in validation studies are partly associated with the validation itself, in particular with scale differences and parallax. This paper aims at establishing standards for validation procedures to minimize these contributions to the differences. To investigate this, LWP values were collected as computed from ground-based microwave radiometer (MWR) summer measurements made at two Cloudnet sites and from the spaceborne Spinning Enhanced Visible and Infrared Imager (SEVIRI) instrument. The large number of all-sky sample pairs (∼2500 after selection) formed an essential condition for the present study. The best validation method was determined by optimum statistical agreement between LWPsat and LWPgr. The method consists of (i) computation of LWPsat by averaging LWP over the pixels surrounding the ground station by means of a Gaussian weight function with a length scale defining the validation area, (ii) computation of LWPgr by averaging the MWR measurements with a Gaussian weight function, by using a time scale that is considerably longer than the time in which the clouds move across the validation area (by a factor of 3–15), and (iii) correcting for parallax. The authors argue that the best length scale for averaging the satellite data is equal to the image resolution. The improvement resulting from the parallax correction was significant at the 99.5% level, but its effect was not significant for a subset of the data for relatively homogeneous cloud fields. Also, there was no significant improvement when, instead of taking a constant, the time scale for averaging the ground data was adjusted to the instantaneous wind field.

Relationship between threshold and suprathreshold perception of position and stereoscopic depth

We seek to determine the relationship between threshold and suprathreshold perception for position offset and stereoscopic depth perception under conditions that elevate their respective thresholds. Two threshold-elevating conditions were used: (1) increasing the interline gap and (2) dioptric blur. Although increasing the interline gap increases position (Vernier) offset and stereoscopic disparity thresholds substantially, the perception of suprathreshold position offset and stereoscopic depth remains unchanged. Perception of suprathreshold position offset also remains unchanged when the Vernier threshold is elevated by dioptric blur. We show that such normalization of suprathreshold position offset can be attributed to the topographical-map-based encoding of position. On the other hand, dioptric blur increases the stereoscopic disparity thresholds and reduces the perceived suprathreshold stereoscopic depth, which can be accounted for by a disparity-computation model in which the activities of absolute disparity encoders are multiplied by a Gaussian weighting function that is centered on the horopter. Overall, the statement "equal suprathreshold perception occurs in threshold-elevated and unelevated conditions when the stimuli are equally above their corresponding thresholds" describes the results better than the statement "suprathreshold stimuli are perceived as equal when they are equal multiples of their respective threshold values."

Monte Carlo sampling of Wigner functions and surface hopping quantum dynamics

The article addresses the achievable accuracy for a Monte Carlo sampling of Wigner functions in combination with a surface hopping algorithm for non-adiabatic quantum dynamics. The approximation of Wigner functions is realized by an adaption of the Metropolis algorithm for real-valued functions with disconnected support. The integration, which is necessary for computing values of the Wigner function, uses importance sampling with a Gaussian weight function. The numerical experiments agree with theoretical considerations and show an error of 2–3%.

Bayesian Wavelet-Based Image Denoising Using the Gauss–Hermite Expansion

The probability density functions (PDFs) of the wavelet coefficients play a key role in many wavelet-based image processing algorithms, such as denoising. The conventional PDFs usually have a limited number of parameters that are calculated from the first few moments only. Consequently, such PDFs cannot be made to fit very well with the empirical PDF of the wavelet coefficients of an image. As a result, the shrinkage function utilizing any of these density functions provides a substandard denoising performance. In order for the probabilistic model of the image wavelet coefficients to be able to incorporate an appropriate number of parameters that are dependent on the higher order moments, a PDF using a series expansion in terms of the Hermite polynomials that are orthogonal with respect to the standard Gaussian weight function, is introduced. A modification in the series function is introduced so that only a finite number of terms can be used to model the image wavelet coefficients, ensuring at the same time the resulting PDF to be non-negative. It is shown that the proposed PDF matches the empirical one better than some of the standard ones, such as the generalized Gaussian or Bessel K-form PDF. A Bayesian image denoising technique is then proposed, wherein the new PDF is exploited to statistically model the subband as well as the local neighboring image wavelet coefficients. Experimental results on several test images demonstrate that the proposed denoising method, both in the subband-adaptive and locally adaptive conditions, provides a performance better than that of most of the methods that use PDFs with limited number of parameters.