Vaclav Skala

Zuzana Majdisova, Vaclav Skala

Approximation of scattered data is often a task in many engineering problems. The Radial Basis Function (RBF) approximation is appropriate for large scattered (unordered) datasets in d-dimensional space. This approach is useful for a higher dimension d>2, because the other methods require the conversion of a scattered dataset to an ordered dataset (i.e. a semi-regular mesh is obtained by using some tessellation techniques), which is computationally expensive. The RBF approximation is non-separable, as it is based on the distance between two points. This method leads to a solution of Linear System of Equations (LSE) Ac=h. In this paper several RBF approximation methods are briefly introduced and a comparison of those is made with respect to the stability and accuracy of computation. The proposed RBF approximation offers lower memory requirements and better quality of approximation.

Zuzana Majdisova, Vaclav Skala

Approximation of scattered data is often a task in many engineering problems. The Radial Basis Function (RBF) approximation is appropriate for large scattered datasets in d-dimensional space. It is non-separable approximation, as it is based on a distance between two points. This method leads to a solution of overdetermined linear system of equations. In this paper a new approach to the RBF approximation of large datasets is introduced and experimental results for different real datasets and different RBFs are presented with respect to the accuracy of computation. The proposed approach uses symmetry of matrix and partitioning matrix into blocks.

Vaclav Skala, Alex Brusi

There are various methods for extracting iso-surfaces from volumetric data. Marching cubes or tetrahedra or raytracing methods are mostly used. There are many specific techniques to increase speed of computation and decrease memory requirements. Although a precision of iso-surface extraction is very important, too, it is not mentioned usually. A comparison of the selected methods was made in different aspects: iso-surface extraction process time, number of triangles generated and estimation of radius, area and volume errors based on approximation of a sphere. Surprisingly, experiments proved that there is no direct relation between precision of extracted and human perception of the extracted iso-surface

Zuzana Majdisova, Vaclav Skala, Michal Smolik

Stationary points of multivariable function which represents some surface have an important role in many application such as computer vision, chemical physics, etc. Nevertheless, the dataset describing the surface for which a sampling function is not known is often given. Therefore, it is necessary to propose an approach for finding the stationary points without knowledge of the sampling function. In this paper, an algorithm for determining a set of stationary points of given sampled surface and detecting the bindings between these stationary points (such as stationary points lie on line segment, circle, etc.) is presented. Our approach is based on the piecewise RBF interpolation of the given dataset.

Vaclav Skala

There are many practical applications based on the Least Square Error (LSE) approximation. It is based on a square error minimization 'on a vertical' axis. The LSE method is simple and easy also for analytical purposes. However, if data span is large over several magnitudes or non-linear LSE is used, severe numerical instability can be expected. The presented contribution describes a simple method for large span of data LSE computation. It is especially convenient if large span of data are to be processed, when the 'standard' pseudoinverse matrix is ill conditioned. It is actually based on a LSE solution using orthogonal basis vectors instead of orthonormal basis vectors. The presented approach has been used for a linear regression as well as for approximation using radial basis functions.

Vaclav Skala

This contribution presents a new analysis of properties of the interpolation using Radial Bases Functions (RBF) related to large data sets interpolation. The RBF application is convenient method for scattered d-dimensional interpolation. The RBF methods lead to a solution of linear system of equations and computational complexity of solution is nearly independent of a dimensionality. However, the RBF methods are usually applied for small data sets with a small span geometric coordinates. This contribution explores properties of the RBF interpolation for large data sets and large span of geometric coordinates of the given data sets with regard to expectable numerical stability of computation.