Michal Smolik

Spherical RBF vector field interpolation: Experimental study

The Radial Basis Function (RBF) interpolation is a common technique for scattered data interpolation. We present and test an approach of RBF interpolation on a sphere which uses the spherical distance on the surface of the sphere instead of the Euclidian distance. We show how the interpolation of vector field data depends on the value of shape parameter of RBF and find the optimal shape parameter for our experiments.

Classification of Critical Points Using a Second Order Derivative

Abstract This article presents a new method for classification of critical points. A vector field is usually classified using only a Jacobian matrix of the approximated vector field. This work shows how an approximation using a second order derivative can be used for more detailed classification. An algorithm for calculation of the curvature of main axes is also presented.

A Point in Non-convex Polygon Location Problem Using the Polar Space Subdivision in E2

The point inside/outside a polygon test is used by many applications in computer graphics, computer games and geographical information systems. When this test is repeated several times with the same polygon a data structure is necessary in order to reduce the linear time needed to obtain an inclusion result. In the literature, different approaches, like grids or quadtrees, have been proposed for reducing the complexity of these algorithms. We propose a new method using a polar space subdivision to reduce the time necessary for this inclusion test. The proposed algorithm is robust and has a performance of \( O(k) \), where \( k \ll N \), \( k \) is the number of tested intersections with polygon edges, and the time complexity of the preprocessing is \( O(N) \), where \( N \) is the number of polygon edges.

Large scattered data interpolation with radial basis functions and space subdivision

We propose a new approach for the radial basis function (RBF) interpolation of large scattered data sets. It uses the space subdivision technique into independent cells allowing processing of large data sets with low memory requirements and offering high computation speed, together with the possibility of parallel processing as each cell can be processed independently. The proposed RBF interpolation was tested on both synthetic and real data sets. It proved its simplicity, robustness and the ability to handle large data sets together with significant speed-up. In the case of parallel processing, speed-up was experimentally proved when 2 and 4 threads were used.

A Comparative Study of LOWESS and RBF Approximations for Visualization

Approximation methods are widely used in many fields and many techniques have been published already. This comparative study presents a comparison of LOWESS (Locally weighted scatterplot smoothing) and RBF (Radial Basis Functions) approximation methods on noisy data as they use different approaches. The RBF approach is generally convenient for high dimensional scattered data sets. The LOWESS method needs finding a subset of nearest points if data are scattered. The experiments proved that LOWESS approximation gives slightly better results than RBF in the case of lower dimension, while in the higher dimensional case

Highly Parallel Algorithm for Large Data In-Core and Out-Core Triangulation in E 2 and E 3 .

Abstract A triangulation of points in E 2 , or a tetrahedronization of points in E 3 , is used in many applications. It is not necessary to fulfill the Delaunay criteria in all cases. For large data (more then 5 · 10 7 points),parallel methods are used for the purpose of decreasingrun–time. A new approach for fast, effective and highly parallel CPU and GPU triangulation, or tetrahedronization, of large data sets in E 2 or E 3 suitable for in–core and out–core memory processing, is proposed. Experimental results proved that the resulting triangulation/tetrahedralization is close to the Delaunay triangulation/tetrahedralization. It also demonstrates the applicability of the methodproposed in applications.

A New Approach to Vector Field Interpolation, Classification and Robust Critical Points Detection Using Radial Basis Functions

Visualization of vector fields plays an important role in many applications. Vector fields can be described by differential equations. For classification null points, i.e. points where derivation is zero, are used. However, if vector field data are given in a discrete form, e.g. by data obtained by simulation or a measurement, finding of critical points is difficult due to huge amount of data to be processed and differential form usually used. This contribution describes a new approach for vector field null points detection and evaluation, which enables data compression and easier fundamental behavior visualization. The approach is based on implicit form representation of vector fields.

Determination of Stationary Points and Their Bindings in Dataset Using RBF Methods

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.

Vector field radial basis function approximation

Abstract Vector field simplification aims to reduce the complexity of the flow by removing features according to their relevance and importance. Our goal is to preserve only the important critical points in the vector field and thus simplify the vector field for the visualization purposes. We use Radial Basis Functions (RBF) approximation with Lagrange multipliers for vector field approximation. The proposed method was experimentally verified on synthetic and real weather forecast data sets. The results proved the quality of the proposed approximation method compared to other existing approaches. A significant contribution of the proposed method is an analytical form of the vector field which can be used in further processing.

Vector Field Second Order Derivative Approximation and Geometrical Characteristics

Vector field is mostly linearly approximated for the purpose of classification and description. This approximation gives us only basic information of the vector field. We will show how to approximate the vector field with second order derivatives, i.e. Hessian and Jacobian matrices. This approximation gives us much more detailed description of the vector field. Moreover, we will show the similarity of this approximation with conic section formula.