# Vaclav SkalaUniversity of West Bohemia · Department of Computer Science and Engineering

Vaclav Skala

prof. Ing. CSc.[Ph.D.] ----- http://www.VaclavSkala.eu

## About

283

Publications

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1,782

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Introduction

Details on
- myself - http://www.VaclavSkala.eu
- WSCG conferences on Computer Graphics, Visualization and Computer Vision - http://www.wscg.eu
Organized every year in Pilsen (Czech Republic) close to Prague since 1992

Additional affiliations

September 1975 - present

## Publications

Publications (283)

This paper describes a novel and fast, simple and robust algorithm with O(N) expected complexity which enables to decrease run time needed to find the maximum distance of two points in E2. It can be easily modified for the E3 case in general. The proposed algorithm has been evaluated experimentally on larger different datasets in order to verify it...

This contribution presents a brief survey of clipping and intersection algorithms in E 2 and E 3 with a nearly complete list of relevant references. Some algorithms use the projective extension of the Euclidean space and vector − vector operations, which supports GPU and SSE use. This survey is intended as help to researchers, students, and practit...

Acceleration of algorithms is becoming a crucial problem, if larger data sets are to be processed. Evaluation of algorithms is mostly done by using computational geometry approach and evaluation of computational complexity. However in todays engineering problems this approach does not respect that number of processed items is always limited and a s...

Many algorithms used are based on geometrical computation. There are several criteria in selecting appropriate algorithm from already known. Recently, the fastest algorithms have been preferred. Nowadays, algorithms with a high stability are preferred. Also technology and computer architecture, like GPU etc., plays a significant role for large data...

There are many applications in which a bounding sphere containing the given triangle E3 is needed, e.g. fast collision detection, ray-triangle intersecting in raytracing etc. This is a typical geometrical problem in E3 and it has also applications in computational problems in general. In this paper a new fast and robust algorithm of circumscribed s...

There are many space subdivision and space partitioning techniques used in many algorithms to speed up computations. They mostly rely on orthogonal space subdivision, resp. using hierarchical data structures, e.g. BSP trees, quadtrees, octrees, kd-trees, bounding volume hierarchies, etc. However in some applications a non-orthogonal space subdivisi...

This paper describes novel and fast, simple and robust algorithm with O(N) expected complexity which enables to decrease run-time needed to find an exact maximum distance of two points in E2. The proposed algorithm has been evaluated experimentally on larger different datasets. The proposed algorithm gives a significant speed-up to applications, wh...

This paper presents a new approach to computation of geometric continuity for parametric bi-cubic patches, based on a simple mathematical reformulation which leads to simple additional conditions to be applied in the patching computation. The paper presents an Hermite formulation of a bicubic parametric patch, but reformulations can be made also fo...

Line intersection with convex and un-convex polygons or polyhedron algorithms are well known as line clipping algorithms and very often used in computer graphics. Rendering of geometrical problems often leads to ray tracing techniques, when an intersection of many lines with spheres or quadrics is a critical issue due to ray-tracing algorithm compl...

This contribution presents a brief survey of clipping and intersection algorithms in E2 and E3 with a nearly complete list of relevant references. Some algorithms use the projective extension of the Euclidean space and vector-vector operations, which supports GPU and SSE use.

Bicubic parametric plates are essential for many geometric applications, especially for CAD/CAM systems used in the automotive industry, mechanical and civil engineering applications. Usually the Hermite, Bézier, Coons or NURBS plates are used. There is always a problem to explain how the Hermit bicubic plate is constructed. This contribution descr...

The smallest enclosing circle is a well-known problem. In this paper, we propose modifications to speed-up the existing Weltzl’s algorithm. We perform the preprocessing to reduce as many input points as possible. The reduction step has lower computational complexity than the Weltzl’s algorithm and thus speed-ups its computation. Next, we propose so...

Finding the smallest enclosing circle of the given points in E2 is a seemingly simple problem. However, already proposed algorithms have high memory requirements or require special solutions due to the great recursion depth or high computational complexity unacceptable for large data sets, etc.
This paper presents a simple and efficient method with...

Many algorithms for clipping a line by a rectangular area or a convex polygon in E2 or by a non-convex or convex polyhedron in E3 have been published. The line segment clipping by the rectangular window in E2 is often restricted to the use of the Cohen-Sutherland (CS) algorithm or its modifications based on some presumptions like small clipping win...

A new algorithm for clipping a line segment against a pyramid in E3 is presented. This algorithm avoids computation of intersection points which are not end-points of the output line segment. It also allows solving all cases more effectively. The performance of this algorithm is shown to be consistently better than existing algorithms, including th...

A new line clipping algorithm against convex polyhedron in E3 with an expected complexity O(1) is presented. The suggested approach is based on two orthogonal projections to E2 co-ordinate system and on pre-processing of the given polyhedron. The pre-processing enables to speed up solution significantly. The proposed method is convenient for those...

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 co...

Cubic parametric curves are used in many applications including the CAD/CAM systems. Especially the Hermite, Bezier and Coons formulations of a cubic parametric curve are used in E2 and E3 space. This paper presents efficient algorithm for the intersection computation of a cubic parametric curve with the Axis Aligned Bounding Box (AAB Box). Usual s...

A comparison of a new algorithm for line clipping in E2 and E3 by convex polygon and/or polyhedron with O(1) processing complexity and Cyrus- Beck algorithm is presented. The new algorithm in E2 is based on dual space representation and space subdivision technique. The principle of algorithm in E3 is based on the projection of polyhedron to three o...

This contribution presents a new coding scheme based on Cohen-Sutherland line segment clipping algorithm, which enables to distinguish all possible cases easily. It leads to more efficient algorithm for a line segment clipping in E2. It also presents importance of a detailed analysis in algorithm development, if the algorithm robustness and efficie...

This paper presents a new approach to line clipping by a convex polygon problem solution. The algorithm is based on a separation function, which separates the polygon vertices to the left or right hand side of the given line. It leads to numerically robust algorithms in comparison to the well-known Cyrus-Beck’s algorithm and its modifications.

So many complain about C^inf RBFs for interpolation. PDEs and IEs as being ill-conditioned. This paper is a simple treatment combining block Gaussian elimination (BGE) and extended precision arithmetic (EPA). The 144x144 Vandermode equation had a condition number ESTIMATED to ne 1e+313, But the rms errors are O(1e-92). This approach is best on para...

This contribution briefly describes some "dangerous" features of the Least Square Error (LSE) methods, which are not generally known, but often used in applications, and researchers are not aware of those. The LSE is usually used in approximations of acquired data to find "the best fit" of the data, especially in financial economics and related fie...

The Taylor expansion is used in many applications for a value estimation of scalar functions of one or two variables in the neighbour point. Usually, only the first two elements of the Taylor expansion are used, i.e. a value in the given point and derivatives estimation. The Taylor expansion can be also used for vector functions, too. The usual for...

We propose a new approach for meshless multi-level radial basis function (ML-RBF) approximation which offers data-sensitive compression and progressive details visualization. It leads to an analytical description of compressed vector fields, too. The proposed approach approximates the vector field at multiple levels of detail. The low-level approxi...

Geometric problems are usually solved in the Euclidean space by using the standard vector algebra techniques. In this study, principles of the projective geometry and geometric algebra will be introduced via a novel method that significantly simplifies
the solution of geometrical problems. Also, it supports the GPU parallel computation application....

The global Radial Basis Functions (RBFs) may lead to ill-conditioned system of linear equations. This contribution analyzes conditionality of the Gauss and the Thin Plate Spline (TPS) functions. Experiments made proved dependency between the shape parameter and number of RBF center points where the matrix is ill-conditioned. The dependency can be f...

Convex hull of points and its diameter computation is a frequent task in many engineering problems, However, in engineering solutions , the asymptotic computational complexity is less important than the computational complexity for the expected data size to be processed. This contribution describes "an engineering solution" of the convex hulls and...

Linear systems of equations and their reliable solution is a key part of nearly all computational systems and in a solution of many engineering problems. Mostly, the estimation of the matrix conditionality is used for an assessment of the solvability of linear systems, which are important for interpolation, approximation , and solution of partial d...

This paper presents a method for efficient Radial basis function (RBF) evaluation if compactly supported radial basis functions (CSRBF) are used. Application of CSRBF leads to sparse matrices, due to limited influence of radial basis functions in the data domain and thus non-zero weights (coefficients) are valid only for some areas in the data doma...

Radial basis functions (RBF) are widely used in many areas especially for interpolation and approximation of scattered data, solution of ordinary and partial differential equations, etc. The RBF methods belong to meshless methods, which do not require tessellation of the data domain, i.e. using Delaunay triangulation, in general. The RBF meshless m...

Interpolation and approximation methods are used in many fields such as in engineering as well as other disciplines for various scientific discoveries. If the data domain is formed by scattered data, approximation methods may become very complicated as well as time-consuming. Usually, the given data is tessellated by some method, not necessarily th...

Interpolation and approximation methods are widely used in many areas. They can be divided to methods based on meshing (tessellation) of the data domain and to meshless (meshfree) methods, which do not require the domain tessellation of scattered data. Scattered n-dimensional data radial basis function (RBF) interpolation and approximation leads to...

Both approximation and interpolation are techniques commonly used in many scientific areas. Many approaches are depending on input data type, result purpose etc. Input data can be formed in a mesh or not (meshless/meshfree data). This contribution is oriented on meshless data approximation and interpolation using Radial Basis Functions (RBFs). Diff...

This paper presents new approaches for Radial basis function (RBF) approximation of 2D height data. The proposed approaches respect local properties of the input data, i.e. stationary points, inflection points, the curvature and other important features of the data. Positions of radial basis functions for RBF approximation are selected according to...

Algorithms for line and line segment clipping are well known algorithms especially in the field of computer graphics. They are formulated for the Euclidean space representation. However, computer graphics uses the projective extension of the Euclidean space and homogeneous coordinates for representation geometric transformations with points in the...

Abstract This paper discusses the functional scattered data interpolation to interpolate the general scattered data. Compared with the previous works, we construct a new cubic Bézier-like triangular basis function controlled by three shape parameters. This is an advantage compared with the existing schemes since it gives more flexibility for the sh...

The Fuzzy regression model provides a good alternative to the standard regression model that existing in statistics as well as engineering based studies. In this study, a new fuzzy regression model is introduced by incorporating the crisp and the spreading for the fuzziness of the data. The fuzzy triangular number is employed to obtain the fuzzy re...

Many problems, not only in signal processing, image processing, digital imaging, computer vision and visualization, lead to the Least Square Error (LSE) problem or Total (Orthogonal) Least Square Error (TLSE) problem computation. Usually the standard least square error approximation method is used due to its simplicity, but it is not an optimal sol...

This chapter discusses the application of the new cubic Timmer triangular patches constructed by Ali et al. [1] to interpolate the irregularly scattered data with C¹ continuity. In order to apply the cubic Timmer triangular patches for scattered data interpolation, the data is first triangulated by using the Delaunay algorithm, and then the suffici...

This paper discusses the application of the new cubic Timmer triangular patches constructed by Ali et al. [1] to interpolate the irregularly scattered data with C ¹ continuity. In order to apply the cubic Timmer triangular patches for scattered data interpolation, the data is first triangulated by using the Delaunay algorithm, and then the sufficie...

Many Radial Basis Functions (RBFs) contain a shape parameter which has an important role to ensure good quality of the RBF approximation. Determination of the optimal shape parameter is a difficult problem. In the majority of papers dealing with the RBF approximation, the shape parameter is set up experimentally or using some ad-hoc method. Moreove...

In this study, a new cubic Timmer triangular patch is constructed by extending the univariate cubic Timmer basis functions. The best scheme that lies towards the control polygon is cubic Timmer curve and surface compared to the other methods. From the best of our knowledge, nobody has extended the univariate cubic Timmer basis to the bivariate tria...

This paper discusses the application of the new cubic Timmer triangular patches constructed by Ali et al. [1] to interpolate the irregularly scattered data with í µí° ¶ 1 continuity. In order to apply the cubic Timmer triangular patches for scattered data interpolation, the data is first triangulated by using the Delaunay algorithm, and then the su...

This paper discusses the constraint data interpolation or range restricted interpolation for surface data arranges on rectangular meshes that lie above or below an arbitrary plane and between two arbitrary planes by using partially blended rational bi-cubic spline function with 12 parameters. Common research in range restricted surface interpolatio...

The approximation of scattered data is known technique in computer science. We propose a new strategy for the placement of radial basis functions respecting points of inflection. The placement of radial basis functions has a great impact on the approximation quality. Due to this fact we propose a new strategy for the placement of radial basis funct...

The Radial basis function (RBF) approximation is an efficient method for scattered scalar and vector data fields. However its application is very difficult in the case of large scattered data. This paper presents RBF approximation together with space subdivision technique for large vector fields.

Finding a maximum distance of points in E² or in E³ is one of those. It is a frequent task required in many applications. In spite of the fact that it is an extremely simple task, the known “Brute force” algorithm is of O(N²) complexity. Due to this complexity the run-time is very long and unacceptable especially if medium or larger data sets are t...

In many technical applications, reconstruction of the scattered data is often task. For big scattered dataset in n-dimensional space, the using some meshless method such as the radial basis function (RBF) approximation is appropriate. RBF approximation is based on the distance computation, and therefore, it is dimensionally non-separable. This appr...

This contribution describes a new approach to a solution of multidimensional dynamical systems using the La-place transform and geometrical product, i.e. using inner product (dot product, scalar product) and outer product (extended cross-product). It leads to a linear system of equations Ax=0 or Ax=b which is equivalent to the outer product if the...

Many problems, not only in signal processing, image processing , digital imaging, computer vision and visualization, lead to the Least Square Error (LSE) problem or Total (Orthogonal) Least Square Error (TLSE) problem computation. Usually the standard least square error approximation method is used due to its simplicity, but it is not an optimal so...

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 p...

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...

This contribution presents a new formulation of Plücker coordinates using geometric algebra and standard linear algebra with projective representation. The Plücker coordinates are usually used for a line representation in space, which is given by two points. However, the line can be also given as an intersection of two planes in space. The principl...

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 mult...

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 equatio...

Approximation of scattered geometric 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 method is useful for a higher dimension d>=2, because the other methods require a conversion of a scattered dataset to a semi-regul...

This contribution presents a new analysis of properties of the Radial Bases Functions (RBF) interpolation and approximation related to data sets with a large data span. The RBF is a convenient method for scattered d-dimensional interpolation and approximation, e.g. for solution of partial differential equations (PDE) etc. The RBF method leads to a...

This contribution describes a new approach for solving linear system of algebraic equations and differential equations using Laplace transform by the extended-cross product. It will be shown that a solution of a linear system of equations Ax = 0 or Ax = b is equivalent to the extended cross-product if the projective extension of the Euclidean syste...

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. Th...

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...

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 nea...

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 pro...

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. Th...

Approximation of scattered data is often a task in many engineering problems. The Radial Basis Function (RBF) approximation is appropriate for big scattered datasets in n–dimensional space. It is a non-separable approximation, as it is based on the distance between two points. This method leads to the solution of an overdetermined linear system of...

This contribution describes relationship between fractions, projective representation, duality, linear algebra and geometry. Many problems lead to a system of linear equations. This paper presents equivalence of the Cross-product operation and solution of a system of linear equations Ax=0 or Ax=b using projective space representation and homogeneou...

Many algorithms used are based on geometrical computation. There are several criteria in selecting appropriate algorithm from already known. Recently, the fastest algorithms have been preferred. On the contrary nowadays, algorithms with a high stability and acceptable algorithm complexity are preferred. Also today’s technology and computer architec...

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...

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 datase...

A convex hull of points in E2 is used in many applications. In spite of low computational complexity O(h logn) it takes considerable time if large data processing is needed. We present a new algorithm to speed up any planar convex hull calculation. It is based on a polar space subdivision and speed up known convex hull algorithms of 3,7 times and m...

This contribution describes relationship between fractions, projective representation, duality, linear algebra and geometry. Many problems lead to a system of linear equations and this paper presents equivalence of the cross–product operation and solution of a system of linear equations í µí±¨í µí² = í µí¿ or í µí±¨í µí² = í µí² using projectiv...

Many problems, not only in computer vision and visualization, lead to a system of linear equations Ax = 0 or Ax = b and fast and robust solution is required. A vast majority of computational problems in computer vision, visualization and computer graphics are three dimensional in principle. This paper presents equivalence of the cross–product opera...

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...

Approximation of scattered geometric 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 method is useful for a higher dimension d ≥2, because the other methods require a conversion of a scattered dataset to a semi-regul...

Many geometrically oriented problems lead to intersection computation or to its dual problems. In many cases the problem is reduced to intersection computation of two planes in E3, e.g. intersection of two triangles. However in several cases triangles are given by vertices in the homogeneous coordinates. The usual approach is to convert coordinates...

Many problems, not only in signal processing, image processing, digital imaging, computer vision and visualization, lead to the Least Square Error (LSE) problem or Total (Orthogonal) Least Square Error (TLSE) problem computation. Mostly the LSE is used due to its simplicity for problems leading to f(x, y) = h, resp. f(x, y, z) = h, i.e. to dependen...