Polygonal blending splines in application to image processing

The paper proposes a novel method of image representation. The basic idea of the method is to transform color images to continuous parametric surfaces. The proposed technique is based on a class of special basis functions, defined on the polygon grid. Besides a exible and symmetric construction, these basis functions are strictly local and Cd-smooth on the entire domain. Having a number of unique features, the proposed representation can be used in various image processing tasks. The main purpose of this paper is to demonstrate the process of the image transformation and discuss possible applications of the presented technique

Representation and application of spline-based finite elements

Isogeometric analysis, as a generalization of the finite element method, employs spline methods to achieve the same representation for both geometric modeling and analysis purpose. Being one of possible tool in application to the isogeometric analysis, blending techniques provide strict locality and smoothness between elements. Motivated by these features, this thesis is devoted to the design and implementation of this alternative type of finite elements. This thesis combines topics in geometry, computer science and engineering. The research is mainly focused on the algorithmic aspects of the usage of the spline-based finite elements in the context of developing generalized methods for solving different model problems. The ability for conversion between different representations is significant for the modeling purpose. Methods for conversion between local and global representations are presented

Isogeometric analysis using a tensor product blending spline construction

In this paper, we introduce a blending spline type construction into the isogeometric analysis. We consider a general algorithm for solving a number of boundary value problems, including heat equation, wave equation, linear elasticity, etc. The usage of blending spline construction in the isogeometric context mixes standard finite element and NURBS-based approaches while accumulating the benefits of both. Since the blending spline construction is locally represented, the finite element discretization can be formulated in a standard way, while the smooth representation of these splines provides an accurate approximation of the computational domain even on a coarse mesh. Besides the standard L 2 -projection algorithm for the domain approximation, we demonstrate a unique scheme for domain construction based on local surfaces and subsequent hp-refinement. In the proposed paper we focus on the features of blending splines in the isogeometric analysis context, identify possible applications, and provide some numerical analysis

Representation and application of spline-based finite elements

Isogeometric analysis, as a generalization of the finite element method, employs spline methods to achieve the same representation for both geometric modeling and analysis purpose. Being one of possible tool in application to the isogeometric analysis, blending techniques provide strict locality and smoothness between elements. Motivated by these features, this thesis is devoted to the design and implementation of this alternative type of finite elements. This thesis combines topics in geometry, computer science and engineering. The research is mainly focused on the algorithmic aspects of the usage of the spline-based finite elements in the context of developing generalized methods for solving different model problems. The ability for conversion between different representations is significant for the modeling purpose. Methods for conversion between local and global representations are presented

Blending spline surfaces over polygon mesh and their application to isogeometric analysis

Finite elements are allowed to be of a shape suitable for the specific problem. This choice defines thereafter the accuracy of the approximated solution. Moreover, flexible element shapes allow for the construction of an arbitrary domain topology. Polygon meshes are a common representation of the domain that cover any choice of the finite element shape. Being an alternative tool for modeling and analysis, blending spline surfaces support representation on polygon grids. The blending splines have a hierarchical structure, which is obtained by generating local surfaces that cover each node support and then blended with a special type of basis functions. This type of splines in their tensor product form is suitable for application to isogeometric analysis problems. A more general representation constructed on polygonal elements can be used on a wider range of domain topology in comparison with tensor product surfaces. In this paper we introduce a novel approach to constructing curvilinear polygon meshes in the blending spline representation in application to the isogeometric analysis context. The focus is on generating a novel special type of basis functions on a connected collection of polygons, with triangles and quadrilaterals as particular cases. The purpose of the proposed paper is to show applications of this construction to various numerical problems, as well as to generalize the approach to evaluating these basis functions on arbitrary planar domains

Finite element application of ERBS extraction

In this paper we explore ERBS extraction in application to a heat conduction problem.Two types of basis functions are considered: B-spline and expo-rational B-spline (ERBS)combined with Bernstein polynomials. The first one is a global basis defined on a subdomain whose size is related to the spline degree, while the second one is a strictlylocal basis under the local geometry. While all the coefficients of the B-spline tensor product surface affect each element of the patch, local surfaces of the blending surface conserve both local manipulation and smoothness of the global surface. We show the conversion between these bases by using an ERBS extraction. The extraction operator allows us to convert the control points of the surface evaluated by B-splines to control points of the ERBS surface and vice versa. This approach is demonstrated on an example of the finite element solution approximation of the heat equation

Regression analysis using a blending type spline construction

Regression analysis allows us to track the dynamics of change in measured data and to investigate their properties. A sufficiently good model allows us to predict the behavior of dependent variables with higher accuracy, and to propose a more precise data generation hypothesis. By using polynomial approximation for big data sets with complex dependencies we get piecewise smooth functions. One way to obtain a smooth spline representation of an entire data set is to use local curves and to blend them using smooth basis functions. This construction allows the computation of derivatives at any point on the spline. Properties such as tangent, velocity, acceleration, curvature and torsion can be computed, which gives us the opportunity to exploit these data in the subsequent analysis. We can adjust the accuracy of the approximation on the different segments of the data set by choosing a suitable knot vector. This article describes a new method for determining the number and location of the knot-points, based on changes in the Frenet frame. We present a method of implementation using generalized expo-rational B-splines (GERBS) for regression problems (in two and three variables) and we evaluate the accuracy of the model using comparison of the residuals