Multivariate polysplines are a new mathematical technique that has arisen from a synthesis of approximation theory and the theory of partial differential equations. It is an invaluable means to interpolate practical data with smooth functions. Multivariate polysplines have applications in the design of surfaces and "smoothing" that are essential in computer aided geometric design (CAGD and CAD/CAM systems), geophysics, magnetism, geodesy, geography, wavelet analysis and signal and image processing. In many cases involving practical data in these areas, polysplines are proving more effective than well-established methods, such as kKriging, radial basis functions, thin plate splines and minimum curvature.
Part 1 assumes no special knowledge of partial differential equations and is intended as a graduate level introduction to the topic Part 2 develops the theory of cardinal Polysplines, which is a natural generalization of Schoenberg's beautiful one-dimensional theory of cardinal splines. Part 3 constructs a wavelet analysis using cardinal Polysplines. The results parallel those found by Chui for the one-dimensional case. Part 4 considers the ultimate generalization of Polysplines - on manifolds, for a wide class of higher-order elliptic operators and satisfying a Holladay variational property.
Applied and pure mathematicians, computer scientists and researchers and engineers in signal and image processing, CAGD and CAD/CAM systems, geophysics, geography, magnetism and related disciplines.
Preface 1 Introduction 1.1 Organization of Material 1.1.1 Part I: Introduction of Polysplines 1.1.2 Part II: Cardinal Polysplines 1.1.3 Part III: Wavelet Analysis Using Polysplines 1.1.4 Part IV: Polysplines on General Interfaces 1.2 Audience 1.3 Statements 1.4 Acknowledgements 1.5 The Polyharmonic Paradigm 1.5.1 The Operator, Object and Data Concepts of the Polyharmonic Paradigm 1.5.2 The Taylor Formula Part I Introduction to Polysplines 2 One-Dimensional Linear and Cubic Splines 2.1 Cubic Splines 2.2 Linear Splines 2.3 Variational (Holladay) Property of the Odd-Degree Splines 2.4 Existence and Uniqueness of Odd-Degree Splines 2.5 The Holladay Theorem 3 The Two-Dimensional Case: Data and Smoothness Concepts 3.1 The Data Concept in Two Dimensions According to the Polyharmonic Paradigm 3.2 The Smoothness Concept According to the Polyharmonic Paradigm 4 The Objects Concept: Harmonic and Polyharmonic Functions in Rectangular Domains in ℝ2 4.1 Harmonic Functions in Strips or Rectangles 4.2 “Parametrization” of the Space of Periodic Harmonic Functions in the Strip: the Dirichlet Problem 4.3 “Parametrization” of the Space of Periodic Polyharmonic Functions in the Strip: the Dirichlet Problem 4.4 Nonperiodicity in y 5 Polysplines on Strips in ℝ2 5.1 Periodic Harmonic Polysplines on Strips, p = 5.2 Periodic Biharmonic Polysplines on Strips, p = 5.3 Computing the Biharmonic Polysplines on Strips 5.4 Uniqueness of the Interpolation Polysplines 6 Application of Polysplines to Magnetism and CAGD 6.1 Smoothing Airborne Magnetic Field Data 6.2 Applications to Computer-Aided Geometric Design 6.3 Conclusions 7 The Objects Concept: Harmonic and Polyharmoni
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- © Academic Press 2001
- 11th June 2001
- Academic Press
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Ognyan Kounchev received his M.S. in partial differential equations from Sofia University, Bulgaria and his Ph.D. in optimal control of partial differential equations and numerical methods from the University of Belarus, Minsk. He was awarded a grant from the Volkswagen Foundation (1996-1999) for studying the applications of partial differential equations in approximation and spline theory. Currently, Dr Kounchev is a Fulbright Scholar at the University of Wisconsin-Madison where he works in the Wavelet Ideal Data Representation Center in the Department of Computer Sciences.
University of Wisconsin, Madison, U.S.A.