Multivariate Polysplines

Multivariate Polysplines

Applications to Numerical and Wavelet Analysis

1st Edition - June 11, 2001

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  • Author: Ognyan Kounchev
  • eBook ISBN: 9780080525006

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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.

Key Features

  • 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

Table of Contents

  • 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 Polyharmonic Functions in Annuli in ℝ2

    7.1 Harmonic Functions in Spherical (Circular) Domains

    7.2 Biharmonic and Polyharmonic Functions

    7.3 “Parametrization” of the Space of Polyharmonic Functions in the Annulus and Ball: the Dirichlet Problem

    8 Polysplines on annuli in ℝ2

    8.1 The Biharmonic Polysplines, p = 2

    8.2 Radially Symmetric Interpolation Polysplines

    8.3 Computing the Polysplines for General (Nonconstant) Data

    8.4 The Uniqueness of Interpolation Polysplines on Annuli

    8.5 The change v = log r and the Operators Mk,p

    8.6 The Fundamental Set of Solutions for the Operator Mk,p(d/dv)

    9 Polysplines on Strips and Annuli in ℝn

    9.1 Polysplines on Strips in ℝn

    9.2 Polysplines on Annuli in ℝn

    10 Compendium on Spherical Harmonics and Polyharmonic Functions

    10.1 Introduction

    10.2 Notations

    10.3 Spherical Coordinates and the Laplace Operator

    10.4 Fourier Series and Basic Properties

    10.5 Finding the Point of View

    10.6 Homogeneous Polynomials in ℝn

    10.7 Gauss Pepresentation of Homogeneous Polynomials

    10.8 Gauss Representation: Analog to the Taylor Series, the Polyharmonic Paradigm

    10.9 The Sets ℋk are Eigenspaces for the Operator ∆θ

    10.10 Completeness of the Spherical Harmonics in L2(𝕊n-1)

    10.11 Solutions of ∆w(x) = 0 with Separated Variables

    10.12 Zonal Harmonics : the Functional Approach

    10.13 The Classical Approach to Zonal Harmonics

    10.14 The Representation of Polyharmonic Functions Using Spherical Harmonics

    10.15 The Operator is Formally Self-Adjoint

    10.16 The Almansi Theorem

    10.17 Bibliographical Notes

    11 Appendix on Chebyshev Splines

    11.1 Differential Operators and Extended Complete Chebyshev Systems

    11.2 Divided Differences for Extended Complete Chebyshev Systems

    11.3 Dual Operator and ECT-System

    11.4 Chebyshev Splines and One-Sided Basis

    11.5 Natural Chebyshev Splines

    12 Appendix on Fourier Series and Fourier Transform

    12.1 Bibliographical Notes

    Bibliography to Part I

    Part II Cardinal Polysplines in ℝn

    13 Cardinal L-Splines According to Micchelli

    13.1 Cardinal L-Splines and the Interpolation Problem

    13.2 Differential Operators and their Solution Sets UZ+1

    13.3 Variation of the Set UZ+1[Λ] with Λ and Other Properties

    13.4 The Green Function (x) of the Operator ℒZ+1

    13.5 The Dictionary: L-Polynomial Case

    13.6 The Generalized Euler Polynomials AZ(x; λ)

    13.7 Generalized Divided Difference Operator

    13.8 Zeros of the Euler–Frobenius Polynomial ΠZ(λ)

    13.9 The Cardinal Interpolation Problem for L-Splines

    13.10 The Cardinal Compactly Supported L-Splines QZ+1

    13.11 Laplace and Fourier Transform of the Cardinal TB-Spline QZ+1

    13.12 Convolution Formula for Cardinal TB-Splines

    13.13 Differentiation of Cardinal TB-Splines

    13.14 Hermite–Gennocchi-Type Formula

    13.15 Recurrence Relation for the TB-Spline

    13.16 The Adjoint Operator ℒ*Z+1 and the TB-Spline Q*Z+1(x)

    13.17 The Euler Polynomial AZ(x; λ) and the TB-Spline QZ+1(x)

    13.18 The Leading Coefficient of the Euler–Frobenius Polynomial ΠZ(λ)

    13.19 Schoenberg’s “Exponential” Euler L-Spline ΦZ(x; λ) and AZ(x; λ)

    13.20 Marsden’s Identity for Cardinal L-Splines

    13.21 Peano Kernel and the Divided Difference Operator in the Cardinal Case

    13.22 Two-Scale Relation (Refinement Equation) for the TB-Splines QZ+1[Λ; h]

    13.23 Symmetry of the Zeros of the Euler–Frobenius Polynomial ΠZ(λ)

    13.24 Estimates of the Functions AZ(x; λ) and QZ+1(x)

    14 Riesz Bounds for the Cardinal L-Splines QZ+1

    14.1 Summary of Necessary Results for Cardinal L-Splines

    14.2 Riesz Bounds

    14.3 The Asymptotic of AZ(0; λ) in k

    14.4 Asymptotic of the Riesz Bounds A, B

    14.5 Synthesis of Compactly Supported Polysplines on Annuli

    15 Cardinal interpolation Polysplines on annuli 287

    15.1 Introduction

    15.2 Formulation of the Cardinal Interpolation Problem for Polysplines

    15.3 α = 0 is good for all L-Splines with L = Mk,p

    15.4 Explaining the Problem

    15.5 Schoenberg’s Results on the Fundamental Spline L(X) in the Polynomial Case

    15.6 Asymptotic of the Zeros of ΠZ(λ; 0)

    15.7 The Fundamental Spline Function L(X) for the Spherical Operators Mk,p

    15.8 Synthesis of the Interpolation Cardinal Polyspline

    15.9 Bibliographical Notes

    Bibliography to Part II

    Part III Wavelet Analysis

    16 Chui’s Cardinal Spline Wavelet Analysis

    16.1 Cardinal Splines and the Sets Vj

    16.2 The Wavelet Spaces Wj

    16.3 The Mother Wavelet ψ

    16.4 The Dual Mother Wavelet ψ

    16.5 The Dual Scaling Function φ

    16.6 Decomposition Relations

    16.7 Decomposition and Reconstruction Algorithms

    16.8 Zero Moments

    16.9 Symmetry and Asymmetry

    17 Cardinal L-Spline Wavelet Analysis

    17.1 Introduction: the Spaces Vj and Wj

    17.2 Multiresolution Analysis Using L-Splines

    17.3 The Two-Scale Relation for the TB-Splines QZ+1(x)

    17.4 Construction of the Mother Wavelet ψh

    17.5 Some Algebra of Laurent Polynomials and the Mother Wavelet ψh

    17.6 Some Algebraic Identities

    17.7 The Function ψh Generates a Riesz Basis of W0

    17.8 Riesz Basis from all Wavelet Functions ψ(x)

    17.9 The Decomposition Relations for the Scaling Function QZ+1

    17.10 The Dual Scaling Function φ and the Dual Wavelet ψ

    17.11 Decomposition and Reconstruction by L-Spline Wavelets and MRA

    17.12 Discussion of the Standard Scheme of MRA

    18 Polyharmonic Wavelet Analysis: Scaling and Rotationally Invariant Spaces

    18.1 The Refinement Equation for the Normed TB-Spline QZ+1

    18.2 Finding the Way: some Heuristics

    18.3 The Sets PVj and Isomorphisms

    18.4 Spherical Riesz Basis and Father Wavelet

    18.5 Polyharmonic MRA

    18.6 Decomposition and Reconstruction for Polyharmonic Wavelets and the Mother Wavelet

    18.7 Zero Moments of Polyharmonic Wavelets

    18.8 Bibliographical Notes

    Bibliography to Part III

    Part IV Polysplines for General Interfaces

    19 Heuristic Arguments

    19.1 Introduction

    19.2 The Setting of the Variational Problem

    19.3 Polysplines of Arbitrary Order p

    19.4 Counting the Parameters

    19.5 Main Results and Techniques

    19.6 Open Problems

    20 Definition of Polysplines and Uniqueness for General Interfaces

    20.1 Introduction

    20.2 Definition of Polysplines

    20.3 Basic Identity for Polysplines of even Order p = 2q

    20.4 Uniqueness of Interpolation Polysplines and Extremal Holladay-Type Property

    21 A Priori Estimates and Fredholm Operators

    21.1 Basic Proposition for Interface on the Real Line

    21.2 A Priori Estimates in a Bounded Domain with Interfaces

    21.3 Fredholm Operator in the Space H2p+r(D\ST ) for r ≥ 0

    22 Existence and Convergence of Polysplines

    22.1 Polysplines of Order 2q for Operator L = L

    22.2 The Case of a General Operator L

    22.3 Existence of Polysplines on Strips with Compact Data

    22.4 Classical Smoothness of the Interpolation Data gj

    22.5 Sobolev Embedding in Ck,α

    22.6 Existence for an Interface which is not C∞

    22.7 Convergence Properties of the Polysplines

    22.8 Bibliographical Notes and Remarks

    23 Appendix on Elliptic Boundary Value Problems in Sobolev and Hölder Spaces

    23.1 Sobolev and Hölder Spaces

    23.2 Regular Elliptic Boundary Value Problems

    23.3 Boundary Operators, Adjoint Problem and Green Formula

    23.4 Elliptic Boundary Value Problems

    23.5 Bibliographical Notes

    24 Afterword

    Bibliography to Part IV


Product details

  • No. of pages: 498
  • Language: English
  • Copyright: © Academic Press 2001
  • Published: June 11, 2001
  • Imprint: Academic Press
  • eBook ISBN: 9780080525006

About the Author

Ognyan Kounchev

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.

Affiliations and Expertise

University of Wisconsin, Madison, U.S.A.

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