Stationary subdivision / Alfred S. Cavaretta, Wolfgang Dahmen, Charles A. Micchelli.

Format:

Book

Author/Creator:

Cavaretta, Alfred S., 1944- author.

Dahmen, Wolfgang, 1949- author.

Micchelli, Charles A., author.

Contributor:

American Mathematical Society.

Series:

Memoirs of the American Mathematical Society ; Volume 93, Number 453.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 93, Number 453

Language:

English

Subjects (All):

Spline theory.

Algorithms.

Computer graphics.

Physical Description:

1 online resource (197 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 1991.

Language Note:

English

Summary:

Subdivision methods in computer graphics constitute a large class of recursive schemes for computing curves and surfaces. They seem to have their origin in the geometric problem of smoothing the corners of a given polyhedral surface - in fact, these methods are sometimes called "wood carver" algorithms because the repeated smoothing operations are analogous to sculpting wood. This book presents a systematic development of the basic mathematical principles and concepts associated with stationary subdivision algorithms. The authors pay special attention to the structure of such algorithms in a multidimensional setting and analyse the convergence issue using appropriate tools from Fourier analysis and functional analysis. The analytic structure of the limiting curves and surfaces is revealed in two ways: the smoothness of these surfaces is determined by certain algebraic properties of the algorithm, while the highest order derivatives of the limiting surfaces are shown to be fractals. Scientists interested in computer graphics, splines, wavelets, and multiresolution analysis will find the analytic and algebraic tools developed here more than useful.

Contents:

""Table of Contents""; ""1. Introduction""; ""2. Subdivision Schemes: Convergence Concepts and the Associated Functional Equation""; ""2.1. The form of the limiting surface of uniformly convergent subdivision schemes""; ""2.2. Consequences of the finite support of the mask""; ""2.3. Other notions of convergence; weakly convergent schemes""; ""2.4. Matrix masks""; ""3. Contractivity of the Subdivision Operator""; ""3.1. Contractivity as a convergence criterion""; ""3.2. Contractivity for masks supported on convex sets""; ""3.3. Contractivity via factorization of the subdivision operator""

""4. Subdivision from Dimension Compression""""4.1. Compression of the refinable function[omitted]""; ""4.2. The algebra of compressed schemes""; ""4.3. Convergence theorems for compressed schemes""; ""4.4. The line average algorithm""; ""5. Solution of the Functional Equation""; ""5.1. Necessary conditions in terms of the geometric mean""; ""5.2. Sufficient conditions based on the Paley�Wiener theorem""; ""5.3. Conditions for convergence of the subdivision scheme suggested by the mean ergodic theorem""; ""6. Algebraic Properties of Subdivision Schemes""

""6.1. The subdivision operator on polynomial sequences""""6.2. Spectral properties of S on polynomial sequence spaces""; ""6.3. Polynomial subspaces generated by convergent subdivision schemes""; ""6.4. Matrix representation for a local convergence analysis of regular subdivision schemes; matrix subdivision schemes""; ""7. Matrix Refinement Equation""; ""7.1. Contractivity for matrix subdivision schemes""; ""7.2. Definition of refinement pairs""; ""7.3. Refinement pairs which produce polynomial surfaces""

""7.4. Necessary and sufficient conditions for the generation of smooth surfaces by a refinement pair""""7.5. The fractal nature of surfaces generated by a refinement pair""; ""8. Smoothness of S�Refinable Functions and Consequences""; ""8.1. Determining smoothness of the refinahle function using differenced subdivision schemes""; ""8.2. Subdivision schemes with smooth refinahle functions generate polynomials""; ""8.3. Univariate subdivision schemes producing piecewise polynomial functions""; ""9. Appendix""; ""References""

Notes:

"September 1991, Volume 93, Number 453 (second of 3 numbers)."

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-0879-1