splines and their impact · pdf file•a practical guide to splines, carl de boor,...
TRANSCRIPT
![Page 1: Splines and Their Impact · PDF file•A Practical Guide to Splines, Carl de Boor, Springer, 1978. •Spline Functions: Basic Theory, Larry L. Schumaker, Wylie, 1981. Applications](https://reader031.vdocument.in/reader031/viewer/2022013110/5ab8e5eb7f8b9a28468d704d/html5/thumbnails/1.jpg)
Larry L. Schumaker
Center for Constructive Approximation
Vanderbilt University
Splines and Their Impact
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I. J. Schoenberg
THE BEGINNING
April 21, 1903 – Feb. 21, 1990
Contributions to the problem of approximation of equidistant data by analytic functions,
Quart. Appl Math. 4 (1946), 45—99 and 112—141.
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Duck
Draftsman’s Spline
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The Mathematical Model
The Natural Cubic Spline
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http://www.cs.wisc.edu/~deboor/bib/bib.html
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Journals
Constructive Approximation
Journal of Approximation Theory
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•Computer-Compatible Functions
•Store•Evaluate using + - * / <
•Polynomials
•Spline = Piecewise Polynomial
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Polynomial Splines•Finite dimensional space
•Stable local basis (B-splines)
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• Easy to store, evaluate, and manipulate
• Good approximation power
•Derivatives•Integrals•Inner-products
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•A Practical Guide to Splines, Carl de Boor, Springer, 1978.
•Spline Functions: Basic Theory, Larry L. Schumaker, Wylie, 1981
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Applications•Interpolation
•Smoothing noisy data
•Approximation of functions
•Solution of ODE’s
•Design of (parametric) curves
•Signal processing (wavelets)
•Image processing (edge detection)
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Parametric Spline with Control Polygon
B-spline demo: http://www.flm.mw.tum.de/mitarbeiter/kraemers/public_html/bspline_demo.html
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Font Design
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•Piecewise solution of ODE
•More general smoothness conditions between pieces
•Similar properties to polynomial splines
•Examples: trigonometric, hyperbolic, exponential splines
Generalized Splines
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Cam Design
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The Zoo of Splines
•Analytic splines•Arc splines•Beta splines•B-splines•Bernoulli splines•Box splines•Cardinal splines•Circular splines•Complete splines•Complex splines•Confined splines•Deficient splines•D^m splines•Discrete splines•Euler splines•Exponential splines•Gamma splines•GB-splines•HB-splines•Hyperbolic Splines
•Monosplines•Nu-splines•Natural splines•L-splines•Lg-splines•Nonlinear splines•One-sided splines•Parabolic splines•Perfect splines•Periodic splines•Poly-splines•Rational splines•Simplex splines•Spherical splines•Taut splines•Tchebycheffian splines•Tension splines•Thin plate splines•Trigonometric splines•Whittaker splines
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Tensor Product Splines
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Bivariate Splines on Triangulations
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Constructive Theory
•Dimension
•Stable local basis
•Approximation power
•Applications
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Local Support
Approximation Power
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Applications
•Interpolation
•Smoothing noisy data
•Approximation of functions
•Solution of PDE’s
•Design of (parametric) surfaces
•Image processing (compression, denoising)
Spline Functions on Triangulations, M.-J. Lai and Larry L. Schumaker, Cambridge University Press, 2007.
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Trivariate Splines•Piecewise polynomials on tetrahedral partition
•Constructive theory
•Volume modelling (CAT scans)
•Finite element method for PDE’s
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• Approximation Theory• Numerical Analysis• Computer Science• Application Areas
Engineering Biosciences, Chemistry, Physics, Geophysics, Meteorology Medicine Business and Social Sciences Imaging and Visualization Computer-aided design and Manufacture
Computer Vision and Robotics
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Books
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Books