Curve and Surface Fitting with Splines. PAUL DIERCKX Professor, Computer Science Department, Katholieke Universiteit Leuven, Belgium

Transcription

1 Curve and Surface Fitting with Splines PAUL DIERCKX Professor, Computer Science Department, Katholieke Universiteit Leuven, Belgium CLARENDON PRESS OXFORD 1995

2 - Preface List of Figures List of Tables PART I: SPLINE FUNCTIONS vii xv xvii 1 UNIVARIATE SPLINES Definitions The B-spline representation Divided differences B-splines: definition and basic properties Splines as linear combinations of B-splines On calculating with 5-splines Evaluation of a spline function Derivatives of a spline function The definite integral of a spline function Knot insertion algorithms Fourier coefficients of a spline function The zeros of a spline function 20 2 BIVARIATE SPLINES Tensor product splines Definitions Evaluation, differentiation and integration The spherical harmonic coefficients of bivariate splines Contour lines of a bivariate spline Splines on triangulations Powell-Sabin splines 33 XI

3 xii Simplex splines 36 PART II: CURVE FITTING 3 CURVE FITTING: AN INTRODUCTION Curve fitting: a constructive approach Curve fitting with splines A survey of methods Some extensions 51 4 LEAST-SQUARES SPLINE CURVE FITTING Least-squares splines with fixed knots The normal equations An orthogonalization method Least-squares splines with variable knots A non-linear constrained minimization problem A specific optimization algorithm An automatic curve fitting routine A starting position for the knots The optimal number of knots Numerical examples 69 5 SMOOTHING SPLINE CURVE FITTING The smoothing approximation criterion The smoothing spline The method of Lagrange The calculation of the B-spline coefficients The properties of the smoothing spline Determination of the smoothing parameter The knot placing strategy The choice of the smoothing factor Numerical results 91 6 MORE SMOOTHING SPLINE CURVES Smoothing with periodic splines Method Numerical results Smoothing with end point derivative constraints Zero end point derivatives Finite end point derivatives Infinite end point derivatives Smoothing with parametric splines 109

4 xiii Method Practical remarks Ill Spline curves for CAGD FITTING WITH CONVEXITY CONSTRAINTS Shape preserving approximation Convex least-squares cubic splines A quadratic programming problem The Theil-Van de Panne procedure The calculation of the B-spline coefficients The organizational aspects Automatic smoothing with convexity constraints Numerical results 129 PART III: SURFACE FITTING 8 SURFACE FITTING: AN INTRODUCTION Surface fitting: a constructive approach Surface fitting with splines Scattered data methods Mesh data methods Extensions Surface fitting: a variational approach SCATTERED DATA SURFACE FITTING The least-squares criterion The observation matrix Rank deficiency Curved knot lines The smoothing criterion The smoothing norm The smoothing spline The knot placing strategy Numerical results MESH DATA SURFACE FITTING Matrix calculus: notation and properties The least-squares criterion The smoothing criterion The new smoothing spline The choice of knots Numerical results 179

5 xiv 10.4 Incomplete grids Statement of the problem Direct method Iterative method Two-stage approximation methods Extremely large data sets MORE SCATTERED DATA SMOOTHING Smoothing using polar coordinates Smogthing splines for the unit disk Generalizations Numerical results Smoothing using spherical coordinates Smoothing splines for the sphere Numerical results MORE MESH DATA SMOOTHING Smoothing over a polar grid The smoothing problem Origin derivatives provided Optimal origin derivatives Practical considerations and results Smoothing over a spherical grid Surface reconstruction from planar contours Introduction Smoothing in cylindrical coordinates Smoothing with parametric surfaces Spline surfaces for CAGD 245 PART IV: FITPACK 13 AVAILABLE SOFTWARE The FITPACK software package Organization Availability and documentation Concluding remarks 266 References 271 Index 281