Shape-preserving, first-derivative-based parametric and nonparametric cubic L1 spline curves

We investigate parametric and nonparametric cubic L"1 interpolating spline curves (''L"1 splines'') in two and three dimensions with the goal of achieving shape-preserving interpolation of irregular data. We introduce five types of parametric L"1 and L"2 splines calculated by minimizing expressions involving L"1 norms, L"2 norms and squares of L"2 norms of second derivatives and five types of parametric L"1 and L"2 splines calculated by minimizing analogous expressions involving first derivatives minus first differences. We compare these splines among themselves, with a simple monotonicity-based interpolant and with the interpolant of Brodlie, Fritsch and Butland. Of all of the parametric splines, first-derivative-based ''interactive-component''L"1 splines preserve the shape of irregular data best. Nonparametric first-derivative-based L"1 splines are introduced and shown to preserve shape better than the previously known nonparametric second-derivative-based L"1 splines, than nonparametric first- and second-derivative-based L"2 splines and than the simple monotonicity-based and Brodlie-Fritsch-Butland interpolants.

[1]  R. Schaback.  On global GC 2 convexity preserving interpolation of planar curves by piecewise Be´zier polynomials , 1989 .

[2]  Robert J. Vanderbei,et al.  A modification of karmarkar's linear programming algorithm , 1986, Algorithmica.

[3]  Tim N. T. Goodman,et al.  Shape-preserving interpolation by splines using vector subdivision , 2005, Adv. Comput. Math..

[4]  R. E. Carlson,et al.  A Bivariate Interpolation Algorithm for Data that are Monotone in One Variable , 1991, SIAM J. Sci. Comput..

[5]  Panagiotis D. Kaklis,et al.  Advanced Course on FAIRSHAPE , 2012, Vieweg+Teubner Verlag.

[6]  R. E. Carlson,et al.  Monotone Piecewise Cubic Interpolation , 1980 .

[7]  Robert J. Vanderbei,et al.  Affine-scaling for linear programs with free variables , 1989, Math. Program..

[8]  Tim N. T. Goodman,et al.  Shape preserving interpolation by curvature continuous parametric curves , 1988, Comput. Aided Geom. Des..

[9]  Renato D. C. Monteiro,et al.  Limiting behavior of the affine scaling continuous trajectories for linear programming problems , 1991, Math. Program..

[10]  S. Fang,et al.  Shape-preserving properties of univariate cubic L 1 splines , 2005 .

[11]  Panagiotis D. Kaklis,et al.  Spatial shape‐preserving interpolation using ν‐splines , 2004, Numerical Algorithms.

[12]  Helmuth Späth,et al.  Two-dimensional spline interpolation algorithms , 1993 .

[13]  F. Fritsch,et al.  A Method for Constructing Local Monotone Piecewise Cubic Interpolants , 1984 .

[14]  Fred N. Fritsch,et al.  Monotonicity preserving bicubic interpolation: A progress report , 1985, Comput. Aided Geom. Des..

[15]  Masakazu Muramatsu,et al.  Global Convergence of a Long-Step Affine Scaling Algorithm for Degenerate Linear Programming Problems , 1995, SIAM J. Optim..

[16]  John C. Clements.  A convexity-preservingC2 parametric rational cubic interpolation , 1992 .

[17]  Helmuth Späth,et al.  One-dimensional spline interpolation algorithms , 1995 .

[18]  Carla Manni,et al.  Shape-Preserving C3 Interpolation: The Curve Case , 2003, Adv. Comput. Math..

[19]  Panagiotis D. Kaklis,et al.  Convexity-preserving interpolatory parametric splines of non-uniform polynomial degree , 1995 .

[20]  R. E. Carlson,et al.  Improved accuracy of multiquadric interpolation using variable shape parameters , 1992 .

[21]  R. E. Carlson,et al.  A Note on Piecewise Monotonic Bivariate Interpolation , 1991, Curves and Surfaces.

[22]  John E. Lavery.  Univariate cubic Lp splines and shape-preserving, multiscale interpolation by univariate cubic L1 splines , 2000, Comput. Aided Geom. Des..

[23]  T. N. T. Goodman.  Shape Preserving Interpolation by Parametric Rational Cubic Splines , 1988 .

[24]  John E. Lavery,et al.  Univariate cubic L1 splines – A geometric programming approach , 2002, Math. Methods Oper. Res..

[25]  Larry L. Schumaker,et al.  Approximation theory X : wavelets, splines, and applications , 2002 .

[26]  John E. Lavery.  Shape-preserving approximation of multiscale univariate data by cubic L1 spline fits , 2004, Comput. Aided Geom. Des..

[27]  Tim N. T. Goodman.  Shape Preserving Interpolation by Planar Curves , 1996, Advanced Course on FAIRSHAPE.

[28]  Panagiotis D. Kaklis,et al.  Fairing spatial B-spline curves , 1996 .

[29]  George Celniker,et al.  Deformable curve and surface finite-elements for free-form shape design , 1991, SIGGRAPH.

[30]  Robert J. Vanderbei,et al.  Two-thirds is sharp for affine scaling , 1993, Oper. Res. Lett..

[31]  Tzvetomir Ivanov Vassilev,et al.  Fair interpolation and approximation of B-splines by energy minimization and points insertion , 1996, Comput. Aided Des..

[32]  Y. Dodge.  on Statistical data analysis based on the L1-norm and related methods , 1987 .

[33]  John E. Lavery,et al.  Shape-preserving, multiscale fitting of univariate data by cubic L1 smoothing splines , 2000, Comput. Aided Geom. Des..

[34]  Roger Koenker,et al.  Elastic and Plastic Splines: Some Experimental Comparisons , 2002 .

[35]  R. E. Carlson,et al.  An algorithm for monotone piecewise bicubic interpolation , 1989 .

[36]  Tim N. T. Goodman,et al.  Shape preserving interpolation by space curves , 1997, Comput. Aided Geom. Des..

[37]  Carla Manni,et al.  Shape‐preserving interpolation by G1 and G2 PH quintic splines , 2003 .

[38]  J. Kjellander.  Smoothing of cubic parametric splines , 1983 .

[39]  John E. Lavery,et al.  Shape-preserving, multiscale interpolation by bi- and multivariate cubic L1 splines , 2001, Comput. Aided Geom. Des..

[40]  R. E. Carlson,et al.  Sparse approximate multiquadric interpolation , 1994 .

[41]  E. Wagner.  International Series of Numerical Mathematics , 1963 .

[42]  John E. Lavery.  Shape-preserving, multiscale interpolation by univariate curvature-based cubic L1 splines in Cartesian and polar coordinates , 2002, Comput. Aided Geom. Des..

[43]  Tim N. T. Goodman,et al.  Automatic interpolation by fair, shape-preserving, G2 space curves , 1998, Comput. Aided Des..

[44]  Gerald E. Farin,et al.  Automatic fairing algorithm for B-spline curves , 1990, Comput. Aided Des..

[45]  R. E. Carlson,et al.  The parameter R2 in multiquadric interpolation , 1991 .

[46]  Panagiotis D. Kaklis,et al.  Shape-preserving interpolation in R3 , 1997 .