Geometric Dual Formulation for First-derivative-based Univariate Cubic $L_1$ Splines

Y.B. Zhao (ybzhaoamss.ac.cn)
S.C. Fang (fangeos.ncsu.edu)
J.E. Lavery (john.lavery2us.army.mil)

Abstract: With the objective of generating ``shape-preserving'' smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing, we study a class of first-derivative-based ${\cal C}^1$-smooth univariate cubic $L_1$ splines. An $L_1$ spline minimizes the $L_1$ norm of the difference between the first-order derivative of the spline and the local divided difference of the data. Calculating the coefficients of an $L_1$ spline is a nonsmooth nonlinear convex program. Via Fenchel's conjugate transformation, the geometric dual program is a smooth convex program with a linear objective function and convex cubic constraints. The dual-to-primal transformation is accomplished by solving a linear program.

Keywords: Conjugate function, convex program, cubic $L_1$ spline, shape-preserving interpolation

Category 1: Applications -- Science and Engineering

Category 2: Convex and Nonsmooth Optimization

Category 3: Nonlinear Optimization

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Entry Submitted: 08/11/2006
Entry Accepted: 08/12/2006
Entry Last Modified: 08/11/2006

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