Theory and algorithms for shape-preserving bivariate cubic L1 splines.
Abstract (Summary)A major objective of modelling geophysical features, biological objects, financial processes and many other irregular surfaces and functions is to develop "shape-preserving" methodologies for smoothly interpolating bivariate data with sudden changes in magnitude or spacing. Shape preservation usually means the elimination of extraneous non-physical oscillations. Classical splines do not preserve shape well in this sense. Empirical experiments have shown that the recently proposed cubic L1 splines are cable of providing C1-smooth, shape-preserving, multi-scale interpolation of arbitrary data, including data with abrupt changes in spacing and magnitude, with no need for node adjustment or other user input. However, a theoretic treatment of the bivariate cubic L1 splines is still lacking. The currently available approximation algorithms are not able to generate the exact coefficients of a bivariate cubic L1 spline. For theoretical treatment and the algorithm development, we propose to solve bivariate cubic L1 spline problems in a generalized geometric programming framework. Our framework includes a primal problem, a geometric dual problem with a linear objective function and convex cubic constraints, and a linear system for dual-to-primal transformation. We show that bivariate cubic L1 splines indeed preserve linearity under some mild conditions. Since solving the dual geometric program involves heavy computation, to improve computational efficiency, we further develop three methods for generating bivariate cubic L1 splines: a tensor-product approach that generates a good approximation for large scale bivariate cubic L1 splines; a primal-dual interior point method that obtains discretized bivariate cubic L1 splines robustly for small and medium size problems; and a compressed primal-dual method that efficiently and robustly generates discretized bivariate cubic L1 splines of large size.
School Location:USA - North Carolina
Source Type:Master's Thesis
Date of Publication:04/12/2005