Shape-preserving, multiscale interpolation by bi- and multivariate cubic L1 splines

We introduce a class of bi- and multivariate cubic L-1 interpolating splines. the coefficients of which are calculated by minimizing the sum of the L-1 norms of second derivatives. The focus is mainly on bivariate cubic L-1 splines for C-1 interpolation of data located at the nodes of a tenser-product grid. These L-1 splines preserve the shape of data even when the data have abrupt changes in magnitude or spacing. Extensions to interpolation of regularly spaced and scattered bi-and multivariate data by cubic and higher-degree surfaces/hypersurfaces on regular and irregular rectangular/quadrilateral/hexahedral and triangular/tetrahedral grids are outlined. (C) 2001 Elsevier Science B.V. All rights reserved.