Scattered Data Approximation on the Bisphere and Application to Texture Analysis

Abstract

The paper deals with the approximation and optimal interpolation of functions defined on the bisphere \(\mathbb {S}^{2}\times \mathbb {S}^{2}\) from scattered data. We demonstrate how the least square approximation to the function can be computed in a stable and efficient manner. The analysis of this problem is based on Marcinkiewicz–Zygmund inequalities for scattered data which we present here for the bisphere. The complementary problem of optimal interpolation is also solved by using well-localized kernels for our setting. Finally, we discuss the application of the developed methods to problems of texture analysis in material science.