Spectral geometry and Riemannian manifold mesh approximations: some autocorrelation lessons from spatial statistics

Abstract

A spectral geometry utility awareness, with specific reference to isospectralisation and art painting analytics, is permeating the academy today, with special interest in its ability to foster interfaces between a range of analytical quantitative disciplines and art, exhibiting popularity in, for example, computer engineering/image processing and GIScience/spatial statistics, among other subject areas. This paper contributes to the emerging literature about such mathematized interdisciplinarities and synergies. It more specifically extends the matrix algebra based 2-D Graph Moranian operator that dominates spatial statistics/econometrics to the 3-D Riemannian manifold sphere whose analysis the general Graph Laplacian (i.e. Laplace-Beltrami) operator monopolizes today. One conclusion is that harmonizing the use of these two operators offers a way to expand knowledge and comprehension. Another is a continuing demonstration that the understanding and analysis of art sculptures dovetails with mathematics-art studies.