2019/07/17 by Johannes Wallner, Wallner, Johannes · 2 citations
Computer Science · Mathematics · #Digital Image Processing Techniques #FOS: Mathematics #Morphological variations and asymmetry #Numerical Analysis (math.NA) #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.1907.07550
openalex publication_date 2019/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Any procedure applied to data, and any quantity derived from data, is required to respect the nature and symmetries of the data. This axiom applies to refinement procedures and multiresolution transforms as well as to more basic operations like averages. This chapter discusses different kinds of geometric structures like metric spaces, Riemannian manifolds, and groups, and in what way we can make elementary operations geometrically meaningful. A nice example of this is the Riemannian metric naturally associated with the space of positive definite matrices and the intrinsic operations on positive definite matrices derived from it. We disucss averages first and then proceed to refinement operations (subdivision) and multiscale transforms. In particular, we report on the current knowledge as regards convergence and smoothness.