On the consistency of a separator

Abstract

A smoother can be called a separator if it is idempotent and co-idempotent, with the motivation coming from a physical analogy. This is a rudementary consistency, and type of “linearity”, which has been found to occur in practice. All the LULU-separator turn out to act linearly on any non-negative combination of the two outputs of such separators. This is arguably a very consistent behaviour. We prove this, and investigate how far this follows for general separators. We also prove that this type of linearity is extendable to composite “separators” that are used in a nonlinear multiresolution decomposition with LULU-operator. This has important implications for image processing.

Keywords: Separator; consistency; near-linearity; cone-linearity; highlight-conjecture; DPT (Discrete Pulse Transforms)

Quaestiones Mathematicae 33(2010), 197–209