Multidimensional possibility/probability domination for extending maxitive kernel based signal processing

Signal and image processing make intensive use of positive, bounded and centered functions that are called kernels. Kernels are used for defining the interplay between discrete and continuous domains, filtering, modeling a system through a point spread function, etc. The possible analogy between kernels and fuzzy sets has led to a wide use of fuzzy set theory for signal and image processing [1]. The possibilistic interpretation of fuzzy sets has recently been exploited to extend signal processing with the aim of accounting for poor knowledge of the appropriate kernel to be used. These imprecise kernels are called maxitive kernels. A maxitive kernel can be seen as a convex set of conventional kernels. Within this framework, the triangular kernel with mode 0 and spread Δ has a specific role since it can be used to represent a convex set of all bounded centered bell-shaped kernels of spread δ≤Δ, i.e. the way kernels are usually imprecisely known (shape unknown, spread imprecise). However, this principle has yet to be extended to more than one dimension despite the fact that it is needed for image processing. An extension to higher dimensions is proposed in this paper.