Morphological approach to smoothing: stochastic morphological sampling theorem

Abstract

From Communication to Pattern Recognition, from low layer signal processing to high layer cognition, from practice to theory of engineering principles, the question of inherent complexities of entities represented as sets in euclidean space is of fundamental interest. In this paper, we present some fundamental theoretical results pertaining to the question of how many randomly selected labelled example points it takes to reconstruct a set in euclidean space, and thereby propose a morphological sampling theorem in the form of Stochastic Morphological Sampling Theorem. Drawing on results and concepts from Mathematical Morphology and Learnability Theory, we pursue a set-theoretic approach and demonstrate some provable performances pertaining to euclidean-set- reconstruction from stochastic samples. In particular, we demonstrate a result towards the formulation of a stochastic (morphological) version of the Nyquist Sampling Theorem -- that, under weak assumptions on the situation under consideration, the number of randomly-drawn (positive) example points needed to reconstruct the target set is at most polynomial in the performance parameters and also the complexity of the target set as loosely captured by size, dimension and surface-area. The reconstruction result of this paper pertaining to the complexity of euclidean sets has natural interpretations for the process of smoothing modelled formally as a dilation and morphological operation. Thus, in this paper, we formulate and demonstrate a certain fundamental (distribution-free) well-behaving aspect of smoothing by proving a fundamental result pertaining to the (set-theoretic) complexity of sets in euclidean space.