Morphological Approach to Multiscale Analysis: From Principles to Equations

Abstract

A numerical image can be modelized as a real function I ₀ (x) defined in ℝ^N (In practice, N = 2 or 3). The main concept of vision theory and image analysis is multiscale analysis (or “scale space”). Multiscale analysis associates with I(0) = I ₀ is a sequence of simplified (smoothed) images I(t, x) which depend upon an abstract parameter t > 0, the scale. The image I(t, x) is called analysis of the image I ₀ at scale t. The formalization of scale-space has received a lot of attention in the past ten years; more than a dozen of theories for image, shape or “texture” multiscale analysis have been proposed and recent mathematical work has permitted a formalization of the whole field. We shall see that a few formal principles (or axioms) are enough to characterize and unify these theories and algorithms and show that some of them simply are equivalent. Those principles are causality (a concept in vision theory which can be led back to a maximum principle), the Euclidean (and/or affine) invariance, which means that image analysis does not depend upon the distance and orientation in space of the analysed image, and the morphological invariance which means that image analysis does not depend upon a contrast change.