Scale and the differential structure of images

Abstract

Why and how one should study a scale-space is prescribed by the universal physical law of scale invariance, expressed by the so-called Pi-theorem. The fact that any image is a physical observable with an inner and outer scale bound, necessarily gives rise to a ‘scale-space representation’, in which a given image is represented by a one-dimensional family of images representing that image on various levels of inner spatial scale. An early vision system is completely ignorant of the geometry of its input. Its primary task is to establish this geometry at any available scale. The absence of geometrical knowledge poses additional constraints on the construction of a scale-space, notably linearity, spatial shift invariance and isotropy, thereby defining a complete hierarchical family of scaled partial differential operators: the Gaussian kernel (the lowest order, resettling operator) and its linear partial derivatives. They enable local image analysis through the detection of local differential structure in a robust way, while at the same time capturing global features through the extra scale degree of freedom. In this paper we show why the operations of scaling and differentiation cannot be separated. This framework permits us to construct in a systematic way multiscale, cartesian differential invariants, i.e. true image descriptors that exhibit manifest invariance with respect to a change of cartesian coordinates. The scale-space operators closely resemble the receptive field profiles found in mammalian frontend visual systems.