Local Image Structure

Abstract

Local image properties are those that can be associated with a single base point at a fixed inner scale, i.e. that can be defined in terms of derivatives taken at a fixed point in scale-space. Multilocal properties take into account multiple local neighbourhoods. (Note that certain multilocal properties may have a local interpretation at a coarser level of scale or at a higher order of differentiation.) Local properties are the building blocks on the basis of which multilocal expertise must be built. Indeed, the very scale-space construct is intended to encompass global structure as a manifestation of purely local entities that are causally connected in a continuous tree-like structure (“dynamic shape” [162]).

The question about the applicability of the assumptions of geometry at infinitesimal distances is connected with the question about the inner reasons of the measure relationships of space. With this question, which one may perhaps still include in the study of space, the previous remark is used, that a discrete manifold contains the principle of the measure relationships already in the concept of this manifold, but it must come from somewhere else for a continuous manifold. Hence, the reality on which space is based must either be a discrete manifold, or the reasons for the measure relationships must be searched outside in attracting forces acting on them.

—Bernhard Riemann