Differential structure of images: accuracy of representation

Abstract

Differentiation is known to be ill-posed in the sense of Hadamard. The theory of regular tempered distributions and the concept of Gaussian convolution filters open the way to a well-posed differentiation process, thereby introducing the notion of scale (or: inverse resolution). There is no a priori fundamental limit to the order of differentiation of images provided they are calculated on a sufficiently high scale (relative to pixel scale and noise correlation width), and provided we have a sufficient dynamic range of intensity values. Constraints in resolution (both in the spatial and in the intensity domain) enforce a scale-dependent restriction to the accuracy with which Gaussian kernels G/sub n/(x; /spl sigma/) can be represented in a physical sense. So at a given scale /spl sigma/ (e.g. in units of the sampling scale) and a given measure of inaccuracy /spl alpha/ there is a maximal order n above which the margin /spl alpha/ is exceeded. In this paper we quantify this relation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>