On the axiomatic foundations of linear scale-space: Combining semi-group structure with causality vs. scale invariance

Tony Lindeberg

Technical report, Department of Numerical Analysis and Computing Science, Royal Institute of Technology, S-100 44 Stockholm, Sweden, Aug 1994. (ISRN KTH NA/P--94/20--SE)

Revised version published as Chapter 6 in J. Sporring, M. Nielsen, L. Florack, and P. Johansen (eds.) {\em Gaussian Scale-Space Theory: Proc. PhD School on\/} {\em Scale-Space Theory\/}, (Copenhagen, Denmark, May 1996), pages 75--98, Kluwer Academic Publishers, 1997.

Abstract

The notion of multi-scale representation is essential to many aspects of early visual processing. This article deals with the axiomatic formulation of the special type of multi-scale representation known as scale-space representation. Specifically, this work is concerned with the problem of how different choices of basic assumptions (scale-space axioms) restrict the class of permissible smoothing operations.

A scale-space formulation previously expressed for discrete signals is adapted to the continuous domain. The basic assumptions are that the scale-space family should be generated by convolution with a one-parameter family of rotationally symmetric smoothing kernels that satisfy a semi-group structure and obey a causality condition expressed as a non-enhancement requirement of local extrema. Under these assumptions, it is shown that the smoothing kernel is uniquely determined to be a Gaussian.

Relations between this scale scale-space formulation and recent formulations based on scale invariance are explained in detail. Connections are also pointed out to approaches based on non-uniform smoothing.

Keywords: scale-space, Gaussian filtering, causality, diffusion, scale invariance, multi-scale representation, computer vision, signal processing

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Tony Lindeberg <tony@bion.kth.se>