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.