On the Axiomatization of Scale-Space Theory

Scale-spaces are a common method for analysis of images. It involves an operator depending on a parameter, called a scale, which acts in some space of real functions. Many different scale-spaces have been defined and used in practice. The most popular example is the Gaussian Scale-Space where the operator is defined as a convolution integral involving the Gaussian distribution function with the standard variation being the scale parameter. There have also been many attempts for axiomatization of scale-space theory. The novelty of our approach is that it does not assume integral representation of the scale-space operator. While the new theory is applicable to, and in fact represents and abstraction of the properties of all existing operators, most importantly that of causality, it uses as an essential example the operator and the respective scale-space provided by the Discrete Pulse Transform.