Scale-Space with Casual Time Direction

This article presents a theory for multi-scale representation of temporal data. Assuming that a real-time vision system should represent the incoming data at different time scales, an additional causality constraint arises compared to traditional scale-space theory—we can only use what has occurred in the past for computing representations at coarser time scales. Based on a previously developed scale-space theory in terms of noncreation of local maxima with increasing scale, a complete classification is given of the scale-space kernels that satisfy this property of non-creation of structure and respect the time direction as causal. It is shown that the cases of continuous and discrete time are inherently different.