The Geometric Approach to the Classification of Signals via a Maximal Set of Signals

Abstract

In this paper we study the scale-space classification of signals via the maximal set of kernels. We use a geometric approach which arises naturally when we consider parameter variations in scale-space. We derive the Fourier transform formulas for quick and efficient computation of zero-crossings and the corresponding classifying trees. General theory of convergence for convolutions is developed, and practically useful properties of scale-space classification are derived as a consequence also give a complete topological description of level curves for convolutions of signals with the maximal set of kernels. We use these results to develop a bifurcation theory for the curves under the parameter changes. This approach leads to a novel set of integer invariants for arbitrary signals.