A hierarchical representation of random waveforms by scale-space filtering

Scale-space filtering is a multiresolution filtering, which is defined as the convolution of a random waveform f(x) with the Gaussian kernel w(x,σ). The set of filtered waveform f(x,σ), called generalized waveform of f(x), is very useful for the structural analysis of waveforms. In this paper, we introduce an analytic line, named structure line, on the surface of generalized waveform f(x,σ). Structure line describes the relation of the convex and concave region of generalized waveform. Structure line is defined by some derivatives of generalized waveform, and has the same topological structure as a trinary tree. The properties of structure line are discussed and some examples are shown. It is confirmed that the structure line is effective to represent waveforms hierarchically.