Scale space filtering

Section II introduces the formal framework for the definition anc description of process trends at all levels of detail qualitative, order-of magnitude, and analytic. A detour through the basic concepts of scale-spact filtering is necessary in order to see the connection between the concept o process trends and the classical material on signal analysis. Within th( framework of scale-space filtering we can then elucidate the notions o episode, scale, local filtering, structure of scale, distinguishec features, and others. 

The procedure described above is a pictorial approximation of a process called scale-space filtering of a function, proposed by Witkin (1983). The surface (e.g., Fig. 6) swept out by a filtered signal as the Gaussian filter s standard deviation is varied, is called scale-space image of the signal and is given by... 

Scale-space filtering provides a multiscale description of a signal s trends in terms of its inflexion points (second-order zero crossings). The only legal sequences of triangles between two adjacent inflexion points are (in terms of triangular episodes) ... 

If a signal is represented by a sequence of triangular episodes, scale-space filtering manipulates the sequences of triangular episodes with very concrete mechanisms. Here is the complete list of syntactic manipulations carried out by scale-space filtering ... 

So, within the context of scale-space filtering, it is more convenient to express a trend as a sequence of scaling episodes, rather than as a sequence of episodes. 

The wavelet interval-tree of scale is constructed fi om log 2 N distinct representations, where N is the number of points in the record of measured data. This is a far more efficient representation than that of scale-space filtering with continuous variation of Gaussian a. 

Witkin, A. P., Scale space filtering A new approach to multi-scale description. In Image Understanding (S. Ullman and W. Richard, eds.), pp. 79-95. Ablex, Norwood, NJ, 1983. 

Within the framework of scale-space filtering, inflexion points of F(t) appear as extrema in dF(t)/dt and zero crossings in d2F(t)/dt2. Thus, filtering a signal by the Laplacian (second derivative) of a Gaussian will generate the inflexion points at various scales (Marr and Hildreth, 1980). In the same spirit, if the wavelet is chosen to be the first derivative of a scaling function, i.e., ip(t) = d(t)/ dt, then from Eqs. (5a) and (11) we... 

A.P. Witkin, Scale-Space Filtering, Proceedings of 8th International Joint Conference on Artificial Intelligence, 1983, pp. 1019-1022. 

---