Scale-Space for N-dimensional discrete signals

Tony Lindeberg

Technical report ISRN KTH/NA/P--92/26--SE.

Shortened version in: Y. O. Ying, A. Toet, D. Foster, H. Heijmanns and P. Meer (eds.) (1994) Shape in Picture: Mathematical Description of Shape in Grey-Level Images, (Proc. of workshop in Driebergen, Netherlands, Sep. 7--11, 1992). NATO ASI Series F, vol. 126, Springer-Verlag, pp. 571--590.

Abstract

This article shows how a (linear) scale-space representation can be defined for discrete signals of arbitrary dimension. The treatment is based upon the assumptions that (i) the scale-space representation should be defined by convolving the original signal with a one-parameter family of symmetric smoothing kernels possessing a semi-group property, and (ii) local extrema must not be enhanced when the scale parameter is increased continuously.

It is shown that given these requirements the scale-space representation must satisfy the differential equation \partial_t L = A L for some linear and shift invariant operator A satisfying locality, positivity, zero sum, and symmetry conditions. Examples in one, two, and three dimensions illustrate that this corresponds to natural semi-discretizations of the continuous (second-order) diffusion equation using different discrete approximations of the Laplacean operator. In a special case the multi-dimensional representation is given by convolution with the one-dimensional discrete analogue of the Gaussian kernel along each dimension.

Keywords: scale, scale-space, diffusion, Gaussian smoothing, multi-scale representation, wavelets, image structure, causality

Full paper: (PostScript 0.1Mb)

See also: (Discrete derivative approximations) (Deep structure of differential feature detectors) (Monograph on scale-space theory) (Other publications on scale-space theory with applications)

Tony Lindeberg <tony@nada.kth.se>