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 \({\partial _t}L = {A_{ScSp}}L\) for some linear and shift invariant operator \({A_{ScSp}}\) 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 multidimensional representation is given by convolution with the one-dimensional discrete analogue of the Gaussian kernel along each dimension.
The support from the Swedish National Board for Industrial and Technical Development, NUTEK, is gratefully acknowledged.
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© 1994 Springer-Verlag Berlin Heidelberg
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Lindeberg, T. (1994). Scale-Space for N-Dimensional Discrete Signals. In: O, YL., Toet, A., Foster, D., Heijmans, H.J.A.M., Meer, P. (eds) Shape in Picture. NATO ASI Series, vol 126. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03039-4_42
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DOI: https://doi.org/10.1007/978-3-662-03039-4_42
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