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Abstract : |
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 @ 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 multi-dimensional representation is given by convolution with the onedimensional discrete analogue of the Gaussian kernel along each dimension., |
