In this paper we address the formulation of a scale-space theory for discrete images. We denote a one-dimensional kernel a scale-space kernel if it reduces the number of local extrema and discuss which discrete kernels are possible scale-space kernels. Unimodality and positivity properties are shown to hold for such kernels as well as their Fourier transforms. An explicit expression characterizing all discrete scale-space kernels is given.
We propose that there is only one reasonable way to define a scale-space family of images L(x; t) for a one-dimensional discrete signal f(x) namely by convolution with the family of discrete kernels T(n; t) = e^(-t) I_nt(t) where I_n is the modified Bessel function of order n.
With this representation, comprising a continuous scale parameter, we are no longer restricted to specific predetermined levels of scale. Further, T(n; t) appears naturally in the solution of a discretized version of the heat equation, both in one and two dimensions.
The family T(n; t) (t >= 0) is the only one-parameter family of discrete symmetric shift-invariant kernels satisfying both necessary scale-space requirements and the semigroup property T(n; s) * T(n; t) = T(n; s+t). Similar arguments applied in the continuous case uniquely lead to the family of Gaussian kernels.
The commonly adapted technique with a sampled Gaussian produces undesirable effects. It is shown that scale-space violations might occur in the family of functions generated by convolution with the sampled Gaussian kernel. The result exemplifies that properties derived in the continuous case might be violated after discretization.
A discussion about the numerical implementation is performed and an algorithm generating the filter coefficients is supplied.
KTH Royal Institute of Technology, 1988. , 51 p.