The Bessel Scale-Space

In this paper we propose a novel type of scales-spaces which is emerging from the family of inhomogeneous pseudodifferential equations (I − τ∆) t 2 u = f with τ ≥ 0 and scale parameter t ≥ 0. Since they are connected to the convolution semi-group of Bessel potentials we call the associated operators {R t,τ | 0 ≤ τ, t} either Bessel scale-space (τ = 1), R t for short, or scaled Bessel scale-space (τ = 1). This is the first concrete example of a family of scale-spaces that is not originating from a PDE of parabolic type and where the Fourier transforms F(R t,τ ) do not have exponential form. These properties make them different from other scale-spaces considered so far in the literature in this field. In contrast to the α-scale-spaces the integral kernels for R t,τ can be given in explicit form for any t, τ ≥ 0 involving the modified Bessel functions of third kind Kν . In theoretical investigations and numerical experiments on 1D and 2D data we compare this new scale-space with the classical Gaussian one.