The Bessel Scale-Space

Abstract

In this paper we propose a novel type of scales-spaces which is emerging from the family of inhomogeneous pseudodifferential equations $(I - \tau\Delta)^{\frac{t}{2}}u$ 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 $^{n}_{t,{ \tau}}$ | 0≤ τ,t} either Bessel scale-space (τ=1), R $^{n}_{t}$ 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 $\mathcal{F}(R^n_{t,\tau})$ 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 $^{n}_{t,{\tau}}$ can be given in explicit form for any t, τ ≥ 0 involving the modified Bessel functions of third kind K _ν . In theoretical investigations and numerical experiments on 1D and 2D data we compare this new scale-space with the classical Gaussian one.