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.
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Burgeth, B., Didas, S., Weickert, J. (2005). The Bessel Scale-Space. In: Fogh Olsen, O., Florack, L., Kuijper, A. (eds) Deep Structure, Singularities, and Computer Vision. DSSCV 2005. Lecture Notes in Computer Science, vol 3753. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11577812_8
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DOI: https://doi.org/10.1007/11577812_8
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