Relativistic Scale-Spaces

In this paper we extend the notion of Poisson scale-space. We propose a generalisation inspired by the linear parabolic pseudodifferential operator √−∆+m2−m, 0 ≤ m, connected with models of relativistic kinetic energy from quantum mechanics. This leads to a new family of operators {Qt | 0 ≤ m, t} which we call relativistic scale-spaces. They provide us with a continuous transition from the Poisson scale-space {Pt | t ≥ 0} (for m = 0) to the identity operator I (for m −→ +∞). For any fixed t0 > 0 the family {Qt0 | m ≥ 0} constitutes a scale-space connecting I and Pt0 . In contrast to the α-scale-spaces the integral kernels for Qt can be given in explicit form for any m, t ≥ 0 enabling us to make precise statements about smoothness and boundary behaviour of the solutions. Numerical experiments on 1D and 2D data demonstrate the potential of the new scale-space setting.