On the Axioms of Scale Space Theory

Authors:
Remco Duits;Luc Florack;Jan De Graaf;Bart Ter Haar Romeny
Affiliations:
Eindhoven University of Technology, Den Dolech 2, NL-5600 MB, The Netherlands. R.Duits@tue.nl;Eindhoven University of Technology, Den Dolech 2, NL-5600 MB, The Netherlands. L.M.J.Florack@tue.nl;Eindhoven University of Technology, Den Dolech 2, NL-5600 MB, The Netherlands. J.d.Graaf@tue.nl;Eindhoven University of Technology, Den Dolech 2, NL-5600 MB, The Netherlands. B.M.terHaarRomeny@tue.nl
Venue:
Journal of Mathematical Imaging and Vision
Year:
2004

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Abstract

We consider alternative scale space representations beyond the well-established Gaussian case that satisfy all “reasonable” axioms. One of these turns out to be subject to a first order pseudo partial differential equation equivalent to the Laplace equation on the upper half plane {(x, s) ∈ \Bbb Rd × \Bbb R | s 0}. We investigate this so-called Poisson scale space and show that it is indeed a viable alternative to Gaussian scale space. Poisson and Gaussian scale space are related via a one-parameter class of operationally well-defined intermediate representations generated by a fractional power of (minus) the spatial Laplace operator.