α Scale Spaces on a Bounded Domain
2003 (English)In: Scale Space Methods in Computer Vision / [ed] Lewis D. Griffin and Martin Lillholm, 2003, Vol. 2695, 502-518 p.Conference paper (Refereed)
We consider alpha scale spaces, a parameterized class (alpha is an element of (0, 1]) of scale space representations beyond the well-established Gaussian scale space, which are generated by the alpha-th power of the minus Laplace operator on a bounded domain using the Neumann boundary condition. The Neumann boundary condition ensures that there is no grey-value flux through the boundary. Thereby no artificial grey-values from outside the image affect the evolution proces, which is the case for the alpha scale spaces on an unbounded domain. Moreover, the connection between the a scale spaces which is not trivial in the unbounded domain case, becomes straightforward: The generator of the Gaussian semigroup extends to a compact, self-adjoint operator on the Hilbert space L-2(Omega) and therefore it has a complete countable set of eigen functions. Taking the alpha-th power of the Gaussian generator simply boils down to taking the alpha-th power of the corresponding eigenvalues. Consequently, all alpha scale spaces have exactly the same eigen-modes and can be implemented simultaneously as scale dependent Fourier series. The only difference between them is the (relative) contribution of each eigen-mode to the evolution proces. By introducing the notion of (non-dimensional) relative scale in each a scale space, we are able to compare the various alpha scale spaces. The case alpha = 0.5, where the generator equals the square root of the minus Laplace operator leads to Poisson scale space, which is at least as interesting as Gaussian scale space and can be extended to a (Clifford) analytic scale space.
Place, publisher, year, edition, pages
2003. Vol. 2695, 502-518 p.
, Lecture Notes in Computer Science, ISSN 1611-3349 ; 2695
Engineering and Technology
IdentifiersURN: urn:nbn:se:liu:diva-21600DOI: 10.1007/3-540-44935-3_34OAI: oai:DiVA.org:liu-21600DiVA: diva2:241564
4th International Conference, Scale Space 2003 Isle of Skye, UK, June 10–12, 2003