α Scale Spaces on a Bounded Domain

Abstract

We consider α scale spaces, a parameterized class (α ∈ (0, 1]) of scale space representations beyond the well-established Gaussian scale space, which are generated by the α-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 α scale spaces on an unbounded domain. Moreover, the connection between the α 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 L₂(Ω) and therefore it has a complete countable set of eigen functions. Taking the α-th power of the Gaussian generator simply boils down to taking the α-th power of the corresponding eigenvalues. Consequently, all α 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 α scale space, we are able to compare the various α scale spaces. The case α = 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.