A Comparison of the Deep Structure of α-Scale Spaces

Abstract

We compare the topology and deep structure of alternative scale space representations, so called α-scale spaces, 1/2 ≤ α ≤ 1, which are subject to a first order pseudo partial differential equation on the upper half plane {(x,s) ∈ ℝ d ×ℝ|s > 0}. In particular, the cases α = 1 and α = 1/2, which correspond to respectively Poisson scale space and Gaussian scale space, are considered. Poisson scale space is equivalent to harmonic extension to the upper half plane, inducing potential physics, whereas Gaussian scale space is generated by the diffusion equation on the upper half plane, inducing heat physics. Despite the continuous connection (by parameter 1/2 ≤ α ≤ 1) between these scale spaces and the similarity between their convolution convolution kernels, we show both theoretically and experimentally that there is a strong difference between the topology in the deep structure of these scale spaces.