Gaussian Scale Space from Insufficient Image Information

Abstract

Gaussian scale space is properly defined and well-developed for images completely knownand defined on the d dimensional Euclidean space ℝ^d. However, as soon as image information is only partly available, say, on a subset V of ℝ^d, the Gaussian scale space paradigm is not readily applicable and one has to resort to different approaches to come to a scale space on V. Examples are the theory dealing with scale space on ℤ^d ⊂ ℝ^d, i.e., discrete scale space; the approach based on the heat equation satisfying certain boundary conditions; and the ad hoc approaches dealing with (hyper)rectangular images, e.g. zero-padding of the area outside of V, or periodic continuation of the image.

We propose to solve the foregoing problem for general V from a Bayesian viewpoint. Assuming that the observed image is obtained by linearly sampling a real underlying image that is actually defined on the complete d dimensional Euclidean space, we can infer this latter image and from that image build the scale space. Re-sampling this scale space then gives rise to the scale space on V. Necessary for inferring the underlying image is knowledge on the linear apertures (or receptive field) used for sampling this image, and information on the prior over the class of all images.