Regularity Classes for Locally Orderless Images

Gaussian scale space permits one to compute image derivatives. The limitation to some finite order is not inherent in the paradigm itself (Gaussian blurred functions are always smooth), but is caused by the interplay of (at least) two external factors. One is the ratio of the Gaussian scale parameter versus the atomic scale that limits physically or perceptually meaningful sizes (e.g. pixel size, or in general any scale at which the image is “locally orderless”). The second factor involved is the fiducial level of tolerance. Together these factors conspire to determine a maximal order beyond which differential structure becomes meaningless. Thus they give rise to the notion of regularity classes for images akin to the conceptual C-classification pertaining to mathematical functions. We study the relationship between the maximal differential order k, the ratio of inner scale to atomic scale, and the prescribed tolerance level, and draw several conclusions that are of practical interest when considering image derivatives.