Regularization and Scale Space

Computational vision often needs to deal with derivatives of digital images. Derivatives are not intrinsic properties of a digital image; a paradigm is required to make them well-defined. Normally, a linear filtering is applied. This can be formulated in terms of scale space, functional minimization or edge detection filters. In this paper, we take regularization (or functional minimization) as a starting point, and show that it boils down to a ordered set of linear filters of which the Gaussian is the first if we require the semi group constraint to be fulfilled. This regularization implies the minimization of a functional which contains terms up to infinite order of differentiation. If the functional is truncated at second order, the Canny-Deriche filter arises. Furthermore, we show that the $n$th order Canny-optimal edge detection filter implements $n$th order regularization. We also show, that higher dimensional regularization in its most general form boils down to a rotation of the one dimensional case, when Cartesian invariance is imposed. This means that results from 1D regularization are easily generalized to higher dimensions. Finally, we show that regularization in its most general form can be implemented as recursive filtering without any approximation.