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-deened. Normally, a linear ltering is applied. This can be formulated in terms of scale space, functional minimization or edge detection lters. 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 lters of which the Gaussian is the rst if we require the semi group constraint to be fulllled. This regularization implies the minimization of a functional which contains terms up to innnite order of diierentiation. If the functional is truncated at second order, the Canny-Deriche lter arises. Furthermore, we show that the nth order Canny-optimal edge detection lter implements nth 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 ltering without any approximation. r ecursifs sans aucune approximation.