Adaptive Convolutions

Abstract

When smoothing a function f via convolution with some kernel, it is often desirable to adapt the amount of smoothing locally to the variation of f. For this purpose, the constant smoothing coefficient of regular convolutions needs to be replaced by an adaptation function μ. This function is matrix-valued which allows for different degrees of smoothing in different directions. The aim of this paper is twofold. The first is to provide a theoretical framework for such adaptive convolutions. The second purpose is to derive a formula for the automatic choice of the adaptation function μ=μf in dependence of the function f to be smoothed. This requires the notion of the \emph{local variation} of f, the quantification of which relies on certain phase space transformations of f. The derivation is guided by meaningful axioms which, among other things, guarantee invariance of adaptive convolutions under shifting and scaling of f.