Feature controlled adaptive difference operators

TL;DR: This work proposes to replace differential operators by the combined action of sets of feature detectors and locally adaptive difference operators, resulting in a more accurate computation of the required derivatives in each pixel neighborhood, to compute the best discrete approximation for the Laplacian differential operator.

Abstract: Differential operators are essential for many image processing applications which require the computation of typical characteristics of continuous surfaces, as e.g. tangents, curvature, flatness, shape descriptors. We propose to replace differential operators by the combined action of sets of feature detectors and locally adaptive difference operators, resulting in a more accurate computation of the required derivatives in each pixel neighborhood. Both the set of feature detectors and the set of difference operators have a rigid mathematical structure, which is described by a set of Groebner bases for each class of fitting functions. This representation allows a systematic description of the hierarchical structure with ordering relations for all different function classes. The explicit computation of fitting functions is avoided by our technique and replaced by a function classification process. A set of simple local feature detectors is used to find the class of fitting functions which locally yields the best approximation for the digitized image surface. By a systematic optimization process, we determine for each fitting function class a difference operator which is an optimal approximation for a particular differential operator. As an example, we describe how to compute the best discrete approximation for the Laplacian differential operator in each pixel neighborhood and illustrate how the Laplacian of Gaussian edge detection method can benefit from these results.