Adaptive and optimal difference operators in image processing

Differential operators are essential in many image processing applications. Previous work has shown how to compute derivatives more accurately by examining the image locally, and by applying a difference operator which is optimal for each pixel neighborhood. The proposed technique avoids the explicit computation of fitting functions, and replaces the function fitting process by a function classification process using a filter bank of feature detection templates. Both the feature detectors and the optimal difference operators have a specific shape and an associated cost, defined by a rigid mathematical structure, which can be described by Gröbner bases. This paper introduces a cost criterion to select the operator of the best approximating function class and the most appropriate template size so that the difference operator can be locally adapted to the digitized function. We describe how to obtain discrete approximates for commonly used differential operators, and illustrate how image processing applications can benefit from the adaptive selection procedure for the operators by means of two example applications: tangent computation for digitized object boundaries and the Laplacian of Gaussian edge detector.