Selecting appropriate difference operators for digital images by local feature detection

Abstract

An often reoccurring problem in digital image processing is the application of operators from differential geometry to discrete representations of curves and surfaces. We propose the use of feature detectors to improve the estimation of differentials of discrete functions. To this end we replace a differential operator by a bank of feature detectors and difference operators. The purpose of the feature detectors is first to examine the local behavior of the function. Next, depending on the outcome, the feature detectors select the most appropriate difference operator. For example, if the function behaves locally as a linear function, they select a difference operator that is well suited for linear functions. We show that this technique can be put on a firm mathematical basis. In particular, when designing a bank of feature detectors, we use Groebner bases for the functional decomposition and combination of the detectors. We illustrate the mathematical results with several practical examples.