A Hausdorff Stable Method for Finding Singularities with Application to the Intersections of Sampled Manifolds

Abstract

We will present a family of novel methods for feature detection and image restoration which have a very clear geometrical interpretation, though the application areas, however, are not only limited to these ones. Our methods rely on the idea of realizing a close smooth approximation of the digital image or of a modified image which creates the singularity at the feature of interest. By close smooth approximation we mean that given the input function, our transformation outputs a smooth function which coincides with the input function in the neighbourhood where the function is smooth. As a result, one then by difference gets a neighbourhood of the singularity. With this respect, we could term them as geometric based methods for singularity detection. By such transformation, we are able to develop multi-scale, parametrised methods for identifying singularities in functions. These tools can then be used, via a numerical implementation, to detect features in images or data (e.g. edges, corner points, blobs, etc.), remove noise from images, identify intersections between surfaces, etc, and thus produce new geometric techniques for image processing, feature extraction and geometric interrogation.