Geometry-Limited Diffusion in the Characterization of Geometric Patches in Images

Abstract

Much of image processing and artificial vision has focused on the detection of edges—particularly, edges that are measured by the gradient magnitude. Higher order geometry can provide a richer variety of information about objects within images and can also yield useful measurements which are invariant to certain kinds of intensity transformations. However, analyzing higher order geometry can be difficult because of the sensitivity of higher order filters to noise. Low pass filters can alleviate the effects of high frequency noise but tend to distort the geometry in ways that make the resulting measurements less useful. This paper suggests a generalization of anisotropic diffusion as a mechanism for making reliable and precise geometric measurements in the presence of blurring and noise. This mechanism is a generalized form of edge-affected diffusion that applies to multi-valued functions. We pursue the interpretation of multi-valued descriptors as positions in a feature space and describe how this premise yields a natural form for a set of coupled anisotropic diffusion equations that depend on one′s choice of distance in the resulting feature space. The appropriate choice of distance allows one to measure areas of the image where the feature positions are changing rapidly and vary the conductance in the diffusion equation accordingly. These features can be the outputs of some multi-valued imaging device, or measurements made (via filters) on a single valued image. Feature spaces that consist of measurements made on single-valued images can reflect geometric properties of the local intensity surface. The anisotropic diffusion can be used to segment images into patches that share local geometric properties so that the boundaries of these patches are geometrically and visually interesting. The appropriate choice of distance in such feature spaces can yield meaningful geometric information. One such geometric feature space consists of the first-order derivatives. This paper presents a distance measure in this space that results in a process for reliable and accurate detection of "creases" and "corners." These ideas can be generalized to other features, including higher order derivatives. The appropriate choice of distance in such feature spaces could yield meaningful higher order geometric information.