These are notes for a very rapid introduction to the basics of exterior differential systems and their connection with what is now known as Lie theory, together with some typical and not-so-typical applications to illustrate their use.

No abstract

Abstract. Representation of object shape by medial structures has been an important aspect of image analysis. Methods for describing objects in a binary image by medial axes are well understood. Many attempts have been made to construct similar medial structures for objects in gray scale images. In particular, researchers have studied images by analyzing the graphs of the intensity data and identifying ridge and valley structures on those surfaces. In this paper we review many of the definitions for ridges. Computational vision models require that medial structures should remain invariant under certain transformations of the spatial locations and intensities. For each ridge definition we point out which invariances the definition satisfies. We also give extensions of the concepts so that we can locate d-dimensional ridge structures within n-dimensional images. A comparison of the ridge structures produced by the different definitions is given both by mathematical examples and by an application to a 2-dimensional MR image of a head. Keywords. 1 The Need for Ridges in Image AnalysisMethods for representing shapes of objects in gray-scale images have typically fallen into two categories: edge-based or region-based. Edgebased algorithms are developed under the assumption that large gradients of image intensity indicate the presence of an edge. The property of edgeness at a pixel is determined by measuring the dissimilarity between the pixel intensity and its neighbors' intensities, for example, by us-*Research supported by National Science Foundation Grant DMS-9003037.~Research supported by NIH grant # P01 CA 47982.ing the magnitude of the gradient of intensity. These algorithms additionally must handle edge orientation, edge strength, and edge connectivity. The method of edge detection essentially consists of following r/dges of edgeness. Figure 1 illustrates this for a simple object.Many edge-based methods are deficient since the presence of noise can make it difficult to detect an edge and determine its orientation.Moreover, the characterization of the global structure and shape of an object by its boundary depends greatly on the correctness of the edge connectivity scheme.Region-based algorithms are developed under

No abstract

No abstract