We extend the particle filtering approach to detecting polygonal structures of fixed shape and variable size in overhead images to the more practically useful case of unconstrained side lengths. Our approach deals with multiple candidate sides by validating and clustering particles based on evidence that corners of proper acuteness and orientation (as constrained by the polygon model) might exist. A queue-regulated tracking algorithm that handles multiple candidates across multiple sides is discussed. Compelling detection results in overhead images that involve entire families of polygonal structure are provided.

For shapes represented as closed planar contours, we introduce a class of functionals which are invariant with respect to the Euclidean group and which are obtained by performing integral operations. While such integral invariants enjoy some of the desirable properties of their differential counterparts, such as locality of computation (which allows matching under occlusions) and uniqueness of representation (asymptotically), they do not exhibit the noise sensitivity associated with differential quantities and, therefore, do not require presmoothing of the input shape. Our formulation allows the analysis of shapes at multiple scales. Based on integral invariants, we define a notion of distance between shapes. The proposed distance measure can be computed efficiently and allows warping the shape boundaries onto each other; its computation results in optimal point correspondence as an intermediate step. Numerical results on shape matching demonstrate that this framework can match shapes despite the deformation of subparts, missing parts and noise. As a quantitative analysis, we report matching scores for shape retrieval from a database.

In this paper we demonstrate the use of an anisotropic PDE to improve the behavior of first and second derivatives of a distance function. We begin by deriving a property of these derivatives and showing a natural relationship between this property and the antigeometric heat flow. We then construct a PDE that combines a known first-order PDE with the antigeometric flow and then demonstrate its effectiveness with a discrete simulation.

We utilize an anisotropic diffusion model, which we call the anti-geometric heat flow, for adaptive thresholding of bimodal images and for segmentation of more general greyscale images. In a departure from most anisotropic diffusion techniques, we select the local diffusion direction that smears edges in the image rather than seeking to preserve them. In this manner, we are able rapidly to detect and discriminate between entire image regions that lie near, but on opposite sides of, a prominent edge. The detection of such regions occurs during the diffusion process rather than afterward, thereby side-stepping the most notorious problem associated with diffusion methods, namely, when diffusion should stop. We initially outline a procedure for adaptive thresholding, but ultimately show how this model may be used in a region splitting procedure which, when combined with energy based region merging procedures, provides a general framework for image segmentation. We discuss a fast implementation of one such framework and demonstrate its effectiveness in segmenting medical, military, and scene imagery.

We present a novel adaptive thresholding technique based upon an anisotropic diffusion model, which may be referred to as the anti-geometric heat flow. In contrast to its more popular counterparts (such as the geometric heat flow) which diffuse parallel to image edges, this model diffuses perpendicular to image edges, yielding surfaces which are naturally suited for adaptive thresholding and segmentation. While it is possible to apply this diffusion for a fixed amount of time to detect features, we discuss how to detect features during the diffusion process, thus avoiding much of the arbitrariness associated with choosing a single scale (and makes the most notorious problem associated with anisotropic diffusion methods, namely "when do you stop?" a moot point). We demonstrate the performance of this technique on both synthetic and real images, showing applications to thresholding written text and segmentation of medical images and scenes.