Integral invariants for shape matching.

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.

Anti-geometric diffusion for adaptive thresholding and fast segmentation.

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.