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Abstract-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.

For shapes represented as closed planar contours, we introduce a class of functionals that are invariant with respect to the Euclidean and similarity group, obtained by performing integral operations. While such integral invariants enjoy some of the desirable properties of their differential cousins, such as locality of computation (which allows matching under occlusions) and uniqueness of representation (in the limit), they are not as sensitive to noise in the data. We exploit the integral invariants to define a unique signature, from which the original shape can be reconstructed uniquely up to the symmetry group, and a notion of scale-space that allows analysis at multiple levels of resolution. The invariant signature can be used as a basis to define various notions of distance between shapes, and we illustrate the potential of the integral invariant representation for shape matching on real and synthetic data.

In this paper, 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 will demonstrate the perfonnance of this technique on both synthetic and real images, showing applications to thresholding written text and segmentation of mehcal images and scenes.

Abstract. We match shapes, even under severe deformations, via a smooth reparametrization of their integral invariant signatures. These robust signatures and correspondences are the foundation of a shape energy functional for variational image segmentation. Integral invariant shape templates do not require registration and allow for significant deformations of the contour, such as the articulation of the object's parts. This enables generalization to multiple instances of a shape from a single template, instead of requiring several templates for searching or training. This paper motivates and presents the energy functional, derives the gradient descent direction to optimize the functional, and demonstrates the method, coupled with a data term, on real image data where the object's parts are articulated.