We have been developing a theory for the generic representation of 2-D shape, where structural descriptions are derived from the shocks (singularities) of a curve evolution process, acting on bounding contours. We now apply the theory to the problem of shape matching. The shocks are organized into a directed, acyclic shock graph, and complexity is managed by attending to the most significant (central) shape components first. The space of all such graphs is highly structured and can be characterized by the rules of a shock graph grammar. The grammar permits a reduction of a shock graph to a unique rooted shock tree. We introduce a novel tree matching algorithm which finds the best set of corresponding nodes between two shock trees in polynomial time. Using a diverse database of shapes, we demonstrate our system's performance under articulation, occlusion, and changes in viewpoint.

This article addresses two important themes in early visual computation: rst it presents a novel theory for learning the universal statistics of natural images { a prior model for typical cluttered scenes of the world { from a set of natural images, second it proposes a general framework of designing reaction-diusion equations for image processing. We start by studying the statistics of natural images including the scale invariant properties, then generic prior models were learned to duplicate the observed statistics, based on the minimax entropy theory studied in two previous papers. The resulting Gibbs distributions have potentials of the form U(I; ; S) = P K I)(x; y)) with S = fF g being a set of lters and = f the potential functions. The learned Gibbs distributions con rm and improve the form of existing prior models such as line-process, but in contrast to all previous models, inverted potentials (i.e. (x) decreasing as a function of jxj) were found to be necessary. We nd that the partial dierential equations given by gradient descent on U(I; ; S) are essentially reaction-diusion equations, where the usual energy terms produce anisotropic diusion while the inverted energy terms produce reaction associated with pattern formation, enhancing preferred image features. We illustrate how these models can be used for texture pattern rendering, denoising, image enhancement and clutter removal by careful choice of both prior and data models of this type, incorporating the appropriate features. Song Chun Zhu is now with the Computer Science Department, Stanford University, Stanford, CA 94305, and David Mumford is with the Division of Applied Mathematics, Brown University, Providence, RI 02912. This work started when the authors were at ...

The eikonal equation and variants of it are of significant interest for problems in computer vision and image processing. It is the basis for continuous versions of mathematical morphology, stereo, shape-from-shading and for recent dynamic theories of shape. Its numerical simulation can be delicate, owing to the formation of singularities in the evolving front and is typically based on level set methods. However, there are more classical approaches rooted in Hamiltonian physics which have yet to be widely used by the computer vision community. In this paper we review the Hamiltonian formulation, which offers specific advantages when it comes to the detection of singularities or shocks. We specialize to the case of Blum's grass fire flow and measure the average outward ux of the vector field that underlies the Hamiltonian system. This measure has very different limiting behaviors depending upon whether the region over which it is computed shrinks to a singular point or a non-singular one. Hence, it is an effective way to distinguish between these two cases. We combine the ux measurement with a homotopy preserving thinning process applied in a discrete lattice. This leads to a robust and accurate algorithm for computing skeletons in 2D as well as 3D, which has low computational complexity. We illustrate the approach with several computational examples.

In this paper, we present a new variational formulation for geometric active contours that forces the level set function to be close to a signed distance function, and therefore completely eliminates the need of the costly re-initialization procedure. Our variational formulation consists of an internal energy term that penalizes the deviation of the level set function from a signed distance function, and an external energy term that drives the motion of the zero level set toward the desired image features, such as object boundaries. The resulting evolution of the level set function is the gradient flow that minimizes the overall energy functional. The proposed variational level set formulation has three main advantages over the traditional level set formulations. First, a significantly larger time step can be used for numerically solving the evolution partial differential equation, and therefore speeds up the curve evolution. Second, the level set function can be initialized with general functions that are more efficient to construct and easier to use in practice than the widely used signed distance function. Third, the level set evolution in our formulation can be easily implemented by simple finite difference scheme and is computationally more efficient. The proposed algorithm has been applied to both simulated and real images with promising results. 1.

What does it mean for a deforming object to be "moving" (see Fig. 1)? How can we separate the overall motion (a finite-dimensional group action) from the more general deformation (a di#eomorphism)? In this paper we propose a definition of motion for a deforming object and introduce a notion of "shape average" as the entity that separates the motion from the deformation. Our definition allows us to derive novel and e#cient algorithms to register non-equivalent shapes using region-based methods, and to simultaneously approximate and register structures in grey-scale images. We also extend the notion of shape average to that of a "moving average" in order to track moving and deforming objects through time.

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There are currently two main types of active contours: 1) parametric active contours, which represent contours explicitly as parameterized curves; and 2) geometric active contours, which represent contours implicitly as level sets of two-dimensional scalar functions. In this paper, we derive an explicit mathematical relationship between the general formulations of parametric and geometric active contours. Based on this relationship and the results of two recent parametric active contours, we propose two new geometric active contours. Using both simulated and real images, we show that the proposed algorithms have an improved performance over both existing parametric and geometric active contours. 1 Introduction Active contours [9], a physically-motivated model that can deform itself to recover object shape from digital images, have been extensively researched in the past decade (see [14] for a recent survey on this topic). Current active contours can be classified as either parametric...

Blum's medial axes have great strengths, in principle, in intuitively describing object shape in terms of a quasihierarchy of figures. But it is well known that, derived from a boundary, they are damagingly sensitive to detail in that boundary. The development of notions of spatial scale has led to some definitions of multiscale medial axes different from the Blum medial axis that considerably overcame the weakness. Three major multiscale medial axes have been proposed: iteratively pruned trees of Voronoi edges [Ogniewicz 1993, Székely 1996, Näf 1996], shock loci of reaction-diffusion equations [Kimia et al. 1995, Siddiqi & Kimia 1996], and height ridges of medialness (cores) [Fritsch et al. 1994, Morse et al. 1993, Pizer et al. 1998]. These are different from the Blum medial axis, and each has different mathematical properties of generic branching and ending properties, singular transitions, and geometry of implied boundary, and they have different strengths and weaknesses for computing object descriptions from images or from object boundaries. These mathematical properties and computational abilities are laid out and compared and contrasted in this paper.

We investigate the geometry of that function in the plane or 3-space, which associates to each point the square of the shortest distance to a given curve or surface. Particular emphasis is put on second order Taylor approximants and other local quadratic approximants. Their key role in a variety of geometric optimization algorithms is illustrated at hand of registration in Computer Vision and surface approximation.

Abstract. This paper presents a novel processing scheme for the automatic and robust computation of a medial shape model, which represents an object population with shape variability. The sensitivity of medial descriptions to object variations and small boundary perturbations are fundamental problems of any skeletonization technique. These problems are approached with the computation of a model with common medial branching topology and grid sampling. This model is then used for a medial shape description of individual objects via a constrained model fit. The process starts from parametric 3D boundary representations with existing point-to-point homology between objects. The Voronoi skeleton of each sampled object boundary is partitioned into non-branching medial sheets and simplified by a novel pruning algorithm using a volumetric contribution criterion. Using the surface homology, medial sheets are combined to form a common medial branching topology. Finally, the medial sheets are sampled and represented as meshes of medial primitives. Results on populations of up to 184 biological objects clearly demonstrate that the common medial branching topology can be described by a small number of medial sheets and that even a coarse sampling leads to a close approximation of individual objects.