Although the classical Blum skeleton has long been considered unstable, many have attempted to alleviate this defect through pruning. Unfortunately, these methods have an arbitrary basis, and, more importantly, they do not prevent internal structural alterations due to slight changes in an object's boundary. The result is a relative lack of development of skeleton representations for indexing object databases, despite a long history. Here we revisit a subset of the skeleton---called ligature by Blum---to demonstrate how the topological sensitivity of the skeleton can be alleviated. In particular, we show how the deletion of ligature regions leads to stable hierarchical descriptions, illustrating this point with several computational examples. We then relate ligature to a natural growth principle to provide an account of the perceptual parts of shape. Finally, we discuss the duality between the problems of shape partitioning and contour fragment grouping. 1

We consider the average outward flux through a Jordan curve of the gradient vector field of the Euclidean distance function to the boundary of a 2D shape. Using an alternate form of the divergence theorem, we show that in the limit as the area of the region enclosed by such a curve shrinks to zero, this measure has very different behaviours at medial points than at non-medial ones, providing a theoretical justification for its use in the Hamilton-Jacobi skeletonization algorithm of [7]. We then specialize to the case of shrinking circular neighborhoods and show that the average outward flux measure also reveals the object angle at skeletal points. Hence, formulae for obtaining the boundary curves, their curvatures, and other geometric quantities of interest, can be written in terms of the average outward flux limit values at skeletal points. Thus this measure can be viewed as a Euclidean invariant for shape description: it can be used to both detect the skeleton from the Euclidean distance function, as well as to explicitly reconstruct the boundary from it. We illustrate our results with several numerical simulations. 1.

Abstract We present a model of curvilinear grouping using piece-wise linear representations of contours and a conditional random field to capture continuity and the frequency of dif-ferent junction types. Potential completions are generated by building a constrained Delaunay triangulation (CDT)over the set of contours found by a local edge detector. Maximum likelihood parameters for the model arelearned from human labeled groundtruth. Using held out test data, we measure how the model, by incorporating con-tinuity structure, improves boundary detection over the local edge detector. We also compare performance with abaseline local classifier that operates on pairs of edgels. Both algorithms consistently dominate the low-levelboundary detector at all thresholds. To our knowledge, this is the first time that curvilinear continuity has been shownquantitatively useful for a large variety of natural images. Better boundary detection has immediate application in theproblem of object detection and recognition.

Blum's medial axes have great strengths, in principle, in intuitively describing object shape in terms of a quasi-hierarchy 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; Szekely, 1996; Naf, 1996), shock loci of reaction-diffusion equations (Kimia et al., 1995; Siddiqi and 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 present a recurrent neural network for feature binding and sensory segmentation, the competitive layer model (CLM). The CLM uses topographically structured competitive and cooperative interactions in a layered network to partition a set of input features into salient groups. The dynamics is formulated within a standard additive recurrent network with linear threshold neurons. Contextual relations among features are coded by pairwise compatibilities which define an energy function to be minimized by the neural dynamics. Due to the usage of dynamical winner-take-all circuits the model gains more flexible response properties than spin models of segmentation by exploiting amplitude information in the grouping process. We prove analytic results on the convergence and stable attractors of the CLM, which generalize earlier results on winner-take-all networks, and incorporate deterministic annealing for robustness against local minima. The piecewise linear dynamics of the CLM allows a linear eigensubspace analysis which we use to analyze the dynamics of binding in conjunction with annealing. For the example of contour detection we show how the CLM can integrate figure-ground segmentation and grouping into a unified model.

Can the organization of local edge measurements into curves be directly related to natural image structure? By viewing curve organization as a statistical estimation problem, we suggest that it can. In particular, the classical Gestalt perceptual organization cues of proximity and good continuation---the basis of many current curve organization systems---can be statistically measured in images. As a prior for our estimation approach we introduce the curve indicator random field. In contrast to other techniques that require contour closure or are based on a sparse set of detected edges, the curve indicator random field emphasizes the short-distance, dense nature of organizing curve elements into (possibly) open curves. Its explicit formulation allows the calculation of its properties such as its autocorrelation. On the one hand, the curve indicator random field leads us to introduce the oriented Wiener filter, capturing the blur and noise inherent in the edge measurement process. On the other, it suggests we seek such correlations in natural images. We present the results of some initial edge correlation measurements that not only confirm the presence of Gestalt cues, but also suggest that curvature has a role in curve organization.

We present a model of curvilinear grouping using piecewise linear representations of contours and a conditional random field to capture continuity and the frequency of different junction types. Potential completions are generated by building a constrained Delaunay triangulation (CDT) over the set of contours found by a local edge detector.

Using a large set of human segmented natural images, we study the statistics of region boundaries. We observe several power law distributions which likely arise from both multi-scale structure within individual objects and from arbitrary viewing distance. Accordingly, we develop a scale-invariant representation of images from the bottom up, using a piecewise linear approximation of contours and constrained Delaunay triangulation to complete gaps. We model curvilinear grouping on top of this graphical/geometric structure using a conditional random field to capture the statistics of continuity and different junction types. Quantitative evaluations on several large datasets show that our contour grouping algorithm consistently dominates and significantly improves on local edge detection.

ABSTRACT—How the visual system comes to bind diverse image regions into whole objects is not well understood. We recently had a unique opportunity to investigate this issue when we met three congenitally blind individuals in India. After providing them treatment, we studied the early stages of their visual skills. We found that prominent figural cues of grouping, such as good continuation and junction structure, were largely ineffective for image parsing. By contrast, motion cues were of profound significance in that they enabled intraobject integration and facilitated the development of object representations that permitted recognition in static images. Following 10 to 18 months of visual experience, the individuals ’ performance improved, and they were able to use the previously ineffective static figural cues to correctly parse many static