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17
Ligature Instabilities in the Perceptual Organization of Shape
 Computer Vision and Image Understanding
, 1999
"... 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&ap ..."
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Cited by 45 (7 self)
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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 skeletoncalled ligature by Blumto 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
Multiscale Medial Loci and Their Properties
"... 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 le ..."
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Cited by 43 (8 self)
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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 reactiondiffusion 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.
Scaleinvariant contour completion using conditional random fields
 In ICCV
, 2005
"... Abstract 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 ..."
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Cited by 37 (7 self)
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Abstract 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. Maximum likelihood parameters for the model arelearned from human labeled groundtruth. Using held out test data, we measure how the model, by incorporating continuity 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 lowlevelboundary 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.
Flux Invariants for Shape
 In CVPR
, 2003
"... 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, ..."
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Cited by 29 (3 self)
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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 nonmedial ones, providing a theoretical justification for its use in the HamiltonJacobi 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.
A Competitive Layer Model for Feature Binding and Sensory Segmentation
 NEURAL COMPUTATION
, 2001
"... 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 fo ..."
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Cited by 22 (11 self)
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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 winnertakeall 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 winnertakeall 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 figureground segmentation and grouping into a unified model.
The Curve Indicator Random Field: Curve Organization Via Edge Correlation
 In Perceptual Organization for Artificial Vision Systems
, 2000
"... 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 ..."
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Cited by 11 (1 self)
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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 continuationthe basis of many current curve organization systemscan 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 shortdistance, 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.
Finding tree structures by grouping symmetries
 In Proceedings of the International Conference on Computer Vision
, 2005
"... The representation of objects in images as tree structures is of great interest to vision, as they can represent articulated objects such as people as well as other structured objects like arteries in human bodies, roads, circuit board patterns, etc. Tree structures are often related to the symmetry ..."
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Cited by 4 (0 self)
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The representation of objects in images as tree structures is of great interest to vision, as they can represent articulated objects such as people as well as other structured objects like arteries in human bodies, roads, circuit board patterns, etc. Tree structures are often related to the symmetry axis representation of shapes, which captures their local symmetries. Algorithms have been introduced to detect (i) open contours in images in quadratic time (ii) closed contours in images in cubic time, and (iii) tree structures from contours in quadratic time. The algorithms are based on dynamic programming and Single Source Shortest Path algorithms. However, in this paper, we show that the problem of finding tree structures in images in a principled manner is a much harder problem. We argue that the optimization problem of finding tree structures in images is essentially equivalent to a variant of the Steiner Tree problem, which is NPhard. Nevertheless, an approximate polynomialtime algorithm for this problem exists: we apply a fast implementation of the GoemansWilliamson approximate algorithm to the problem of finding a tree representation after an image is transformed by a local symmetry mapping. Examples of extracting tree structures from images illustrate the idea and applicability of the approximate method. 1.
Perceptual Organization of Visual Flows
, 2003
"... Locally parallel dense patterns visual flows define a perceptually coherent structure of particular significance to perceptual organization. Geometrically, it is argued that a proper way to investigate these structures requires the frame field approach from differential geometry, a study that lead ..."
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Cited by 4 (3 self)
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Locally parallel dense patterns visual flows define a perceptually coherent structure of particular significance to perceptual organization. Geometrically, it is argued that a proper way to investigate these structures requires the frame field approach from differential geometry, a study that leads to the notion of visual flow curvatures and to constraints on their mutual behavior. These curvatures are then used to develop a theory, and a rigorous model, of visual flow “good continuation ” that extends common terminology from Gestalt psychology and from computational studies of curves. The geometrical theory is then applied in three ways. Firstly, psychophysical exploration of the role of visual flow curvatures in human perception shows that sensitivity to these curvatures greatly affects orientationbased texture segmentation. Secondly, a contextual framework for the computation of coherent visual flows from images is developed and applied to texture, shading, and color analysis. Unlike existing approaches, the proposed framework is able to handle both sparse, dense, and multivalued data sets, while preserving line and point singularities and rejecting large scale nonflow structures. Lastly, the geometrical theory is linked to the functional organization of primary visual cortex to accurately predict the distribution of long range horizontal connections and to support their identification with those obtained mathematically.
Adaptive Pseudo Dilation for Gestalt Edge Grouping and Contour Detection
"... Abstract—We consider the problem of detecting object contours in natural images. In many cases, local luminance changes turn out to be stronger in textured areas than on object contours. Therefore, local edge features, which only look at a small neighborhood of each pixel, cannot be reliable indicat ..."
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Cited by 2 (1 self)
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Abstract—We consider the problem of detecting object contours in natural images. In many cases, local luminance changes turn out to be stronger in textured areas than on object contours. Therefore, local edge features, which only look at a small neighborhood of each pixel, cannot be reliable indicators of the presence of a contour, and some global analysis is needed. We introduce a new morphological operator, called adaptive pseudodilation (APD), which uses context dependent structuring elements in order to identify long curvilinear structure in the edge map. We show that grouping edge pixels as the connected components of the output of APD results in a good agreement with the gestalt law of good continuation. The novelty of this operator is that dilation is limited to the Voronoi cell of each edge pixel. An efficient implementation of APD is presented. The grouping algorithm is then embedded in a multithreshold contour detector. At each threshold level, small groups of edges are removed, and contours are completed by means of a generalized reconstruction from markers. The use of different thresholds makes the algorithm much less sensitive to the values of the input parameters. Both qualitative and quantitative comparison with existing approaches prove the superiority of the proposed contour detector in terms of larger amount of suppressed texture and more effective detection of lowcontrast contours. Index Terms—Edge and boundary detection, Gestalt grouping, morphological analysis methods. I.
Graphspectral methods for Computer Vision
, 2003
"... This thesis describes a family of graphspectral methods for computer vision that exploit the properties of the first eigenvector of the adjacency matrix of a weighted graph. The algorithms are applied to segmentation and grouping, shapefromshading and graphmatching. In Chapter 3, we cast the prob ..."
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Cited by 1 (0 self)
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This thesis describes a family of graphspectral methods for computer vision that exploit the properties of the first eigenvector of the adjacency matrix of a weighted graph. The algorithms are applied to segmentation and grouping, shapefromshading and graphmatching. In Chapter 3, we cast the problem of grouping into an evidence combining setting where the number of clusters is determined by the modes of the adjacency matrix. With the number of clusters to hand, we model the grouping process using two sets of variables. These are the cluster memberships and the pairwise affinities or linkweights for the nodes of a graph. From a simple probability distribution for these parameters, we show how they may be estimated using the apparatus of the expectationmaximisation (EM) algorithm. The new method is demonstrated on the problems of linesegment grouping and grayscale image segmentation. The method is shown to outperform a noniterative eigenclustering method. In Chapter 4, we present a more direct graphspectral method for segmentation and grouping by developing an iterative maximum likelihood framework for perceptual clustering. Here, we focuss in more detail on the likelihood function that results from the