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66
Quick shift and kernel methods for mode seeking
 In European Conference on Computer Vision, volume IV
, 2008
"... Abstract. We show that the complexity of the recently introduced medoidshift algorithm in clustering N points is O(N 2), with a small constant, if the underlying distance is Euclidean. This makes medoid shift considerably faster than mean shift, contrarily to what previously believed. We then explo ..."
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Cited by 94 (6 self)
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Abstract. We show that the complexity of the recently introduced medoidshift algorithm in clustering N points is O(N 2), with a small constant, if the underlying distance is Euclidean. This makes medoid shift considerably faster than mean shift, contrarily to what previously believed. We then exploit kernel methods to extend both mean shift and the improved medoid shift to a large family of distances, with complexity bounded by the effective rank of the resulting kernel matrix, and with explicit regularization constraints. Finally, we show that, under certain conditions, medoid shift fails to cluster data points belonging to the same mode, resulting in overfragmentation. We propose remedies for this problem, by introducing a novel, simple and extremely efficient clustering algorithm, called quick shift, that explicitly trades off under and overfragmentation. Like medoid shift, quick shift operates in nonEuclidean spaces in a straightforward manner. We also show that the accelerated medoid shift can be used to initialize mean shift for increased efficiency. We illustrate our algorithms to clustering data on manifolds, image segmentation, and the automatic discovery of visual categories. 1
Image segmentation using active contours driven by the Bhattacharyya gradient flow
, 2007
"... The present paper addresses the problem of image segmentation by means of active contours, whose evolution is driven by the gradient flow derived from an energy functional that is based on the Bhattacharyya distance. In particular, given the values of a photometric variable (or of a set thereof), wh ..."
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Cited by 54 (10 self)
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The present paper addresses the problem of image segmentation by means of active contours, whose evolution is driven by the gradient flow derived from an energy functional that is based on the Bhattacharyya distance. In particular, given the values of a photometric variable (or of a set thereof), which is to be used for classifying the image pixels, the active contours are designed to converge to the shape that results in maximal discrepancy between the empirical distributions of the photometric variable inside and outside of the contours. The above discrepancy is measured by means of the Bhattacharyya distance that proves to be an extremely useful tool for solving the problem at hand. The proposed methodology can be viewed as a generalization of the segmentation methods, in which the active contours maximize the difference between a finite number of empirical moments of the “inside” and “outside” distributions. Furthermore, it is shown that the proposed methodology is very versatile and flexible in the sense that it allows one to easily accommodate a diversity of the image features based on which the segmentation should be performed. As additional contribution, a method for automatically adjusting the smoothness properties of the empirical distributions is proposed. Such a procedure is crucial in the situations when the number of data samples (supporting a certain segmentation class) varies considerably in the course of the evolution of the active contour. In this case, the smoothness properties of the empirical distributions have to be properly adjusted to avoid either overor underestimation artifacts. Finally, a number of relevant segmentations results are demonstrated and some further research directions are discussed.
Generalized gradients: priors on minimization flows
 INTERNATIONAL JOURNAL OF COMPUTER VISION
, 2007
"... This paper tackles an important aspect of the variational problem underlying active contours: optimization by gradient flows. Classically, the definition of a gradient depends directly on the choice of an inner product structure. This consideration is largely absent from the active contours literatu ..."
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Cited by 31 (4 self)
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This paper tackles an important aspect of the variational problem underlying active contours: optimization by gradient flows. Classically, the definition of a gradient depends directly on the choice of an inner product structure. This consideration is largely absent from the active contours literature. Most authors, explicitely or implicitely, assume that the space of admissible deformations is ruled by the canonical L 2 inner product. The classical gradient flows reported in the literature are relative to this particular choice. Here, we investigate the relevance of using (i) other inner products, yielding other gradient descents, and (ii) other minimizing flows not deriving from any inner product. In particular, we show how to induce different degrees of spatial consistency into the minimizing flow, in order to decrease the probability of getting trapped into irrelevant local minima. We report numerical experiments indicating that the sensitivity of the active contours method to initial conditions, which seriously limits its applicability and efficiency, is alleviated by our applicationspecific spatially coherent minimizing flows. We show that the choice of the inner product can be seen as a prior on the deformation fields and we present an extension of the definition of the gradient toward more general priors.
Generalized Surface Flows for Mesh Processing
, 2007
"... Geometric flows are ubiquitous in mesh processing. Curve and surface evolutions based on functional minimization have been used in the context of surface diffusion, denoising, shape optimization, minimal surfaces, and geodesic paths to mention a few. Such gradient flows are nearly always, yet often ..."
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Cited by 30 (2 self)
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Geometric flows are ubiquitous in mesh processing. Curve and surface evolutions based on functional minimization have been used in the context of surface diffusion, denoising, shape optimization, minimal surfaces, and geodesic paths to mention a few. Such gradient flows are nearly always, yet often implicitly, based on the canonical L 2 inner product of vector fields. In this paper, we point out that changing this inner product provides a simple, powerful, and untapped approach to extend current flows. We demonstrate the value of such a norm alteration for regularization and volumepreservation purposes and in the context of shape matching, where deformation priors (ranging from rigid motion to articulated motion) can be incorporated into a gradient flow to drastically improve results. Implementation details, including a differentiable approximation of the Hausdorff distance between irregular meshes, are presented.
Variational BSpline LevelSet: A Linear Filtering Approach for Fast Deformable Model Evolution
"... Abstract—In the field of image segmentation, most levelsetbased activecontour approaches take advantage of a discrete representation of the associated implicit function. We present in this paper a different formulation where the implicit function is modeled as a continuous parametric function expr ..."
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Cited by 20 (0 self)
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Abstract—In the field of image segmentation, most levelsetbased activecontour approaches take advantage of a discrete representation of the associated implicit function. We present in this paper a different formulation where the implicit function is modeled as a continuous parametric function expressed on a Bspline basis. Starting from the activecontour energy functional, we show that this formulation allows us to compute the solution as a restriction of the variational problem on the space spanned by the Bsplines. As a consequence, the minimization of the functional is directly obtained in terms of the Bspline coefficients. We also show that each step of this minimization may be expressed through a convolution operation. Because the Bspline functions are separable, this convolution may in turn be performed as a sequence of simple 1D convolutions, which yields an efficient algorithm. As a further consequence, each step of the levelset evolution may be interpreted as a filtering operation with a Bspline kernel. Such filtering induces an intrinsic smoothing in the algorithm, which can be controlled explicitly via the degree and the scale of the chosen Bspline kernel. We illustrate the behavior of this approach on simulated as well as experimental images from various fields. Index Terms—Active contours, Bspline, levelsets, segmentation, variational method.
The Piecewise Smooth MumfordShah Functional on an Arbitrary Graph
"... Abstract—The MumfordShah functional has had a major impact on a variety of image analysis problems including image segmentation and filtering and, despite being introduced over two decades ago, it is still in widespread use. Present day optimization of the MumfordShah functional is predominated by ..."
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Cited by 19 (8 self)
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Abstract—The MumfordShah functional has had a major impact on a variety of image analysis problems including image segmentation and filtering and, despite being introduced over two decades ago, it is still in widespread use. Present day optimization of the MumfordShah functional is predominated by active contour methods. Until recently, these formulations necessitated optimization of the contour by evolving via gradient descent, which is known for its overdependence on initialization and the tendency to produce undesirable local minima. In order to reduce these problems, we reformulate the corresponding MumfordShah functional on an arbitrary graph and apply the techniques of combinatorial optimization to produce a fast, lowenergy solution. In contrast to traditional optimization methods, use of these combinatorial techniques necessitates consideration of the reconstructed image outside of its usual boundary, requiring additionally the inclusion of regularization for generating these values. The energy of the solution provided by this graph formulation is compared with the energy of the solution computed via traditional gradient descentbased narrowband level set methods. This comparison demonstrates that our graph formulation and optimization produces lower energy solutions than the traditional gradient descent based contour evolution methods in significantly less time. Finally, we demonstrate the usefulness of the graph formulation to apply the MumfordShah functional to new applications such as point clustering and filtering of nonuniformly sampled images. Index Terms—Level sets, active contours, piecewise smooth MumfordShah, combinatorial optimization, graph reformulation I.
A New Geometric Metric in the Space of Curves, and Applications to Tracking Deforming Objects by Prediction and Filtering
, 2010
"... We define a novel metric on the space of closed planar curves. According to this metric centroid translations, scale changes and deformations are orthogonal, and the metric is also invariant with respect to reparameterizations of the curve. The Riemannian structure that is induced on the space of cu ..."
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Cited by 19 (1 self)
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We define a novel metric on the space of closed planar curves. According to this metric centroid translations, scale changes and deformations are orthogonal, and the metric is also invariant with respect to reparameterizations of the curve. The Riemannian structure that is induced on the space of curves is a smooth Riemannian manifold, which is isometric to a classical wellknown manifold. As a consequence, geodesics and gradients of energies defined on the space can be computed using fast closedform formulas, and this has obvious benefits in numerical applications. The obtained Riemannian manifold of curves is apt to address complex problems in computer vision; one such example is the tracking of highly deforming objects. Previous works have assumed that the object deformation is smooth, which is realistic for the tracking problem, but most have restricted the deformation to belong to a finitedimensional group – such as affine motions – or to finitelyparameterized models. This is too restrictive for highly deforming objects such as the contour of a beating heart. We adopt the smoothness assumption implicit in previous work, but we lift the restriction to finitedimensional motions/deformations. We define a dynamical model in this Riemannian manifold of curves, and use it to perform filtering and prediction to infer and extrapolate not just the pose (a finitely parameterized quantity) of an object, but its deformation (an infinitedimensional quantity) as well. We illustrate these ideas using a simple firstorder dynamical model, and show that it can be effective even on data sets where existing methods fail. 1
Coarsetofine segmentation and tracking using Sobolev active contours
 IEEE TPAMI
, 2008
"... Recently proposed Sobolev active contours introduced a new paradigm for minimizing energies defined on curves by changing the traditional cost of perturbing a curve and thereby redefining gradients associated to these energies. Sobolev active contours evolve more globally and are less attracted to ..."
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Cited by 18 (5 self)
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Recently proposed Sobolev active contours introduced a new paradigm for minimizing energies defined on curves by changing the traditional cost of perturbing a curve and thereby redefining gradients associated to these energies. Sobolev active contours evolve more globally and are less attracted to certain intermediate local minima than traditional active contours, and it is based on a wellstructured Riemannian metric, which is important for shape analysis and shape priors. In this paper, we analyze Sobolev active contours using scalespace analysis in order to understand their evolution across different scales. This analysis shows an extremely important and useful behavior of Sobolev contours, namely, that they move successively from coarse to increasingly finer scale motions in a continuous manner. This property illustrates that one justification for using the Sobolev technique is for applications where coarsescale deformations are preferred over finescale deformations. Along with other properties to be discussed, the coarsetofine observation reveals that Sobolev active contours are, in particular, ideally suited for tracking algorithms that use active contours. We will also justify our assertion that the Sobolev metric should be used over the traditional metric for active contours in tracking problems by experimentally showing how a variety of activecontourbased tracking methods can be significantly improved merely by evolving the active contour according to the Sobolev method.
New possibilities with Sobolev active contours
 In Scale Space Variational Methods 07
, 2007
"... Abstract. Recently, the Sobolev metric was introduced to define gradient flows of various geometric active contour energies. It was shown that the Sobolev metric outperforms the traditional metric for the same energy in many cases such as for tracking where the coarse scale changes of the contour ar ..."
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Cited by 16 (7 self)
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Abstract. Recently, the Sobolev metric was introduced to define gradient flows of various geometric active contour energies. It was shown that the Sobolev metric outperforms the traditional metric for the same energy in many cases such as for tracking where the coarse scale changes of the contour are important. Some interesting properties of Sobolev gradient flows include that they stabilize certain unstable traditional flows, and the order of the evolution PDEs are reduced when compared with traditional gradient flows of the same energies. In this paper, we explore new possibilities for active contours made possible by Sobolev metrics. The Sobolev method allows one to implement new energybased active contour models that were not otherwise considered because the traditional minimizing method render them illposed or numerically infeasible. In particular, we exploit the stabilizing and the order reducing properties of Sobolev gradients to implement the gradient descent of these new energies. We give examples of this class of energies, which include some simple geometric priors and new edgebased energies. We also show that these energies can be quite useful for segmentation and tracking. We also show that the gradient flows using the traditional metric are either illposed or numerically difficult to implement, and then show that the flows can be implemented in a stable and numerically feasible manner using the Sobolev gradient.
Tracking with sobolev active contours
 In Proceedings of International Conference on Computer Vision and Pattern Recognition, volume I
, 2006
"... Recently proposed Sobolev active contours introduced a new paradigm for minimizing energies defined on curves by changing the traditional cost of perturbing a curve and thereby redefining their gradients. Sobolev active contours evolve more globally and are less attracted to certain intermediate loc ..."
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Cited by 14 (6 self)
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Recently proposed Sobolev active contours introduced a new paradigm for minimizing energies defined on curves by changing the traditional cost of perturbing a curve and thereby redefining their gradients. Sobolev active contours evolve more globally and are less attracted to certain intermediate local minima than traditional active contours. In this paper we analyze Sobolev active contours in the Fourier domain in order to understand their evolution across different scales. This analysis shows an important and useful behavior of Sobolev contours, namely, that they move successively from coarse to increasingly finer scale motions in a continuous manner. Along with other properties, the previous observation reveals that Sobolev active contours are ideally suited for tracking problems that use active contours. Our purpose in this work is to show how a variety of active contour based tracking methods can be significantly improved merely by evolving the active contours according to the Sobolev method. 1.