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Yezzi,
Anthony
Yezzi,
Anthony
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ItemA New Geometric Metric in the Space of Curves, and Applications To Tracking Deforming Objects by Prediction and Filtering(Georgia Institute of Technology, 2011-02) Sundaramoorthi, Ganesh ; Mennucci, Andrea C. ; Soatto, Stefano ; Yezzi, AnthonyWe define a novel metric on the space of closed planar curves which decomposes into three intuitive components. 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. While earlier related Sobolev metrics for curves exhibit some general similarities to the novel metric proposed in this work, they lacked this important three-way orthogonal decomposition, which has particular relevance for tracking in computer vision. Another positive property of this new metric is that the Riemannian structure that is induced on the space of curves is a smooth Riemannian manifold, which is isometric to a classical well-known manifold. As a consequence, geodesics and gradients of energies defined on the space can be computed using fast closed-form formulas, and this has obvious benefits in numerical applications. The obtained Riemannian manifold of curves is ideal for addressing 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 finite-dimensional group—such as affine motions—or to finitely parameterized 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 finite-dimensional 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 infinite-dimensional quantity) as well. We illustrate these ideas using a simple first-order dynamical model and show that it can be effective even on image sequences where existing methods fail.
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ItemCoarse-to-Fine Segmentation and Tracking Using Sobolev Active Contours(Georgia Institute of Technology, 2008-05) Sundaramoorthi, Ganesh ; Yezzi, Anthony ; Mennucci, Andrea C.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 scale-space 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 coarse-scale deformations are preferred over fine-scale deformations. Along with other properties to be discussed, the coarse-to-fine 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 showinghow a variety of active-contour-based tracking methods can be significantly improved merely by evolving the active contour according to the Sobolev method.
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ItemProperties of Sobolev-type metrics in the space of curves(Georgia Institute of Technology, 2008) Mennucci, Andrea C. ; Yezzi, Anthony ; Sundaramoorthi, GaneshWe define a manifold M where objects c ϵ M are curves, which we parameterize as c : S¹ → R ⁿ (n ≥2, S¹ is the circle). Given a curve c, we define the tangent space TcM of M at c including in it all deformations h : S¹ → R ⁿ of c. We study geometries on the manifold of curves, provided by Sobolev–type Riemannian metrics H[superscript j]. We initially present some mathematical examples to show how the metrics H[superscript j] simplify or regularize gradient flows used in Computer Vision applications. We then provide some basilar results of Hj metrics; and, for the cases j = 1, 2, we characterize the completion of the space of smooth curves; we call this completion(s) “H¹ and H² Sobolev–type Riemannian Manifolds of Curves”. As a byproduct, we prove that the Fréchet distance of curves (see [MM06b]) coincides with the distance induced by the “Finsler L H [superscript ∞] metric” defined in §2.2 in [YM04b].
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ItemGlobal regularizing flows with topology preservation for active contours and polygons(Georgia Institute of Technology, 2007-03) Sundaramoorthi, Ganesh ; Yezzi, AnthonyActive contour and active polygon models have been used widely for image segmentation. In some applications, the topology of the object(s) to be detected from an image is known a priori, despite a complex unknown geometry, and it is important that the active contour or polygon maintain the desired topology. In this work, we construct a novel geometric flow that can be added to image-based evolutions of active contours and polygons in order to preserve the topology of the initial contour or polygon. We emphasize that, unlike other methods for topology preservation, the proposed geometric flow continually adjusts the geometry of the original evolution in a gradual and graceful manner so as to prevent a topology change long before the curve or polygon becomes close to topology change. The flow also serves as a global regularity term for the evolving contour, and has smoothness properties similar to curvature flow. These properties of gradually adjusting the original flow and global regularization prevent geometrical inaccuracies common with simple discrete topology preservation schemes. The proposed topology preserving geometric flow is the gradient flow arising from an energy that is based on electrostatic principles. The evolution of a single point on the contour depends on all other points of the contour, which is different from traditional curve evolutions in the computer vision literature.