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171
Kernel Density Estimation and Intrinsic Alignment for Knowledgedriven Segmentation: Teaching Level Sets to Walk
 International Journal of Computer Vision
, 2004
"... We address the problem of image segmentation with statistical shape priors in the context of the level set framework. Our paper makes two contributions: Firstly, we propose to generate invariance of the shape prior to certain transformations by intrinsic registration of the evolving level set fun ..."
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Cited by 114 (16 self)
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We address the problem of image segmentation with statistical shape priors in the context of the level set framework. Our paper makes two contributions: Firstly, we propose to generate invariance of the shape prior to certain transformations by intrinsic registration of the evolving level set function. In contrast to existing approaches to invariance in the level set framework, this closedform solution removes the need to iteratively optimize explicit pose parameters. Moreover, we will argue that the resulting shape gradient is more accurate in that it takes into account the e#ect of boundary variation on the object's pose.
Approximations of shape metrics and application to shape warping and empirical shape statistics
 FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
, 2004
"... This paper proposes a framework for dealing with several problems related to the analysis of shapes. Two related such problems are the definition of the relevant set of shapes and that of defining a metric on it. Following a recent research monograph by Delfour and Zolésio [11], we consider the cha ..."
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Cited by 94 (20 self)
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This paper proposes a framework for dealing with several problems related to the analysis of shapes. Two related such problems are the definition of the relevant set of shapes and that of defining a metric on it. Following a recent research monograph by Delfour and Zolésio [11], we consider the characteristic functions of the subsets of R 2 and their distance functions. The L² norm of the difference of characteristic functions, the L∞ and the W 1,2 norms of the difference of distance functions define interesting topologies, in particular the wellknown Hausdorff distance. Because of practical considerations arising from the fact that we deal with
Shape representation and classification using the poisson equation
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2006
"... Silhouettes contain rich information about the shape of objects that can be used for recognition and classification. We present a novel approach that allows us to reliably compute many useful properties of a silhouette. Our approach assigns for every internal point of the silhouette a value reflecti ..."
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Cited by 82 (8 self)
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Silhouettes contain rich information about the shape of objects that can be used for recognition and classification. We present a novel approach that allows us to reliably compute many useful properties of a silhouette. Our approach assigns for every internal point of the silhouette a value reflecting the mean time required for a random walk beginning at the point to hit the boundaries. This function can be computed by solving Poisson’s equation, with the silhouette contours providing boundary conditions. We show how this function can be used to reliably extract various shape properties including part structure and rough skeleton, local orientation and aspect ratio of different parts, and convex and concave sections of the boundaries. In addition to this we discuss properties of the solution and show how to efficiently compute this solution using multigrid algorithms. We demonstrate the utility of the extracted properties by using them for shape classification. 1.
Towards a coherent statistical framework for dense deformable template estimation
 J.R. Statist. Soc.B
, 2006
"... Abstract. The problem of estimating probabilistic deformable template models in the field of computer vision or of probabilistic atlases in the field of computational anatomy has not yet received a coherent statistical formulation and remains a challenge. In this paper, we provide a careful definiti ..."
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Cited by 81 (9 self)
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Abstract. The problem of estimating probabilistic deformable template models in the field of computer vision or of probabilistic atlases in the field of computational anatomy has not yet received a coherent statistical formulation and remains a challenge. In this paper, we provide a careful definition and analysis of a well defined statistical model based on dense deformable templates for gray level images of deformable objects. We propose a rigorous Bayesian framework for which we can derived an iterative algorithm for the effective estimation of the geometric and photometric parameters of the model in a small sample setting, together with an asymptotic consistency proof. The model is extended to mixtures of finite numbers of such components leading to a fine description of the photometric and geometric variations. We illustrate some of the ideas with images of handwritten digits, and apply the estimated models to classification through maximum likelihood. 1.
Statistical Shape Analysis: Clustering, Learning, and Testing
 IEEE Trans. Pattern Anal. Mach. Intell
, 2005
"... Using a recently proposed geometric representation of planar shapes, we present algorithmic tools for: (i) hierarchical clustering of imaged objects according to the shapes of their boundaries, (ii) learning of probability models for clustered shapes, and (iii) testing of observed shapes under co ..."
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Cited by 78 (13 self)
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Using a recently proposed geometric representation of planar shapes, we present algorithmic tools for: (i) hierarchical clustering of imaged objects according to the shapes of their boundaries, (ii) learning of probability models for clustered shapes, and (iii) testing of observed shapes under competing probability models. Clustering at any level of hierarchy is performed using a mimimum dispersion criterion and a Markov search process. Statistical means of clusters provide shapes to be clustered at the next higher level, thus building a hierarchy of shapes.
Geometric modeling in shape space
 In Proc. SIGGRAPH
, 2007
"... Figure 1: Geodesic interpolation and extrapolation. The blue input poses of the elephant are geodesically interpolated in an asisometricaspossible fashion (shown in green), and the resulting path is geodesically continued (shown in purple) to naturally extend the sequence. No semantic information, ..."
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Cited by 77 (10 self)
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Figure 1: Geodesic interpolation and extrapolation. The blue input poses of the elephant are geodesically interpolated in an asisometricaspossible fashion (shown in green), and the resulting path is geodesically continued (shown in purple) to naturally extend the sequence. No semantic information, segmentation, or knowledge of articulated components is used. We present a novel framework to treat shapes in the setting of Riemannian geometry. Shapes – triangular meshes or more generally straight line graphs in Euclidean space – are treated as points in a shape space. We introduce useful Riemannian metrics in this space to aid the user in design and modeling tasks, especially to explore the space of (approximately) isometric deformations of a given shape. Much of the work relies on an efficient algorithm to compute geodesics in shape spaces; to this end, we present a multiresolution framework to solve the interpolation problem – which amounts to solving a boundary value problem – as well as the extrapolation problem – an initial value problem – in shape space. Based on these two operations, several classical concepts like parallel transport and the exponential map can be used in shape space to solve various geometric modeling and geometry processing tasks. Applications include shape morphing, shape deformation, deformation transfer, and intuitive shape exploration.
On the Shape of Plane Elastic Curves
 International Journal of Computer Vision
, 2005
"... We study shapes of planar arcs and closed contours modeled on elastic curves obtained by bending, stretching or compressing line segments nonuniformly along their extensions. Shapes are represented as elements of a quotient space of curves obtained by identifying those that differ by shapepreservi ..."
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Cited by 61 (11 self)
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We study shapes of planar arcs and closed contours modeled on elastic curves obtained by bending, stretching or compressing line segments nonuniformly along their extensions. Shapes are represented as elements of a quotient space of curves obtained by identifying those that differ by shapepreserving transformations. The elastic properties of the curves are encoded in Riemannian metrics on these spaces. Geodesics in shape spaces are used to quantify shape divergence and to develop morphing techniques. The shape spaces and metrics constructed are novel and offer an environment for the study of shape statistics. Elasticity leads to shape correspondences and deformations that are more natural and intuitive than those obtained in several existing models. Applications of shape geodesics to the definition and calculation of mean shapes and to the development of shape clustering techniques are also investigated.
A metric on shape spaces with explicit geodesics
, 2007
"... Abstract. This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space ..."
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Cited by 56 (18 self)
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Abstract. This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space of curves (closedopen, modulo rotation etc...) Using these isometries, we are able to explicitely describe the geodesics, first in the parametric case, then by modding out the paremetrization and considering horizontal vectors. We also compute the sectional curvature for these spaces, and show, in particular, that the space of closed curves modulo rotation and change of parameter has positive curvature. Experimental results that explicitly compute minimizing geodesics between two closed curves are finally provided
Metamorphoses Through Lie Group Action
 FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
, 2005
"... We formally analyze a computational problem which has important applications in image understanding and shape analysis. The problem can be summarized as follows. Starting from a group action on a Riemannian manifold M, we introduce a modification of the metric by partly expressing displacements on M ..."
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Cited by 47 (10 self)
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We formally analyze a computational problem which has important applications in image understanding and shape analysis. The problem can be summarized as follows. Starting from a group action on a Riemannian manifold M, we introduce a modification of the metric by partly expressing displacements on M as an effect of the action of some group element. The study of this new structure relates to evolutions on M under the combined effect of the action and of residual displacements, called metamorphoses. This can and has been applied to image processing problems, providing in particular diffeomorphic matching algorithms for pattern recognition.
A novel representation for riemannian analysis of elastic curves in Rn
 In: Proceedings of IEEE CVPR
, 2007
"... We propose a novel representation of continuous, closed curves in Rn that is quite efficient for analyzing their shapes. We combine the strengths of two important ideas elastic shape metric and pathstraightening methodsin shape analysis and present a fast algorithm for finding geodesics in shape ..."
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Cited by 45 (12 self)
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We propose a novel representation of continuous, closed curves in Rn that is quite efficient for analyzing their shapes. We combine the strengths of two important ideas elastic shape metric and pathstraightening methodsin shape analysis and present a fast algorithm for finding geodesics in shape spaces. The elastic metric allows for optimal matching of features while pathstraightening provides geodesics between curves. Efficiency results from the fact that the elastic metric becomes the simple L2 metric in the representation proposed here. We present the stepbystep algorithms for computing geodesics in this framework and demonstrate them with 2D as well as 3D examples. 1.