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One point isometric matching with the heat kernel
 Computer Graphics Forum
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 69 (4 self)
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
StyleContent Separation by Anisotropic Part Scales
"... We perform coanalysis of a set of manmade 3D objects to allow the creation of novel instances derived from the set. We analyze the objects at the part level and treat the anisotropic part scales as a shape style. The coanalysis then allows style transfer to synthesize new objects. The key to coa ..."
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Cited by 36 (23 self)
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We perform coanalysis of a set of manmade 3D objects to allow the creation of novel instances derived from the set. We analyze the objects at the part level and treat the anisotropic part scales as a shape style. The coanalysis then allows style transfer to synthesize new objects. The key to coanalysis is part correspondence, where a major challenge is the handling of large style variations and diverse geometric content in the shape set. We propose stylecontent separation as a means to address this challenge. Specifically, we define a correspondencefree style signature for style clustering. We show that confining analysis to within a style cluster facilitates tasks such as cosegmentation, content classification, and deformationdriven part correspondence. With part correspondence between each pair of shapes in the set, style transfer can be easily performed. We demonstrate our analysis and synthesis results on several sets of manmade objects with style and content variations.
TopologyInvariant Similarity of Nonrigid Shapes
, 2009
"... This paper explores the problem of similarity criteria between nonrigid shapes. Broadly speaking, such criteria are divided into intrinsic and extrinsic, the first referring to the metric structure of the object and the latter to how it is laid out in the Euclidean space. Both criteria have their ..."
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Cited by 33 (3 self)
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This paper explores the problem of similarity criteria between nonrigid shapes. Broadly speaking, such criteria are divided into intrinsic and extrinsic, the first referring to the metric structure of the object and the latter to how it is laid out in the Euclidean space. Both criteria have their advantages and disadvantages: extrinsic similarity is sensitive to nonrigid deformations, while intrinsic similarity is sensitive to topological noise. In this paper, we approach the problem from the perspective of metric geometry. We show that by unifying the extrinsic and intrinsic similarity criteria, it is possible to obtain a stronger topologyinvariant similarity, suitable for comparing deformed shapes with different topology. We construct this new joint criterion as a tradeoff between the extrinsic and intrinsic similarity and use it as a setvalued distance. Numerical results demonstrate the efficiency of our approach in cases where using either extrinsic or intrinsic criteria alone would fail.
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
1 Interactive Shape Interpolation through Controllable Dynamic Deformation
"... Abstract — In this paper, we introduce an interactive approach to generate physicallybased shape interpolation between poses. We extend linear modal analysis to offer an efficient and robust numerical technique to generate physicallyplausible dynamics even for very large deformation. Our method al ..."
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Cited by 12 (2 self)
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Abstract — In this paper, we introduce an interactive approach to generate physicallybased shape interpolation between poses. We extend linear modal analysis to offer an efficient and robust numerical technique to generate physicallyplausible dynamics even for very large deformation. Our method also provides a rich set of intuitive editing tools with realtime feedback, including control over vibration frequencies, amplitudes, and damping of the resulting interpolation sequence. We demonstrate the versatility of our approach through a series of complex dynamic shape interpolations.
Interactive Spacetime Control of Deformable Objects
"... Creating motions of objects or characters that are physically plausible and follow an animator’s intent is a key task in computer animation. The spacetime constraints paradigm is a valuable approach to this problem, but it suffers from high computational costs. Based on spacetime constraints, we pro ..."
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Cited by 11 (5 self)
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Creating motions of objects or characters that are physically plausible and follow an animator’s intent is a key task in computer animation. The spacetime constraints paradigm is a valuable approach to this problem, but it suffers from high computational costs. Based on spacetime constraints, we propose a framework for controlling the motion of deformable objects that offers interactive response times. This is achieved by a model reduction of the underlying variational problem, which combines dimension reduction, multipoint linearization, and decoupling of ODEs. After a preprocess, the cost for creating or editing a motion is reduced to solving a number of onedimensional spacetime problems, whose solutions are the wiggly splines introduced by Kass and Anderson [2008]. We achieve interactive response times through a new fast and robust numerical scheme for solving the onedimensional problems that is based on a closedform representation of the wiggly splines. time can have a large impact on the state of the system at a later time. Control over a simulation can be achieved by computing optimal physical trajectories that are solutions of a variational spacetime problem. Such techniques calculate acting forces that minimize an objective functional while guaranteeing that the resulting motion satisfies prescribed spacetime constraints, e.g. interpolates a set of keyframes. Resulting forces are optimally distributed over the whole animation and show effects like squashandstretch, timing, or anticipation that are desired in animation. However, the computational cost for obtaining these results is that of solving a spacetime optimization problem. To date, recent methods, even those that use dimension reduction techniques, still require at least several minutes to solve the optimization problem for an interesting motion of an object or a character.
Lie bodies: A manifold representation of 3D human shape
 in ECCV
, 2012
"... Abstract. Threedimensional object shape is commonly represented in terms of deformations of a triangular mesh from an exemplar shape. Existing models, however, are based on a Euclidean representation of shape deformations. In contrast, we argue that shape has a manifold structure: For example, sum ..."
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Abstract. Threedimensional object shape is commonly represented in terms of deformations of a triangular mesh from an exemplar shape. Existing models, however, are based on a Euclidean representation of shape deformations. In contrast, we argue that shape has a manifold structure: For example, summing the shape deformations for two people does not necessarily yield a deformation corresponding to a valid human shape, nor does the Euclidean difference of these two deformations provide a meaningful measure of shape dissimilarity. Consequently, we define a novel manifold for shape representation, with emphasis on body shapes, using a new Lie group of deformations. This has several advantages. First we define triangle deformations exactly, removing nonphysical deformations and redundant degrees of freedom common to previous methods. Second, the Riemannian structure of Lie Bodies enables a more meaningful definition of body shape similarity by measuring distance between bodies on the manifold of body shape deformations. Third, the group structure allows the valid composition of deformations. This is important for models that factor body shape deformations into multiple causes or represent shape as a linear combination of basis shapes. Finally, body shape variation is modeled using statistics on manifolds. Instead of modeling Euclidean shape variation with Principal Component Analysis we capture shape variation on the manifold using Principal Geodesic Analysis. Our experiments show consistent visual and quantitative advantages of Lie Bodies over traditional Euclidean models of shape deformation and our representation can be easily incorporated into existing methods.
A computational model of multidimensional shape
 International Journal of Computer Vision, Online First
, 2010
"... We develop a computational model of shape that extends existing Riemannian models of shape of curves to multidimensional objects of general topological type. We construct shape spaces equipped with geodesic metrics that measure how costly it is to interpolate two shapes through elastic deformations. ..."
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We develop a computational model of shape that extends existing Riemannian models of shape of curves to multidimensional objects of general topological type. We construct shape spaces equipped with geodesic metrics that measure how costly it is to interpolate two shapes through elastic deformations. The model employs a representation of shape based on the discrete exterior derivative of parametrizations over a finite simplicial complex. We develop algorithms to calculate geodesics and geodesic distances, as well as tools to quantify local shape similarities and contrasts, thus obtaining a localglobal formulation that accounts for regional shape differences and integrates them into a global measure of dissimilarity. The Riemannian shape spaces provide a common framework to treat numerous problems such as the statistical modeling of shapes, the comparison of shapes associated with different individuals and groups, and modeling and simulation of dynamical shapes. We give multiple examples of geodesic interpolations and illustrations of the use of the model in brain mapping, particularly, the analysis of anatomical variation based on neuroimaging data. 1
A Continuum Mechanical Approach to Geodesics in Shape Space
"... In this paper concepts from continuum mechanics are used to define geodesic paths in the space of shapes, where shapes are implicitly described as boundary contours of objects. The proposed shape metric is derived from a continuum mechanical notion of viscous dissipation. A geodesic path is defined ..."
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In this paper concepts from continuum mechanics are used to define geodesic paths in the space of shapes, where shapes are implicitly described as boundary contours of objects. The proposed shape metric is derived from a continuum mechanical notion of viscous dissipation. A geodesic path is defined as the family of shapes such that the total amount of viscous dissipation caused by an optimal material transport along the path is minimized. The approach can easily be generalized to shapes given as segment contours of multilabeled images and to geodesic paths between partially occluded objects. The proposed computational framework for finding such a minimizer is based on the time discretization of a geodesic path as a sequence of pairwise matching problems, which is strictly invariant with respect to rigid body motions and ensures a 11 correspondence along the induced flow in shape space. When decreasing the time step size, the proposed model leads to the minimization of the actual geodesic length, where the Hessian of the pairwise matching energy reflects the chosen Riemannian metric on the underlying shape space. If the constraint of pairwise shape correspondence is replaced by the volume of the shape mismatch as a penalty functional, one obtains for decreasing time step size an optical flow term controlling the transport of the shape by the underlying motion field. The method is implemented via a level set representation of shapes, and a finite element approximation is employed as spatial discretization both for the pairwise matching deformations and for the level set representations. The numerical relaxation of the energy is performed via an efficient multiscale procedure in space and time. Various examples for 2D and 3D shapes underline the effectiveness and robustness of the proposed approach. 1