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47
Functional Maps: A Flexible Representation of Maps Between Shapes
"... Figure 1: Horse algebra: the functional representation and map inference algorithm allow us to go beyond pointtopoint maps. The source shape (top left corner) was mapped to the target shape (left) by posing descriptorbased functional constraints which do not disambiguate symmetries (i.e. without ..."
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Figure 1: Horse algebra: the functional representation and map inference algorithm allow us to go beyond pointtopoint maps. The source shape (top left corner) was mapped to the target shape (left) by posing descriptorbased functional constraints which do not disambiguate symmetries (i.e. without landmark constraints). By further adding correspondence constraints, we obtain a near isometric map which reverses orientation, mapping left to right (center). The representation allows for algebraic operations on shape maps, so we can subtract this map from the ambivalent map, to retrieve the orientation preserving nearisometry (right). Each column shows the first 20x20 block of the functional map representation (bottom), and the action of the map by transferring colors from the source shape to the target shape (top). We present a novel representation of maps between pairs of shapes that allows for efficient inference and manipulation. Key to our approach is a generalization of the notion of map that puts in correspondence realvalued functions rather than points on the shapes. By choosing a multiscale basis for the function space on each shape, such as the eigenfunctions of its LaplaceBeltrami operator, we obtain a representation of a map that is very compact, yet fully suitable for global inference. Perhaps more remarkably, most
Coupled quasiharmonic bases
 In ACM SIGGRAPH
, 2004
"... The use of Laplacian eigenbases has been shown to be fruitful in many computer graphics applications. Today, stateoftheart approaches to shape analysis, synthesis, and correspondence rely on these natural harmonic bases that allow using classical tools from harmonic analysis on manifolds. However ..."
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The use of Laplacian eigenbases has been shown to be fruitful in many computer graphics applications. Today, stateoftheart approaches to shape analysis, synthesis, and correspondence rely on these natural harmonic bases that allow using classical tools from harmonic analysis on manifolds. However, many applications involving multiple shapes are obstacled by the fact that Laplacian eigenbases computed independently on different shapes are often incompatible with each other. In this paper, we propose the construction of common approximate eigenbases for multiple shapes using approximate joint diagonalization algorithms. We illustrate the benefits of the proposed approach on tasks from shape editing, pose transfer, correspondence, and similarity. 1
Sparse modeling of intrinsic correspondences
 Computer Graphics Forum
"... We present a novel sparse modeling approach to nonrigid shape matching using only the ability to detect repeatable regions. As the input to our algorithm, we are given only two sets of regions in two shapes; no descriptors are provided so the correspondence between the regions is not know, nor we ..."
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We present a novel sparse modeling approach to nonrigid shape matching using only the ability to detect repeatable regions. As the input to our algorithm, we are given only two sets of regions in two shapes; no descriptors are provided so the correspondence between the regions is not know, nor we know how many regions correspond in the two shapes. We show that even with such scarce information, it is possible to establish very accurate correspondence between the shapes by using methods from the field of sparse modeling, being this, the first nontrivial use of sparse models in shape correspondence. We formulate the problem of permuted sparse coding, in which we solve simultaneously for an unknown permutation ordering the regions on two shapes and for an unknown correspondence in functional representation. We also propose a robust variant capable of handling incomplete matches. Numerically, the problem is solved efficiently by alternating the solution of a linear assignment and a sparse coding problem. The proposed methods are evaluated qualitatively and quantitatively on standard benchmarks containing both synthetic and scanned objects. 1
D.: PoseConsistent 3D Shape Segmentation Based on a Quantum Mechanical Feature Descriptor
 In Pattern Recognition (Proc. DAGM
"... Abstract. We propose a novel method for poseconsistent segmentation of nonrigid 3D shapes into visually meaningful parts. The key idea is to study the shape in the framework of quantum mechanics and to group points on the surface which have similar probability of presence for quantum mechanical pa ..."
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Abstract. We propose a novel method for poseconsistent segmentation of nonrigid 3D shapes into visually meaningful parts. The key idea is to study the shape in the framework of quantum mechanics and to group points on the surface which have similar probability of presence for quantum mechanical particles. For each point on an object’s surface these probabilities are encoded by a feature vector, the Wave Kernel Signature (WKS). Mathematically, the WKS is an expression in the eigenfunctions of the Laplace–Beltrami operator of the surface. It characterizes the relation of surface points to the remaining surface at various spatial scales. Gaussian mixture clustering in the feature space spanned by the WKS signature for shapes in several poses leads to a grouping of surface points into different and meaningful segments. This enables us to perform consistent and robust segmentation of new versions of the shape. Experimental results demonstrate that the detected subdivision agrees with the human notion of shape decomposition (separating hands, arms, legs and head from the torso for example). We show that the method is robust to data perturbed by various kinds of noise. Finally we illustrate the usefulness of a poseconsistent segmentation for the purpose of shape retrieval. 1
Partial shape matching without pointwise correspondence
"... Partial similarity of shapes in a challenging problem arising in many important applications in computer vision, shape analysis, and graphics, e.g. when one has to deal with partial information and acquisition artifacts. The problem is especially hard when the underlying shapes are nonrigid and ar ..."
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Partial similarity of shapes in a challenging problem arising in many important applications in computer vision, shape analysis, and graphics, e.g. when one has to deal with partial information and acquisition artifacts. The problem is especially hard when the underlying shapes are nonrigid and are given up to a deformation. Partial matching is usually approached by computing local descriptors on a pair of shapes and then establishing a pointwise nonbijective correspondence between the two, taking into account possibly different parts. In this paper, we introduce an alternative correspondenceless approach to matching fragments to an entire shape undergoing a nonrigid deformation. We use diffusion geometric descriptors and optimize over the integration domains on which the integral descriptors of the two parts match. The problem is regularized using the MumfordShah functional. We show an efficient discretization based on the AmbrosioTortorelli approximation generalized to triangular meshes and point clouds, and present experiments demonstrating the success of the proposed method.
Spectral generalized multidimensional scaling
, 2013
"... Multidimensional scaling (MDS) is a family of methods that embed a given set of points into a simple, usually flat, domain. The points are assumed to be sampled from some metric space, and the mapping attempts to preserve the distances between each pair of points in the set. Distances in the target ..."
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Multidimensional scaling (MDS) is a family of methods that embed a given set of points into a simple, usually flat, domain. The points are assumed to be sampled from some metric space, and the mapping attempts to preserve the distances between each pair of points in the set. Distances in the target space can be computed analytically in this setting. Generalized MDS is an extension that allows mapping one metric space into another, that is, multidimensional scaling into target spaces in which distances are evaluated numerically rather than analytically. Here, we propose an efficient approach for computing such mappings between surfaces based on their natural spectral decomposition, where the surfaces are treated as sampled metricspaces. The resulting spectralGMDS procedure enables efficient embedding by implicitly incorporating smoothness of the mapping into the problem, thereby substantially reducing the complexity involved in its solution while practically overcoming its nonconvex nature. The method is compared to existing techniques that compute dense correspondence between shapes. Numerical experiments of the proposed method demonstrate its efficiency and accuracy compared to stateoftheart approaches. 1
Supervised Descriptor Learning for NonRigid Shape Matching
"... Abstract. We present a novel method for computing correspondences between pairs of nonrigid shapes. Unlike the majority of existing techniques that assume a deformation model, such as intrinsic isometries, a priori and use a predefined set of point or part descriptors, we consider the problem of ..."
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Abstract. We present a novel method for computing correspondences between pairs of nonrigid shapes. Unlike the majority of existing techniques that assume a deformation model, such as intrinsic isometries, a priori and use a predefined set of point or part descriptors, we consider the problem of learning a correspondence model given a collection of reference pairs with known mappings between them. Our formulation is purely intrinsic and does not rely on a consistent parametrization or spatial positions of vertices on the shapes. Instead, we consider the problem of finding the optimal set of descriptors that can be jointly used to reproduce the given reference maps. We show how this problem can be formalized and solved for efficiently by using the recently proposed functional maps framework. Moreover, we demonstrate how to extract the functional subspaces that can be mapped reliably across shapes. This gives us a way to not only obtain better functional correspondences, but also to associate a confidence value to the different parts of the mappings. We demonstrate the efficiency and usefulness of the proposed approach on a variety of challenging shape matching tasks.
Graph matching with anchor nodes: A learning approach
 In CVPR
, 2013
"... In this paper, we consider the weighted graph matching problem with partially disclosed correspondences between a number of anchor nodes. Our construction exploits recently introduced node signatures based on graph Laplacians, namely the Laplacian family signature (LFS) on the nodes, and the pairw ..."
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In this paper, we consider the weighted graph matching problem with partially disclosed correspondences between a number of anchor nodes. Our construction exploits recently introduced node signatures based on graph Laplacians, namely the Laplacian family signature (LFS) on the nodes, and the pairwise heat kernel map on the edges. In this paper, without assuming an explicit form of parametric dependence nor a distance metric between node signatures, we formulate an optimization problem which incorporates the knowledge of anchor nodes. Solving this problem gives us an optimized proximity measure specific to the graphs under consideration. Using this as a first order compatibility term, we then set up an integer quadratic program (IQP) to solve for a near optimal graph matching. Our experiments demonstrate the superior performance of our approach on randomly generated graphs and on two widelyused image sequences, when compared with other existing signature and adjacency matrix based graph matching methods. 1.
Schrödinger Diffusion for Shape Analysis with Texture
"... Abstract. In recent years, quantities derived from the heat equation have become popular in shape processing and analysis of triangulated surfaces. Such measures are often robust with respect to different kinds of perturbations, including nearisometries, topological noise and partialities. Here, we ..."
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Abstract. In recent years, quantities derived from the heat equation have become popular in shape processing and analysis of triangulated surfaces. Such measures are often robust with respect to different kinds of perturbations, including nearisometries, topological noise and partialities. Here, we propose to exploit the semigroup of a Schrödinger operator in order to deal with texture data, while maintaining the desirable properties of the heat kernel. We define a family of Schrödinger diffusion distances analogous to the ones associated to the heat kernels, and show that they are continuous under perturbations of the data. As an application, we introduce a method for retrieval of textured shapes through comparison of Schrödinger diffusion distance histograms with the earth’s mover distance, and present some numerical experiments showing superior performance compared to an analogous method that ignores the texture.
Persistencebased structural recognition
 In CVPR
, 2014
"... This paper presents a framework for object recognition using topological persistence. In particular, we show that the socalled persistence diagrams built from functions defined on the objects can serve as compact and informative descriptors for images and shapes. Complementary to the bagoffeatur ..."
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This paper presents a framework for object recognition using topological persistence. In particular, we show that the socalled persistence diagrams built from functions defined on the objects can serve as compact and informative descriptors for images and shapes. Complementary to the bagoffeatures representation, which captures the distribution of values of a given function, persistence diagrams can be used to characterize its structural properties, reflecting spatial information in an invariant way. In practice, the choice of function is simple: each dimension of the feature vector can be viewed as a function. The proposed method is general: it can work on various multimedia data, including 2D shapes, textures and triangle meshes. Extensive experiments on 3D shape retrieval, hand gesture recognition and texture classification demonstrate the performance of the proposed method in comparison with stateoftheart methods. Additionally, our approach yields higher recognition accuracy when used in conjunction with the bagoffeatures. 1.