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Continuous Inference in Graphical Models with Polynomial Energies
"... In this paper, we tackle the problem of performing inference in graphical models whose energy is a polynomial function of continuous variables. Our energy minimization method follows a dual decomposition approach, where the global problem is split into subproblems defined over the graph cliques. Th ..."
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In this paper, we tackle the problem of performing inference in graphical models whose energy is a polynomial function of continuous variables. Our energy minimization method follows a dual decomposition approach, where the global problem is split into subproblems defined over the graph cliques. The optimal solution to these subproblems is obtained by making use of a polynomial system solver. Our algorithm inherits the convergence guarantees of dual decomposition. To speed up optimization, we also introduce a variant of this algorithm based on the augmented Lagrangian method. Our experiments illustrate the diversity of computer vision problems that can be expressed with polynomial energies, and demonstrate the benefits of our approach over existing continuous inference methods. 1.
Nonrigid Surface Registration and Completion from RGBD Images
"... Abstract. Nonrigid surface registration is a challenging problem that suffers from many ambiguities. Existing methods typically assume the availability of full volumetric data, or require a global model of the surface of interest. In this paper, we introduce an approach to nonrigid registration tha ..."
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Abstract. Nonrigid surface registration is a challenging problem that suffers from many ambiguities. Existing methods typically assume the availability of full volumetric data, or require a global model of the surface of interest. In this paper, we introduce an approach to nonrigid registration that performs on relatively lowquality RGBD images and does not assume prior knowledge of the global surface shape. To this end, we model the surface as a collection of patches, and infer the patch deformations by performing inference in a graphical model. Our representation lets us fill in the holes in the input depth maps, thus essentially achieving surface completion. Our experimental evaluation demonstrates the effectiveness of our approach on several sequences, as well as its robustness to missing data and occlusions.
Good Vibrations: A Modal Analysis Approach for Sequential NonRigid Structure from Motion
"... We propose an online solution to nonrigid structure from motion that performs camera pose and 3D shape estimation of highly deformable surfaces on a framebyframe basis. Our method models nonrigid deformations as a linear combination of some mode shapes obtained using modal analysis from continu ..."
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We propose an online solution to nonrigid structure from motion that performs camera pose and 3D shape estimation of highly deformable surfaces on a framebyframe basis. Our method models nonrigid deformations as a linear combination of some mode shapes obtained using modal analysis from continuum mechanics. The shape is first discretized into linear elastic triangles, modelled by means of finite elements, which are used to pose the force balance equations for an undamped free vibrations model. The shape basis computation comes down to solving an eigenvalue problem, without the requirement of a learning step. The camera pose and time varying weights that define the shape at each frame are then estimated on the fly, in an online fashion, using bundle adjustment over a sliding window of image frames. The result is a low computational cost method that can run sequentially in realtime. We show experimental results on synthetic sequences with ground truth 3D data and real videos for different scenarios ranging from sparse to dense scenes. Our system exhibits a good tradeoff between accuracy and computational budget, it can handle missing data and performs favourably compared to competing methods. 1.
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"... This paper describes a sequential solution to dense nonrigid structure from motion that recovers the camera motion and 3D shape of nonrigid objects by processing a monocular image sequence as the data arrives. We propose to model the timevarying shape with a probabilistic linear subspace of mode ..."
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This paper describes a sequential solution to dense nonrigid structure from motion that recovers the camera motion and 3D shape of nonrigid objects by processing a monocular image sequence as the data arrives. We propose to model the timevarying shape with a probabilistic linear subspace of mode shapes obtained from continuum mechanics. To efficiently encode the deformations of dense 3D shapes that contain a large number of mesh vertexes, we propose to compute the deformation modes on a downsampled rest shape using finite element modal analysis at a low computational cost. This sparse shape basis is then grown back to dense exploiting the shape functions within a finite element. With this probabilistic lowrank constraint, we estimate camera pose and nonrigid shape in each frame using expectation maximization over a sliding window of frames. Since the timevarying weights are marginalized out, our approach only estimates a small number of parameters per frame, and hence can potentially run in real time. We evaluate our algorithm on both synthetic and real sequences with 3D ground truth data for different objects ranging from inextensible to extensible deformations and from sparse to dense shapes. We show the advantages of our approach with respect to competing sequential methods. 1
Efficient Inference of Continuous Markov Random Fields with Polynomial Potentials
"... In this paper, we prove that every multivariate polynomial with even degree can be decomposed into a sum of convex and concave polynomials. Motivated by this property, we exploit the concaveconvex procedure to perform inference on continuous Markov random fields with polynomial potentials. In parti ..."
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In this paper, we prove that every multivariate polynomial with even degree can be decomposed into a sum of convex and concave polynomials. Motivated by this property, we exploit the concaveconvex procedure to perform inference on continuous Markov random fields with polynomial potentials. In particular, we show that the concaveconvex decomposition of polynomials can be expressed as a sumofsquares optimization, which can be efficiently solved via semidefinite programing. We demonstrate the effectiveness of our approach in the context of 3D reconstruction, shape from shading and image denoising, and show that our method significantly outperforms existing techniques in terms of efficiency as well as quality of the retrieved solution. 1
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"... My research falls into the areas of machine learning and computer vision with the goal of identifying meaningful structures in highdimensional data. Fundamental problems such as clustering, segmentation, matching, and classification are often formulated as energy minimization tasks. However, the r ..."
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My research falls into the areas of machine learning and computer vision with the goal of identifying meaningful structures in highdimensional data. Fundamental problems such as clustering, segmentation, matching, and classification are often formulated as energy minimization tasks. However, the resulting optimization is usually nonconvex due to the inherent complexity or combinatorial nature of