SGP17: Eurographics Symposium on Geometry Processing - Posters

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https://diglib.eg.org/handle/10.2312/2631655

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    Sequentially-Defined Compressed Modes via ADMM
    (The Eurographics Association, 2017) Houston, Kevin; Jakob Andreas Bærentzen and Klaus Hildebrandt
    The eigenfunctions of the discrete Laplace-Beltrami operator have played an important role in many aspects of geometry processing. Given the success of sparse representation methods in areas such as compressive sensing it is reasonable to find a sparse analogue of LBO eigenfunctions. This has been done by Ozolinš et al for Euclidean spaces and Neumann et al for surfaces where the resulting analogues are called compressed modes. In this short report we show that the method of Alternating Direction Method of Multipliers can be used to efficiently calculate compressed modes and that this compares well with a recent method to calculate them with an Iteratively Reweighted Least Squares method.
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    A Primal-to-Primal Discretization of Exterior Calculus on Polygonal Meshes
    (The Eurographics Association, 2017) Ptackova, Lenka; Velho, Luiz; Jakob Andreas Bærentzen and Klaus Hildebrandt
    Discrete exterior calculus (DEC) offers a coordinate-free discretization of exterior calculus especially suited for computations on curved spaces. We present an extended version of DEC on surface meshes formed by general polygons that bypasses the construction of any dual mesh and the need for combinatorial subdivisions. At its core, our approach introduces a polygonal wedge product that is compatible with the discrete exterior derivative in the sense that it obeys the Leibniz rule. Based on this wedge product, we derive a novel primal-primal Hodge star operator, which then leads to a discrete version of the contraction operator. We show preliminary results indicating the numerical convergence of our discretization to each one of these operators.
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    Symposium on Geometry Processing 2017: Frontmatter
    (Eurographics Association, 2017) Bærentzen, Jakob Andreas; Hildebrandt, Klaus; Bærentzen, Jakob Andreas and Hildebrandt, Klaus
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    Localized Manifold Harmonics for Spectral Shape Analysis
    (The Eurographics Association, 2017) Melzi, Simone; Rodolà, Emanuele; Castellani, Umberto; Bronstein, Michael M.; Jakob Andreas Bærentzen and Klaus Hildebrandt
    The use of Laplacian eigenfunctions is ubiquitous in a wide range of computer graphics and geometry processing applications. In particular, Laplacian eigenbases allow generalizing the classical Fourier analysis to manifolds. A key drawback of such bases is their inherently global nature, as the Laplacian eigenfunctions carry geometric and topological structure of the entire manifold. In this paper, we introduce a new framework for local spectral shape analysis. We show how to efficiently construct localized orthogonal bases by solving an optimization problem that in turn can be posed as the eigendecomposition of a new operator obtained by a modification of the standard Laplacian. We study the theoretical and computational aspects of the proposed framework and showcase our new construction on the classical problems of shape approximation and correspondence.
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    PCR: A Geometric Cocktail for Triangulating Point Clouds Beautifully Without Angle Bounds
    (The Eurographics Association, 2017) Leitão, Gonçalo N. V.; Gomes, Abel J. P.; Jakob Andreas Bærentzen and Klaus Hildebrandt
    Reconstructing a triangulated surface from a point cloud through a mesh growing algorithm is a difficult problem, in largely because they use bounds for the admissible dihedral angle to decide on the next triangle to be attached to the mesh front. This paper proposes a solution to this problem by combining three geometric properties: proximity, co-planarity, and regularity; hence, the PCR cocktail. The PCR cocktail-based algorithm works well even for point clouds with non-uniform point density, holes, high curvature regions, creases, apices, and noise.
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    DepthCut: Improved Depth Edge Estimation Using Multiple Unreliable Channels
    (The Eurographics Association, 2017) Guerrero, Paul; Winnemöller, Holger; Li, Wilmot; Mitra, Niloy J.; Jakob Andreas Bærentzen and Klaus Hildebrandt
    In the context of scene understanding, a variety of methods exists to estimate different information channels from mono or stereo images, including disparity, depth, and normals. Although several advances have been reported in the recent years for these tasks, the estimated information is often imprecise particularly near depth contours or creases. Studies have however shown that precisely such depth edges carry critical cues for the perception of shape, and play important roles in tasks like depth-based segmentation or foreground selection. Unfortunately, the currently extracted channels often carry conflicting signals, making it difficult for subsequent applications to effectively use them. In this paper, we focus on the problem of obtaining high-precision depth edges by jointly analyzing such unreliable information channels. We propose DEPTHCUT, a data-driven fusion of the channels using a convolutional neural network trained on a large dataset with known depth. The resulting depth edges can be used for segmentation, decomposing a scene into segments with relatively smooth depth, or improving the accuracy of the depth estimate near depth edges by constraining its gradients to agree with these edges. Quantitative experiments show that our depth edges result in an improved segmentation performance compared to a more naive channel fusion. Qualitatively, we demonstrate that the depth edges can be used for superior segmentation and an improved depth estimate near depth edges.
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    Schrödinger Operator for Sparse Approximation of 3D Meshes
    (The Eurographics Association, 2017) Choukroun, Yoni; Pai, Gautam; Kimmel, Ron; Jakob Andreas Bærentzen and Klaus Hildebrandt
    We introduce a Schrödinger operator for spectral approximation of meshes representing surfaces in 3D. The operator is obtained by modifying the Laplacian with a potential function which defines the rate of oscillation of the harmonics on different regions of the surface. We design the potential using a vertex ordering scheme which modulates the Fourier basis of a 3D mesh to focus on crucial regions of the shape having high-frequency structures and employ a sparse approximation framework to maximize compression performance. The combination of the spectral geometry of the Hamiltonian in conjunction with a sparse approximation approach outperforms existing spectral compression schemes.