31-Issue 5

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https://diglib.eg.org/handle/10.2312/174

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Browsing 31-Issue 5 by Subject "and systems"

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    Can Mean-Curvature Flow be Modified to be Non-singular?
    (The Eurographics Association and Blackwell Publishing Ltd., 2012) Kazhdan, Michael; Solomon, Jake; Ben-Chen, Mirela; Eitan Grinspun and Niloy Mitra
    This work considers the question of whether mean-curvature flow can be modified to avoid the formation of singularities. We analyze the finite-elements discretization and demonstrate why the original flow can result in numerical instability due to division by zero. We propose a variation on the flow that removes the numerical instability in the discretization and show that this modification results in a simpler expression for both the discretized and continuous formulations. We discuss the properties of the modified flow and present empirical evidence that not only does it define a stable surface evolution for genus-zero surfaces, but that the evolution converges to a conformal parameterization of the surface onto the sphere.
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    Shape-Up: Shaping Discrete Geometry with Projections
    (The Eurographics Association and Blackwell Publishing Ltd., 2012) Bouaziz, Sofien; Deuss, Mario; Schwartzburg, Yuliy; Weise, Thibaut; Pauly, Mark; Eitan Grinspun and Niloy Mitra
    We introduce a unified optimization framework for geometry processing based on shape constraints. These constraints preserve or prescribe the shape of subsets of the points of a geometric data set, such as polygons, one-ring cells, volume elements, or feature curves. Our method is based on two key concepts: a shape proximity function and shape projection operators. The proximity function encodes the distance of a desired least-squares fitted elementary target shape to the corresponding vertices of the 3D model. Projection operators are employed to minimize the proximity function by relocating vertices in a minimal way to match the imposed shape constraints. We demonstrate that this approach leads to a simple, robust, and efficient algorithm that allows implementing a variety of geometry processing applications, simply by combining suitable projection operators. We show examples for computing planar and circular meshes, shape space exploration, mesh quality improvement, shape-preserving deformation, and conformal parametrization. Our optimization framework provides a systematic way of building new solvers for geometry processing and produces similar or better results than state-of-the-art methods.
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    Smooth Shape-Aware Functions with Controlled Extrema
    (The Eurographics Association and Blackwell Publishing Ltd., 2012) Jacobson, Alec; Weinkauf, Tino; Sorkine, Olga; Eitan Grinspun and Niloy Mitra
    Functions that optimize Laplacian-based energies have become popular in geometry processing, e.g. for shape deformation, smoothing, multiscale kernel construction and interpolation. Minimizers of Dirichlet energies, or solutions of Laplace equations, are harmonic functions that enjoy the maximum principle, ensuring no spurious local extrema in the interior of the solved domain occur. However, these functions are only C0 at the constrained points, which often causes smoothness problems. For this reason, many applications optimize higher-order Laplacian energies such as biharmonic or triharmonic. Their minimizers exhibit increasing orders of continuity but lose the maximum principle and show oscillations. In this work, we identify characteristic artifacts caused by spurious local extrema, and provide a framework for minimizing quadratic energies on manifolds while constraining the solution to obey the maximum principle in the solved region. Our framework allows the user to specify locations and values of desired local maxima and minima, while preventing any other local extrema. We demonstrate our method on the smoothness energies corresponding to popular polyharmonic functions and show its usefulness for fast handle-based shape deformation, controllable color diffusion, and topologically-constrained data smoothing.
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    Soft Maps Between Surfaces
    (The Eurographics Association and Blackwell Publishing Ltd., 2012) Solomon, Justin; Nguyen, Andy; Butscher, Adrian; Ben-Chen, Mirela; Guibas, Leonidas; Eitan Grinspun and Niloy Mitra
    The problem of mapping between two non-isometric surfaces admits ambiguities on both local and global scales. For instance, symmetries can make it possible for multiple maps to be equally acceptable, and stretching, slippage, and compression introduce difficulties deciding exactly where each point should go. Since most algorithms for point-to-point or even sparse mapping struggle to resolve these ambiguities, in this paper we introduce soft maps, a probabilistic relaxation of point-to-point correspondence that explicitly incorporates ambiguities in the mapping process. In addition to explaining a continuous theory of soft maps, we show how they can be represented using probability matrices and computed for given pairs of surfaces through a convex optimization explicitly trading off between continuity, conformity to geometric descriptors, and spread. Given that our correspondences are encoded in matrix form, we also illustrate how low-rank approximation and other linear algebraic tools can be used to analyze, simplify, and represent both individual and collections of soft maps.