35-Issue 5

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https://diglib.eg.org/handle/10.2312/15441

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Browsing 35-Issue 5 by Subject "Geometric algorithms"

Now showing 1 - 5 of 5
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    Advection-Based Function Matching on Surfaces
    (The Eurographics Association and John Wiley & Sons Ltd., 2016) Azencot, Omri; Vantzos, Orestis; Ben-Chen, Mirela; Maks Ovsjanikov and Daniele Panozzo
    A tangent vector field on a surface is the generator of a smooth family of maps from the surface to itself, known as the flow. Given a scalar function on the surface, it can be transported, or advected, by composing it with a vector field's flow. Such transport is exhibited by many physical phenomena, e.g., in fluid dynamics. In this paper, we are interested in the inverse problem: given source and target functions, compute a vector field whose flow advects the source to the target. We propose a method for addressing this problem, by minimizing an energy given by the advection constraint together with a regularizing term for the vector field. Our approach is inspired by a similar method in computational anatomy, known as LDDMM, yet leverages the recent framework of functional vector fields for discretizing the advection and the flow as operators on scalar functions. The latter allows us to efficiently generalize LDDMM to curved surfaces, without explicitly computing the flow lines of the vector field we are optimizing for. We show two approaches for the solution: using linear advection with multiple vector fields, and using non-linear advection with a single vector field. We additionally derive an approximated gradient of the corresponding energy, which is based on a novel vector field transport operator. Finally, we demonstrate applications of our machinery to intrinsic symmetry analysis, function interpolation and map improvement.
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    Construction of Topologically Correct and Manifold Isosurfaces
    (The Eurographics Association and John Wiley & Sons Ltd., 2016) Grosso, Roberto; Maks Ovsjanikov and Daniele Panozzo
    We present a simple method to describe the geometry and topologically classify the intersection of level sets of trilinear interpolants with a reference unit cell. The solutions of three quadratic equations are used to correctly triangulate the level set within the cell satisfying the conditions imposed by the asymptotic decider. This way the triangulation of isosurfaces across cells borders is manifold and topologically correct. The algorithm presented is intuitive and easy to implement.
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    Mesh Statistics for Robust Curvature Estimation
    (The Eurographics Association and John Wiley & Sons Ltd., 2016) Váša, Libor; Vaněček, Petr; Prantl, Martin; Skorkovská, Věra; Martínek, Petr; Kolingerová, Ivana; Maks Ovsjanikov and Daniele Panozzo
    While it is usually not difficult to compute principal curvatures of a smooth surface of sufficient differentiability, it is a rather difficult task when only a polygonal approximation of the surface is available, because of the inherent ambiguity of such representation. A number of different approaches has been proposed in the past that tackle this problem using various techniques. Most papers tend to focus on a particular method, while an comprehensive comparison of the different approaches is usually missing. We present results of a large experiment, involving both common and recently proposed curvature estimation techniques, applied to triangle meshes of varying properties. It turns out that none of the approaches provides reliable results under all circumstances. Motivated by this observation, we investigate mesh statistics, which can be computed from vertex positions and mesh connectivity information only, and which can help in deciding which estimator will work best for a particular case. Finally, we propose a meta-estimator, which makes a choice between existing algorithms based on the value of the mesh statistics, and we demonstrate that such meta-estimator, despite its simplicity, provides considerably more robust results than any existing approach.
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    Near-Isometric Level Set Tracking
    (The Eurographics Association and John Wiley & Sons Ltd., 2016) Tao, Michael; Solomon, Justin; Butscher, Adrian; Maks Ovsjanikov and Daniele Panozzo
    Implicit representations of geometry have found applications in shape modeling, simulation, and other graphics pipelines. These representations, however, do not provide information about the paths of individual points as shapes move and undergo deformation. For this reason, we reconsider the problem of tracking points on level set surfaces, with the goal of designing an algorithm that - unlike previous work - can recover rotational motion and nearly isometric deformation. We track points on level sets of a time-varying function using approximate Killing vector fields (AKVFs), the velocity fields of near-isometric motions. To this end, we provide suitable theoretical and discrete constructions for computing AKVFs in a narrow band surrounding an animated level set surface. Furthermore, we propose time integrators well-suited to integrating AKVFs in time to track points. We demonstrate the theoretical and practical advantages of our proposed algorithms on synthetic and practical tasks.
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    Polycube Simplification for Coarse Layouts of Surfaces and Volumes
    (The Eurographics Association and John Wiley & Sons Ltd., 2016) Cherchi, Gianmarco; Livesu, Marco; Scateni, Riccardo; Maks Ovsjanikov and Daniele Panozzo
    Representing digital objects with structured meshes that embed a coarse block decomposition is a relevant problem in applications like computer animation, physically-based simulation and Computer Aided Design (CAD). One of the key ingredients to produce coarse block structures is to achieve a good alignment between the mesh singularities (i.e., the corners of each block). In this paper we improve on the polycube-based meshing pipeline to produce both surface and volumetric coarse block-structured meshes of general shapes. To this aim we add a new step in the pipeline. Our goal is to optimize the positions of the polycube corners to produce as coarse as possible base complexes. We rely on re-mapping the positions of the corners on an integer grid and then using integer numerical programming to reach the optimal. To the best of our knowledge this is the first attempt to solve the singularity misalignment problem directly in polycube space. Previous methods for polycube generation did not specifically address this issue. Our corner optimization strategy is efficient and requires a negligible extra running time for the meshing pipeline. In the paper we show that our optimized polycubes produce coarser block structured surface and volumetric meshes if compared with previous approaches. They also induce higher quality hexahedral meshes and are better suited for spline fitting because they reduce the number of splines necessary to cover the domain, thus improving both the efficiency and the overall level of smoothness throughout the volume.