36-Issue 5

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https://diglib.eg.org/handle/10.2312/2631654

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Browsing 36-Issue 5 by Subject "I.3.3 [Computer Graphics]"

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    Adjoint Map Representation for Shape Analysis and Matching
    (The Eurographics Association and John Wiley & Sons Ltd., 2017) Huang, Ruqi; Ovsjanikov, Maks; Bærentzen, Jakob Andreas and Hildebrandt, Klaus
    In this paper, we propose to consider the adjoint operators of functional maps, and demonstrate their utility in several tasks in geometry processing. Unlike a functional map, which represents a correspondence simply using the pull-back of function values, the adjoint operator reflects both the map and its distortion with respect to given inner products. We argue that this property of adjoint operators and especially their relation to the map inverse under the choice of different inner products, can be useful in applications including bi-directional shape matching, shape exploration, and pointwise map recovery among others. In particular, in this paper, we show that the adjoint operators can be used within the cycle-consistency framework to encode and reveal the presence or lack of consistency between distortions in a collection, in a way that is complementary to the previously used purely map-based consistency measures.We also show how the adjoint can be used for matching pairs of shapes, by accounting for maps in both directions, can help in recovering point-to-point maps from their functional counterparts, and describe how it can shed light on the role of functional basis selection.
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    Modeling and Exploring Co-variations in the Geometry and Configuration of Man-made 3D Shape Families
    (The Eurographics Association and John Wiley & Sons Ltd., 2017) Laga, Hamid; Tabia, Hedi; Bærentzen, Jakob Andreas and Hildebrandt, Klaus
    We introduce co-variation analysis as a tool for modeling the way part geometries and configurations co-vary across a family of man-made 3D shapes. While man-made 3D objects exhibit large geometric and structural variations, the geometry, structure, and configuration of their individual components usually do not vary independently from each other but in a correlated fashion. The size of the body of an airplane, for example, constrains the range of deformations its wings can undergo to ensure that the entire object remains a functionally-valid airplane. These co-variation constraints, which are often non-linear, can be either physical, and thus they can be explicitly enumerated, or implicit to the design and style of the shape family. In this article, we propose a data-driven approach, which takes pre-segmented 3D shapes with known component-wise correspondences and learns how various geometric and structural properties of their components co-vary across the set. We demonstrate, using a variety of 3D shape families, the utility of the proposed co-variation analysis in various applications including 3D shape repositories exploration and shape editing where the propagation of deformations is guided by the co-variation analysis. We also show that the framework can be used for context-guided orientation of objects in 3D scenes.
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    Restricting Voronoi Diagrams to Meshes Using Corner Validation
    (The Eurographics Association and John Wiley & Sons Ltd., 2017) Sainlot, Maxime; Nivoliers, Vincent; Attali, Dominique; Bærentzen, Jakob Andreas and Hildebrandt, Klaus
    Restricted Voronoi diagrams are a fundamental geometric structure used in many applications such as surface reconstruction from point sets or optimal transport. Given a set of sites V and a mesh X with vertices in Rd connected by triangles, the restricted Voronoi diagram partitions X by computing for each site the portion of X for which the site is the nearest. The restricted Voronoi diagram is the intersection between the regular Voronoi diagram and the mesh. Depending on the site distribution or the ambient space dimension computing the regular Voronoi diagram may not be feasible using classical algorithms. In this paper, we extend Lévy and Bonneel's approach [LB12] based on nearest neighbor queries. We show that their method is limited when the sites are not located on X. We propose a new algorithm for computing restricted Voronoi which reduces the number of sites considered for each triangle of the mesh and scales smoothly when the sites are far from the surface.