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Modeling shapes with skeletons : scaffolds & anisotropic convolution

Alvaro Javier Fuentes Suárez 1
1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , NKUA | UoA - National and Kapodistrian University of Athens = University of Athens
Abstract : Skeletons have proved to be a successful tool in modeling complex shapes. They provide a basis for many processes ranging from implicit modeling, to deformation and animations. In this work we advance in two topics related with skeleton modeling: quad dominant skeleton-based meshes and smooth implicit surfaces generated from a skeleton. Given a skeleton made of line segments we describe how to obtain a coarse quad mesh of a surface that tightly encloses the skeleton and follows its structure - the scaffold. We formalize as an Integer Linear Program the problem of constructing an optimal scaffold that minimizes the total number of quads on the mesh. We prove the feasibility of the Integer Linear Program for any skeleton. In particular we can generate these scaffolds for skeletons with cycles. We additionally show how to obtain regular scaffolds, i.e. scaffolds with the same number of quad patches around each line segment, and symmetric scaffolds that respect the symmetries of the skeleton. Applications to polygonization of skeleton-based implicit surfaces are also presented. Convolution surfaces with ID skeletons have been limited to close-to-circular normal sections. The new formalism we present here increases the modeling freedom since it allows for ellipsoidal normal sections. The new anisotropy for Gl skeletal curves, chosen as circular splines, is interpolated from the rotation angles and three radii of ellipsoids at each extremity, given as user input. This lightweight model creates smooth shapes that previously required tweaking the skeleton or supplementing it with 2D pieces. The scale invariance of our formalism achieves excellent radii control and thus lends itself to approximate a variety of shapes. The construction of a scaffold is extended to skeletons with G l branches. It projects onto the convolution surface as a quad mesh with skeleton bound edge-flow. In addition to the two main contributions described above we develop further topics related to scaffolding and convolution surfaces. We discuss how spherical Laguerre diagrams may be used to improve the scaffold shapes when different incident radii is allowed at joints, and we describe how to construct volumetric hexahedral meshes for a skeleton-based model starting from a scaffold. We also introduce Creative Telescoping techniques for the computation of closed form formulas through recurrence. Finally we present PySkelton - a Python library for skeleton based modeling that implements our algorithms and provides an academic friendly programming interface.
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Alvaro Javier Fuentes Suárez. Modeling shapes with skeletons : scaffolds & anisotropic convolution. General Mathematics [math.GM]. COMUE Université Côte d'Azur (2015 - 2019), 2019. English. ⟨NNT : 2019AZUR4057⟩. ⟨tel-02420728v2⟩

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