Polycube maps of triangle meshes have proved useful in a wide range of applications including texture mapping and hexahedral mesh generation. However, constructing either fully automatically or with limited user control a low-distortion polycube from a detailed surface remains challenging in practice. We propose a variational method for deforming an input triangle mesh into a polycube shape through minimization of the 1 -norm of the mesh normals, regularized via an as-rigid-as-possible volumetric distortion energy. Unlike previous work, our approach makes no assumption on the orientation, or on the presence of features in the input model. User-guided control over the resulting polycube map is also offered to increase design flexibility. We demonstrate the robustness, efficiency and controllability of our method on a variety of examples, and explore applications in hexahedral remeshing and quadrangulation.

Figure 1: Adjoint invariant distance to orbit. Our Lie algebra voting approach to symmetry and orbit detection maps SE(3) transformations into points in a logarithmic space composed of a rotation part ω ∈ R 3 and a translation part u ∈ R 3 . The rotational orbit of the church and the translational orbit of the side railing (a) are mapped into collinear blue and red spheres respectively (a few transformations within these two orbits are marked with circled numbers to enhance comprehension). When the scene is centered, the two lines are orthogonal to each other and easy to distinguish (b). However, after a rigid translation of the scene, the rotational orbit now has u-values near the translation orbit points, making it impossible to automatically distinguish these two orbits using a Euclidean distance (d), while our adjoint invariant distance for orbit shows no discernible difference in results as evidenced by a binning of detected orbit sizes for both situations (e). AbstractIn this paper, we formulate an automatic approach to the detection of partial, local, and global symmetries and orbits in arbitrary 3D datasets. We improve upon existing voting-based symmetry detection techniques by leveraging the Lie group structure of geometric transformations. In particular, we introduce a logarithmic mapping that ensures that orbits are mapped to linear subspaces, hence unifying and extending many existing mappings in a single Lie-algebra voting formulation. Compared to previous work, our resulting method offers significantly improved robustness as it guarantees that our symmetry detection of an input model is frame, scale, and reflection invariant. As a consequence, we demonstrate that our approach efficiently and reliably discovers symmetries and orbits of geometric datasets without requiring heavy parameter tuning.

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