SIGGRAPH 2018

Despite high practical demand, algorithmic hexahedral meshing with guarantees on robustness and quality remains unsolved. A promising direction follows the idea of integer-grid maps, which pull back the Cartesian hexahedral grid formed by integer isoplanes from a parametric domain to a surface-conforming hexahedral mesh of the input object. Since directly optimizing for a high-quality integer-grid map is mathematically challenging, the construction is usually split into two steps: (1) generation of a surface-aligned octahedral field and (2) generation of an integer-grid map that best aligns to the octahedral field. The main robustness issue stems from the fact that smooth octahedral fields frequently exhibit singularity graphs that are not appropriate for hexahedral meshing and induce heavily degenerate integer-grid maps. The first contribution of this work is an enumeration of all local configurations that exist in hex meshes with bounded edge valence, and a generalization of the Hopf-Poincaré formula to octahedral fields, leading to necessary local and global conditions for the hex-meshability of an octahedral field in terms of its singularity graph. The second contribution is a novel algorithm to generate octahedral fields with prescribed hex-meshable singularity graphs, which requires the solution of a large non-linear mixed-integer algebraic system. This algorithm is an important step toward robust automatic hexahedral meshing since it enables the generation of a hex-meshable octahedral field.

Computer Graphics Forum (Proc. EUROGRAPHICS 2018)

Feature curves on 3D shapes provide important hints about significant parts of the geometry and reveal their underlying structure. However, when we process real world data, automatically detected feature curves are affected by measurement uncertainty, missing data, and sampling resolution, leading to noisy, fragmented, and incomplete feature curve networks. These artifacts make further processing unreliable. In this paper we analyze the global co-occurrence information in noisy feature curve networks to fill in missing data and suppress weakly supported feature curves. For this we propose an unsupervised approach to find meaningful structure within the incomplete data by detecting multiple occurrences of feature curve configurations (co-occurrence analysis). We cluster and merge these into feature curve templates, which we leverage to identify strongly supported feature curve segments as well as to complete missing data in the feature curve network. In the presence of significant noise, previous approaches had to resort to user input, while our method performs fully automatic feature curve co-completion. Finding feature reoccurrences however, is challenging since naive feature curve comparison fails in this setting due to fragmentation and partial overlaps of curve segments. To tackle this problem we propose a robust method for partial curve matching. This provides us with the means to apply symmetry detection methods to identify co-occurring configurations. Finally, Bayesian model selection enables us to detect and group re-occurrences that describe the data well and with low redundancy.

Eurographics Symposium on Geometry Processing 2018

Functional maps have gained popularity as a versatile framework for representing intrinsic correspondence between 3D shapes using algebraic machinery. A key ingredient for this framework is the ability to find pairs of corresponding functions (typically, feature descriptors) across the shapes. This is a challenging problem on its own, and when the shapes are strongly non-isometric, nearly impossible to solve automatically. In this paper, we use feature curve correspondences to provide flexible abstractions of semantically similar parts of non-isometric shapes. We design a user interface implementing an interactive process for constructing shape correspondence, allowing the user to update the functional map at interactive rates by introducing feature curve correspondences. We add feature curve preservation constraints to the functional map framework and propose an efficient numerical method to optimize the map with immediate feedback. Experimental results show that our approach establishes correspondences between geometrically diverse shapes with just a few clicks.