Section: New Results

Approximation

Zometool Shape Approximation

Participants : Henrik Zimmer, Florent Lafarge, Pierre Alliez.

In collaboration with Leif Kobbelt.

We contributed an algorithm that approximates 2-manifold surfaces with Zometool models while preserving their topology. Zometool is a popular hands-on mathematical modeling system used in teaching, research and for recreational model assemblies at home. This construction system relies on a single node type with a small, fixed set of directions and only nine different edge types in its basic form. While being naturally well suited for modeling symmetries, various polytopes or visualizing molecular structures, the inherent discreteness of the system poses difficult constraints on any algorithmic approach to support the modeling of freeform shapes. We contribute a set of local, topology preserving Zome mesh modification operators enabling the efficient exploration of the space of 2-manifold Zome models around a given input shape. Starting from a rough initial approximation, the operators are iteratively selected within a stochastic framework guided by an energy functional measuring the quality of the approximation. We demonstrate our approach on a number of designs and also describe parameters which are used to explore different complexities and enable coarse approximations [15] .

CGALmesh: a Generic Framework for Delaunay Mesh Generation

Participants : Pierre Alliez, Clement Jamin.

In collaboration with Jean-Daniel Boissonnat and Mariette Yvinec.

CGALmesh is the mesh generation software package of the Computational Geometry Algorithm Library (CGAL). It generates isotropic simplicial meshes – surface triangular meshes or volume tetrahedral meshes – from input surfaces, 3D domains as well as 3D multi-domains, with or without sharp features. The underlying meshing algorithm relies on restricted Delaunay triangulations to approximate domains and surfaces, and on Delaunay refinement to ensure both approximation accuracy and mesh quality. CGALmesh provides guarantees on approximation quality as well as on the size and shape of the mesh elements. It provides four optional mesh optimization algorithms to further improve the mesh quality. A distinctive property of CGALmesh is its high flexibility with respect to the input domain representation. Such a flexibility is achieved through a careful software design, gathering into a single abstract concept, denoted by the oracle, all required interface features between the meshing engine and the input domain. We already provide oracles for domains defined by polyhedral and implicit surfaces [5] .

Level-of-Detail Quad Meshing

Participant : David Bommes.

In collaboration with Hans-Christian Ebke and Leif Kobbelt from RWTH Aachen.

The most effective and popular tools for obtaining feature aligned quad meshes from triangular input meshes are based on cross field guided parametrization. These methods are incarnations of a conceptual three-step pipeline: (1) cross field computation, (2) field- guided surface parametrization, (3) quad mesh extraction. While in most meshing scenarios the user prescribes a desired target quad size or edge length, this information is typically taken into account from step 2 onwards only, but not in the cross field computation step. This turns into a problem in the presence of small scale geometric or topological features or noise in the input mesh: closely placed singularities are induced in the cross field, which are not properly reproducible by vertices in a quad mesh with the pre- scribed edge length, causing severe distortions or even failure of the meshing algorithm. We reformulate the construction of cross fields as well as field-guided parameterizations in a scale-aware manner which effectively suppresses densely spaced features and noise of geometric as well as topological kind. Dominant large-scale features are adequately preserved in the output by relying on the unaltered input mesh as the computational domain [4] .

Mesh Watermarking based on a Constrained Optimization Framework

Participants : Xavier Rolland-Nevière, Pierre Alliez.

In collaboration with Technicolor and Gwenael Doerr.

A watermarking strategy for triangle surface meshes consists in modifying the vertex positions along the radial directions, in order to adjust the distribution of radial distances and thereby encode the desired payload. To guarantee that watermark embedding does not alter the center of mass, prior work formulated this task as a quadratic programming problem. We contribute a generalization of this formulation with: (i) integral reference primitives, (ii) arbitrary relocation directions to alter the vertex positions, and (iii) alternate distortion metrics to minimize the perceptual impact of the embedding process. These variants are evaluated against a range of attacks and we report both improved robustness performances, in particular for simplification attacks, and improved control over the embedding distortion [9] .

Robust 3D Watermarking

Participants : Xavier Rolland-Nevière, Pierre Alliez.

In collaboration with Technicolor, thesis co-advised by Pierre Alliez and Gwenael Doerr.

3D models are valuable assets widely used in the industry and likely to face piracy issues. This dissertation deals with robust mesh watermarking that is used for traitor-tracing. Following a review of state-of-the-art 3D watermarking systems, the robustness of several content adaptation transforms are evaluated. An embedding domain robust against pose is investigated, with a thickness estimation based on a robust distance function to a point cloud constructed from some mesh diameters. A benchmark showcases the performance of this domain that provides a basis for robust watermarking in 3D animations. For static meshes, modulating the radial distances is an efficient approach to watermarking. It has been formulated as a quadratic programming problem minimizing the geometric distortion while embedding the payload in the radial distances. This formulation is leveraged to create a robust watermarking framework, with the integration of the spread-transform, integral reference primitives, arbitrarily selected relocation directions and alternate metrics to minimize the distortion perceived. Benchmarking results showcase the benefits of these add-ons w.r.t the fidelity vs. robustness watermarking trade-off. The watermark security is then investigated with two obfuscation mechanisms and a series of attacks that highlight the remaining limitations. A resynchronization approach is finally integrated to deal with cropping attacks. The resynchronization embeds landmarks in a configuration that conveys synchronization information that will be lost after cropping. During the decoding, this information is blindly retrieved and significant robustness improvements are achieved [2] .

Spread Transform and Roughness-based Shaping to Improve 3D Watermarking based on Quadratic Programming

Participants : Xavier Rolland-Nevière, Pierre Alliez.

In collaboration with Technicolor and Gwenael Doerr.

Modulating the distances between the vertices and the center of mass of a triangular mesh is a popular approach to watermark 3D objects. Prior work has formulated this approach as a quadratic programming problem which minimizes the geometric distortion while embedding the watermark payload in the histogram of distances. To enhance this framework, we introduce two watermarking components, namely the spread transform and perceptual shaping based on roughness information. Benchmarking results showcase the benefits of these add-ons with respect to the fidelity-robustness trade-off [13] .