Abstract

Space coordinates offer an elegant, scalable and versatile framework to propagate (multi-)scalar functions from the boundary vertices of a 3-manifold, often called a cage, within its volume. These generalizations of the barycentric coordinate system have progressively expanded the range of eligible cages to triangle and planar polygon surface meshes with arbitrary topology, concave regions and a spatially-varying sampling ratio, while preserving a smooth diffusion of the prescribed on-surface functions. In spite of their potential for major computer graphics applications such as freeform deformation or volume texturing, current space coordinate systems have only found a moderate impact in applications. This follows from the constraint of having only triangles in the cage most of the time, while many application scenarios favor arbitrary (non-planar) quad meshes for their ability to align the surface structure with features and to naturally cope with anisotropic sampling. In order to use space coordinates with arbitrary quad cages currently, one must triangulate them, which results in large propagation distortion. Instead, we propose a generalization of a popular coordinate system -- Mean Value Coordinates -- to quad and tri-quad cages, bridging the gap between high-quality coarse meshing and volume diffusion through space coordinates. Our method can process non-planar quads, comes with a closed-form solution free from global optimization and reproduces the expected behavior of Mean Value Coordinates, namely smoothness within the cage volume and continuity everywhere. As a result, we show how these coordinates compare favorably to classical space coordinates on triangulated quad cages, in particular for freeform deformation.

Video

Downloads

• Manuscript: full res. (46.3MB) and low res. (6.69MB)
• Presentation: SIGGRAPH Asia 2018 slides (55.2MB)
• Source code: the reference implementation on GitHub.

Bibtex

@article{TMB:2018:QMVC,
    author = {Jean-Marc Thiery and Pooran Memari and Tamy Boubekeur},
    title = {Mean value coordinates for quad cages in 3D},
    journal = {ACM Transactions on Graphics (Proc. SIGGRAPH 2018)},
    year = {2018}
} 

Related projects

• our Bounding Proxies (SIGGRAPH 2017) can be used to design interactively low-poly cages from an high-poly models,
• we derived jacobians and hessians (TVCJ 2013) for mean value coordinates,
• our CageR system (Computer Graphics Forum 2012) helps modeling the motion of raw hihg resolution performance capture data using cages,
• our VoxMorph system (Shape Modeling International 2012) helps deforming large voxel grids interactively.