Mean value coordinates for quad cages in 3D

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.