Tutte Embeddings of Tetrahedral Meshes

Tutte's embedding theorem states that every 3-connected graph without a $K_5$ or $K_{3,3}$ minor (i.e. a planar graph) is embedded in the plane if the outer face is in convex position and the interior vertices are convex combinations of their neighbors. We show that this result extends to simply connected tetrahedral meshes in a natural way: for the tetrahedral mesh to be embedded if the outer polyhedron is in convex position and the interior vertices are convex combination of their neighbors it is sufficient (but not necessary) that the graph of the tetrahedral mesh contains no $K_6$ and no $K_{3,3,1}$, and all triangles incident on three boundary vertices are boundary triangles

Differentiable Shadow Mapping for Efficient Inverse Graphics

We show how shadows can be efficiently generated in differentiable rendering of triangle meshes. Our central observation is that pre-filtered shadow mapping, a technique for approximating shadows based on rendering from the perspective of a light, can be combined with existing differentiable rasterizers to yield differentiable visibility information. We demonstrate at several inverse graphics problems that differentiable shadow maps are orders of magnitude faster than differentiable light transport simulation with similar accuracy -- while differentiable rasterization without shadows often fails to converge.Comment: CVPR 2023, project page: https://mworchel.github.io/differentiable-shadow-mappin

An improved adjacency data structure for fast triangle stripping

To speed up the rendering of polygonal meshes, triangle strips are commonly used to reduce the number of vertices sent to the graphics subsystem by exploiting the fact that adjacent triangles share an edge. In this paper, we present an improved adjacency data structure for fast triangle stripping algorithms. There are three major contributions: first, the data structure can be created quickly and robustly from any indexed face set; second, its cache-friendly layout is specifically designed to efficiently answer common stripping queries, such as neighbor finding and least-degree triangle finding, in constant time; third, the stripping algorithm operates in-place, since strips are created by simply relinking pointers. An implementation of a stripping algorithm shows a significant speed-up compared to other implementations. Our implementation is publicly available as part of OpenSG [9].

The mean point of vergence is biased under projection

The point of interest in three-dimensional space in eye tracking is often computed based on intersecting the lines of sight with geometry, or finding the point closest to the two lines of sight. We first start by theoretical analysis with synthetic simulations. We show that the mean point of vergence is generally biased for centrally symmetric errors and that the bias depends on the horizontal vs. vertical error distribution of the tracked eye positions. Our analysis continues with an evaluation on real experimental data. The error distributions seem to be different among individuals but they generally leads to the same bias towards the observer. And it tends to be larger with an increased viewing distance. We also provided a recipe to minimize the bias, which applies to general computations of eye ray intersection. These findings not only have implications for choosing the calibration method in eye tracking experiments and interpreting the observed eye movements data; but also suggest to us that we shall consider the mathematical models of calibration as part of the experiment

Multi-scale geometry interpolation

Interpolating vertex positions among triangle meshes with identical vertex-edge graphs is a fundamental part of many geometric modelling systems. Linear vertex interpolation is robust but fails to preserve local shape. Most recent approaches identify local affine transformations for parts of the mesh, model desired interpolations of the affine transformations, and then optimize vertex positions to conform with the desired transformations. However, the local interpolation of the rotational part is non-trivial for more than two input configurations and ambiguous if the meshes are deformed significantly. We propose a solution to the vertex interpolation problem that starts from interpolating the local metric (edge lengths) and mean curvature (dihedral angles) and makes consistent choices of local affine transformations using shape matching applied to successively larger parts of the mesh. The local interpolation can be applied to any number of input vertex configurations and due to the hierarchical scheme for generating consolidated vertex positions, the approach is fast and can be applied to very large meshes