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We exhibit differential geometric structures that arise in numerical methods, based on the construction of Cauchy sequences, that are currently used to prove explicitly the existence of weak solutions to functional equations. We describe…
Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, computer vision, medical imaging and robotics. This report may be viewed…
Voronoi diagrams are a fundamental geometric data structure for obtaining proximity relations. We consider collections of axis-aligned orthogonal polyhedra in two and three-dimensional space under the max-norm, which is a particularly…
We introduce a new class of dynamic point process models with simple and intuitive dynamics that are based on the Voronoi tessellations generated by the processes. Under broad conditions, these processes prove to be ergodic and produce, on…
Reconstructions of sea level prior to the satellite altimeter era are usually derived from tide gauge records; however most algorithms for this assume that modes of sea level variability are stationary which is not true over several…
In this paper, we propose the differentiable voxelization of 3D meshes via the winding number and solid angles. The proposed approach achieves fast, flexible, and accurate voxelization of 3D meshes, admitting the computation of gradients…
Motivated by recent experiments of exceptional accuracy, we study numerically the spin-glass dynamics in a film geometry. We cover all the relevant time regimes, from picoseconds to equilibrium, at temperatures at and below the 3D critical…
Regression is one of the core problems tackled in supervised learning. Rectified linear unit (ReLU) neural networks generate continuous and piecewise-linear (CPWL) mappings and are the state-of-the-art approach for solving regression…
The original models of causal dynamical triangulations construct space-time by arranging a set of simplices in layers separated by a fixed time-like distance. The importance of the foliation structure in the 2+1 dimensional model is studied…
We study the set of image tuples arising from fixed cameras observing varying planar 3-dimensional point configurations. We derive a formula for the number of complex critical points of the triangulation problem, which seeks to reconstruct…
We extend a model of four-dimensional simplicial quantum gravity to include degenerate triangulations in addition to combinatorial triangulations traditionally used. Relaxing the constraint that every 4-simplex is uniquely defined by a set…
Ballroom dancing is a structured yet expressive motion category. Its highly diverse movement and complex interactions between leader and follower dancers make the understanding and synthesis challenging. We demonstrate that the three-point…
We present medial parametrization, a new approach to parameterizing any compact planar domain bounded by simple closed curves. The basic premise behind our proposed approach is to use two close Voronoi sites, which we call dipoles, to…
We develop a new approach to vector quantization, which guarantees an intrinsic stationarity property that also holds, in contrast to regular quantization, for non-optimal quantization grids. This goal is achieved by replacing the usual…
We derive a combined analytical and numerical scheme to solve the (1+1)-dimensional differential Kirchhoff system. Here the object is to obtain an accurate as well as an efficient solution process. Purely numerical algorithms typically have…
We present a general framework for computing two-dimensional Voronoi diagrams of different classes of sites under various distance functions. The framework is sufficiently general to support diagrams embedded on a family of two-dimensional…
We describe an efficient algorithm to compute a pseudotriangulation of a finite planar family of pairwise disjoint convex bodies presented by its chirotope. The design of the algorithm relies on a deepening of the theory of visibility…
We consider defining the embedding of a triangle mesh into $R^3$, up to translation, rotation, and scale, by its vector of dihedral angles. Theoretically, we show that locally, almost everywhere, the map from realizable vectors of dihedrals…
Dense 3D visual mapping estimates as many as possible pixel depths, for each image. This results in very dense point clouds that often contain redundant and noisy information, especially for surfaces that are roughly planar, for instance,…
We develop a discrete-time version of the blended dynamics theorem for the use of designing distributed computation algorithms. The blended dynamics theorem enables to predict the behavior of heterogeneous multi-agent systems. Therefore,…