Constructing Delaunay triangulations along space-filling curves

Incremental construction con BRIO using a space-filling curve order for insertion is a popular algorithm for constructing Delaunay triangulations. So far, it has only been analyzed for the case that a worst-case optimal point location data structure is used which is often avoided in implementations. In this paper, we analyze its running time for the more typical case that points are located by walking. We show that in the worst-case the algorithm needs quadratic time, but that this can only happen in degenerate cases. We show that the algorithm runs in O(n logn) time under realistic assumptions. Furthermore, we show that it runs in expected linear time for many random point distributions. This research was supported by the Deutsche Forschungsgemeinschaft within the European graduate program ’Combinatorics, Geometry, and Computation’ (No. GRK 588/2) and by the Netherlands’ Organisation for Scientific Research (NWO) under BRICKS/FOCUS grant number 642.065.503 and project no. 639.022.707

Sparse representation of video data by adaptive tetrahedralizations.

Natural videos are composed of a superposition of moving objects, usually resulting from anisotropic motions into different directions. By discretization with respect to time, a video may be regarded as a sequence of consecutive natural still images. Alternatively, when considering time as one dimension, a video may be viewed as a 3d scalar field. In this case, customized methods are needed for capturing both the evolution of moving contours along the time axis and the geometrical distortions of the resulting sweep surfaces. Moreover, it is desirable to work with sparse representations. Indeed, already for basic motions (e.g. rotations, translations), customized methods for the construction of well-adapted sparse video data representations are required. To this end, we propose a novel adaptive approximation algorithm for video data. The utilized nonlinear approximation scheme is based on anisotropic tetrahedralizations of the 3d video domain, whose tetrahedra are adapted locally in space (for contour-like singularities) and locally in time (for anisotropic motions). The key ingredients of our approximation method, 3AT, are adaptive thinning, a recursive pixel removal scheme, and least squares approximation by linear splines over anisotropic tetrahedralizations. The approximation algorithm 3AT yields a new concept for the compression of video data. We apply the proposed approximation method first to prototypical geometrical motions, before numerical simulations concerning one natural video are presented