Detection, tracking and event localization of jet stream features in 4-D atmospheric data

We introduce a novel algorithm for the efficient detection and tracking of features in spatiotemporal atmospheric data, as well as for the precise localization of the occurring genesis, lysis, merging and splitting events. The algorithm works on data given on a four-dimensional structured grid. Feature selection and clustering are based on adjustable local and global criteria, feature tracking is predominantly based on spatial overlaps of the feature's full volumes. The resulting 3-D features and the identified correspondences between features of consecutive time steps are represented as the nodes and edges of a directed acyclic graph, the event graph. Merging and splitting events appear in the event graph as nodes with multiple incoming or outgoing edges, respectively. The precise localization of the splitting events is based on a search for all grid points inside the initial 3-D feature that have a similar distance to two successive 3-D features of the next time step. The merging event is localized analogously, operating backward in time. As a first application of our method we present a climatology of upper-tropospheric jet streams and their events, based on four-dimensional wind speed data from European Centre for Medium-Range Weather Forecasts (ECMWF) analyses. We compare our results with a climatology from a previous study, investigate the statistical distribution of the merging and splitting events, and illustrate the meteorological significance of the jet splitting events with a case study. A brief outlook is given on additional potential applications of the 4-D data segmentation technique

Neural Deformable Cone Beam CT

In oral and maxillofacial cone beam computed tomography (CBCT), patient motion is frequently observed and, if not accounted for, can severely affect the usability of the acquired images. We propose a highly flexible, data driven motion correction and reconstruction method which combines neural inverse rendering in a CBCT setting with a neural deformation field. We jointly optimize a lightweight coordinate based representation of the 3D volume together with a deformation network. This allows our method to generate high quality results while accurately representing occurring patient movements, such as head movements, separate jaw movements or swallowing. We evaluate our method in synthetic and clinical scenarios and are able to produce artefact-free reconstructions even in the presence of severe motion. While our approach is primarily developed for maxillofacial applications, we do not restrict the deformation field to certain kinds of motion. We demonstrate its flexibility by applying it to other scenarios, such as 4D lung scans or industrial tomography settings, achieving state-of-the art results within minutes with only minimal adjustments

Computing a largest empty anchored cylinder, and related problems

Let $S$ be a set of $n$ points in $R^d$, and let each point $p$ of $S$ have a positive weight $w(p)$. We consider the problem of computing a ray $R$ emanating from the origin (resp.\ a line $l$ through the origin) such that $\min_{p\in S} w(p) \cdot d(p,R)$ (resp. $\min_{p\in S} w(p) \cdot d(p,l)$) is maximal. If all weights are one, this corresponds to computing a silo emanating from the origin (resp.\ a cylinder whose axis contains the origin) that does not contain any point of $S$ and whose radius is maximal. For $d=2$, we show how to solve these problems in $O(n \log n)$ time, which is optimal in the algebraic computation tree model. For $d=3$, we give algorithms that are based on the parametric search technique and run in $O(n \log^5 n)$ time. The previous best known algorithms for these three-dimensional problems had almost quadratic running time. In the final part of the paper, we consider some related problem

About the Algebraic Solutions of Smallest Enclosing Cylinders Problems

Given n points in Euclidean space E^d, we propose an algebraic algorithm to compute the best fitting (d-1)-cylinder. This algorithm computes the unknown direction of the axis of the cylinder. The location of the axis and the radius of the cylinder are deduced analytically from this direction. Special attention is paid to the case d=3 when n=4 and n=5. For the former, the minimal radius enclosing cylinder is computed algebrically from constrained minimization of a quartic form of the unknown direction of the axis. For the latter, an analytical condition of existence of the circumscribed cylinder is given, and the algorithm reduces to find the zeroes of an one unknown polynomial of degree at most 6. In both cases, the other parameters of the cylinder are deduced analytically. The minimal radius enclosing cylinder is computed analytically for the regular tetrahedron and for a trigonal bipyramids family with a symmetry axis of order 3.Comment: 13 pages, 0 figure; revised version submitted to publication (previous version is a copy of the original one of 2010

The Convex Hull of Ellipsoids (Video)

The treatment of curved algebraic surfaces becomes more and more the focus of attention in Computational Geometry. We present a video that illustrates the computation of the convex hull of a set of ellipsoids. The underlying algorithm is an application of our work on determining a cell in a 3-dimensional arrangement of quadrics, see \cite{ghs-ccaq-01}. In the video, the main emphasis is on a simple and comprehensible visualization of the geometric aspects of the algorithm. In addition, we give some insights into the underlying mathematical problems

Effects of a modular filter on geometric applications

We report on the effects of a filter based on modular arithmetic that has been introduced recently into the EXACUS library. Our experiments with planar arrangements for curves up to degree four show that the exact construction and comparison of real algebraic numbers are some of the most time consuming operations when solving intersection problems for curved objects. In our experiments the modular filter accelerated the computation for arrangements of cubic curves by a factor of about 6.

Sweeping Arrangements of Cubic Segments Exactly and Efficiently

A method is presented to compute the planar arrangement induced by segments of algebraic curves of degree three (or less), using an improved Bentley-Ottmann sweep-line algorithm. Our method is exact (it provides the mathematically correct result), complete (it handles all possible geometric degeneracies), and efficient (the implementation can handle hundreds of segments). The range of possible input segments comprises conic arcs and cubic splines as special cases of particular practical importance