Trinocular stereovision using figural continuity, dealing with curved objects

A method to build a dense and reliable 3-D description of a scene from three digital images by means of passive stereovision is presented. This method uses figural continuity to improve the results of a previously developed algorithm. In particular, it copes much better with curved objects and produces results which are organized as 3-D chains of segments

Inner and Outer Rounding of Boolean Operations on Lattice Polygonal Regions

Robustness problems due to the substitution of the exact computation on real numbers by the rounded floating point arithmetic are often an obstacle to obtain practical implementation of geometric algorithms. If the adoption of the --exact computation paradigm--[Yap et Dube] gives a satisfactory solution to this kind of problems for purely combinatorial algorithms, this solution does not allow to solve in practice the case of algorithms that cascade the construction of new geometric objects. In this report, we consider the problem of rounding the intersection of two polygonal regions onto the integer lattice with inclusion properties. Namely, given two polygonal regions A and B having their vertices on the integer lattice, the inner and outer rounding modes construct two polygonal regions with integer vertices which respectively is included and contains the true intersection. We also prove interesting results on the Hausdorff distance, the size and the convexity of these polygonal regions

Balanced partitions of 3-colored geometric sets in the plane

Let SS be a finite set of geometric objects partitioned into classes or colors . A subset S'¿SS'¿S is said to be balanced if S'S' contains the same amount of elements of SS from each of the colors. We study several problems on partitioning 33-colored sets of points and lines in the plane into two balanced subsets: (a) We prove that for every 3-colored arrangement of lines there exists a segment that intersects exactly one line of each color, and that when there are 2m2m lines of each color, there is a segment intercepting mm lines of each color. (b) Given nn red points, nn blue points and nn green points on any closed Jordan curve ¿¿, we show that for every integer kk with 0=k=n0=k=n there is a pair of disjoint intervals on ¿¿ whose union contains exactly kk points of each color. (c) Given a set SS of nn red points, nn blue points and nn green points in the integer lattice satisfying certain constraints, there exist two rays with common apex, one vertical and one horizontal, whose union splits the plane into two regions, each one containing a balanced subset of SS.Peer ReviewedPostprint (published version

Shortest Path in a Polygon using Sublinear Space

\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)} We resolve an open problem due to Tetsuo Asano, showing how to compute the shortest path in a polygon, given in a read only memory, using sublinear space and subquadratic time. Specifically, given a simple polygon \Polygon with $n$ vertices in a read only memory, and additional working memory of size \Space, the new algorithm computes the shortest path (in \Polygon) in O( n^2 /\, \Space ) expected time. This requires several new tools, which we believe to be of independent interest

On the complexity of range searching among curves

Modern tracking technology has made the collection of large numbers of densely sampled trajectories of moving objects widely available. We consider a fundamental problem encountered when analysing such data: Given $n$ polygonal curves $S$ in $\mathbb{R}^d$, preprocess $S$ into a data structure that answers queries with a query curve $q$ and radius $\rho$ for the curves of $S$ that have \Frechet distance at most $\rho$ to $q$. We initiate a comprehensive analysis of the space/query-time trade-off for this data structuring problem. Our lower bounds imply that any data structure in the pointer model model that achieves $Q(n) + O(k)$ query time, where $k$ is the output size, has to use roughly $\Omega\left((n/Q(n))^2\right)$ space in the worst case, even if queries are mere points (for the discrete \Frechet distance) or line segments (for the continuous \Frechet distance). More importantly, we show that more complex queries and input curves lead to additional logarithmic factors in the lower bound. Roughly speaking, the number of logarithmic factors added is linear in the number of edges added to the query and input curve complexity. This means that the space/query time trade-off worsens by an exponential factor of input and query complexity. This behaviour addresses an open question in the range searching literature: whether it is possible to avoid the additional logarithmic factors in the space and query time of a multilevel partition tree. We answer this question negatively. On the positive side, we show we can build data structures for the \Frechet distance by using semialgebraic range searching. Our solution for the discrete \Frechet distance is in line with the lower bound, as the number of levels in the data structure is $O(t)$, where $t$ denotes the maximal number of vertices of a curve. For the continuous \Frechet distance, the number of levels increases to $O(t^2)$

Unbounded Orbits for Outer Billiards

Outer billiards is a basic dynamical system, defined relative to a planar convex shape. This system was introduced in the 1950's by B.H. Neumann and later popularized in the 1970's by J. Moser. All along, one of the central questions has been: is there an outer billiards system with an unbounded orbit. We answer this question by proving that outer billiards defined relative to the Penrose Kite has an unbounded orbit. The Penrose kite is the quadrilateral that appears in the famous Penrose tiling. We also analyze some of the finer orbit structure of outer billiards on the penrose kite. This analysis shows that there is an uncountable set of unbounded orbits. Our method of proof relates the problem to self-similar tilings, polygon exchange maps, and arithmetic dynamics.Comment: 65 pages, computer-aided proof. Auxilliary program, Billiard King, available from author's website. Latest version is essentially the same as earlier versions, but with minor improvements and many typos fixe