Optimal Mesh Algorithms for the Voronoi Diagram of Line Segments, Visibility Graphs and Motion Planning in the Plane

The motion planning problem for an object with two degrees of freedom moving in the plane can be stated as follows: Given a set of polygonal obstacles in the plane, and a two-dimensional mobile object B with two degrees of freedom, determine if it is possible to move B from a start position to a final position while avoiding the obstacles. If so, plan a path for such a motion. Techniques from computational geometry have been used to develop exact algorithms for this fundamental case of motion planning. In this paper we obtain optimal mesh implementations of two different methods for planning motion in the plane. We do this by first presenting optimal mesh algorithms for some geometric problems that, in addition to being important substeps in motion planning, have numerous independent applications in computational geometry. In particular, we first show that the Voronoi diagram of a set of n nonintersecting (except possibly at endpoints) line segments in the plane can be constructed in O(√n) time on a √n x √n mesh, which is optimal for the mesh. Consequently, we obtain an optimal mesh implementation of the sequential motion planning algorithm described in [14]; in other words, given a disc B and a polygonal obstacle set of size n, we can plan a path (if it exists) for the motion of B from a start position to a final position in O (√n) time on a mesh of size n. Next we show that given a set of n line segments and a point p, the set of segment endpoints that are visible from p can be computed in O (√n) mesh-optimal time on a √n x √n mesh. As a result, the visibility graph of a set of n line segments can be computed in O(n) time on an n x n mesh. This result leads to an O(n) algorithm on an n x n mesh for planning the shortest path motion between a start position and a final position for a convex object B (of constant size) moving among convex polygonal obstacles of total size n

Algorithmic Motion Planning and Related Geometric Problems on Parallel Machines (Dissertation Proposal)

The problem of algorithmic motion planning is one that has received considerable attention in recent years. The automatic planning of motion for a mobile object moving amongst obstacles is a fundamentally important problem with numerous applications in computer graphics and robotics. Numerous approximate techniques (AI-based, heuristics-based, potential field methods, for example) for motion planning have long been in existence, and have resulted in the design of experimental systems that work reasonably well under various special conditions [7, 29, 30]. Our interest in this problem, however, is in the use of algorithmic techniques for motion planning, with provable worst case performance guarantees. The study of algorithmic motion planning has been spurred by recent research that has established the mathematical depth of motion planning. Classical geometry, algebra, algebraic geometry and combinatorics are some of the fields of mathematics that have been used to prove various results that have provided better insight into the issues involved in motion planning [49]. In particular, the design and analysis of geometric algorithms has proved to be very useful for numerous important special cases. In the remainder of this proposal we will substitute the more precise term of algorithmic motion planning by just motion planning

Geometry Processing of Conventionally Produced Mouse Brain Slice Images

Brain mapping research in most neuroanatomical laboratories relies on conventional processing techniques, which often introduce histological artifacts such as tissue tears and tissue loss. In this paper we present techniques and algorithms for automatic registration and 3D reconstruction of conventionally produced mouse brain slices in a standardized atlas space. This is achieved first by constructing a virtual 3D mouse brain model from annotated slices of Allen Reference Atlas (ARA). Virtual re-slicing of the reconstructed model generates ARA-based slice images corresponding to the microscopic images of histological brain sections. These image pairs are aligned using a geometric approach through contour images. Histological artifacts in the microscopic images are detected and removed using Constrained Delaunay Triangulation before performing global alignment. Finally, non-linear registration is performed by solving Laplace's equation with Dirichlet boundary conditions. Our methods provide significant improvements over previously reported registration techniques for the tested slices in 3D space, especially on slices with significant histological artifacts. Further, as an application we count the number of neurons in various anatomical regions using a dataset of 51 microscopic slices from a single mouse brain. This work represents a significant contribution to this subfield of neuroscience as it provides tools to neuroanatomist for analyzing and processing histological data.Comment: 14 pages, 11 figure

Dense point sets have sparse Delaunay triangulations

The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of n points in R^3 with spread D has complexity O(D^3). This bound is tight in the worst case for all D = O(sqrt{n}). In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to regular triangulations of k-ply systems of balls, unions of several dense point sets, and uniform samples of smooth surfaces. On the other hand, for any n and D=O(n), we construct a regular triangulation of complexity Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm