Approximating the Maximum Overlap of Polygons under Translation

Let $P$ and $Q$ be two simple polygons in the plane of total complexity $n$, each of which can be decomposed into at most $k$ convex parts. We present an $(1-\varepsilon)$-approximation algorithm, for finding the translation of $Q$, which maximizes its area of overlap with $P$. Our algorithm runs in $O(c n)$ time, where $c$ is a constant that depends only on $k$ and $\varepsilon$. This suggest that for polygons that are "close" to being convex, the problem can be solved (approximately), in near linear time

Probabilistic Matching of Planar Regions

We analyze a probabilistic algorithm for matching shapes modeled by planar regions under translations and rigid motions (rotation and translation). Given shapes $A$ and $B$, the algorithm computes a transformation $t$ such that with high probability the area of overlap of $t(A)$ and $B$ is close to maximal. In the case of polygons, we give a time bound that does not depend significantly on the number of vertices

matching, interpolation, and approximation ; a survey

In this survey we consider geometric techniques which have been used to measure the similarity or distance between shapes, as well as to approximate shapes, or interpolate between shapes. Shape is a modality which plays a key role in many disciplines, ranging from computer vision to molecular biology. We focus on algorithmic techniques based on computational geometry that have been developed for shape matching, simplification, and morphing

Geometric Optimization Problem Solving: Matching Sets of Line Segments and Multi-robot Path Planning

Department of Computer Science and EngineeringWe study two geometric optimization problems: Line segments pattern matching and multi-robot path planning. We give approximation algorithms for matching two sets of line segments in constant dimension. We consider several versions of the problem: Hausdorff distance, bottleneck distance and largest common subset. We study these similarity measures under several sets of transformations: translations in arbitrary dimension, rotations about a fixed point and rigid motions in two dimensions. As opposed to previous theoretical work on this problem, we match segments individually, in other words we regard our two input sets as sets of segments rather than unions of segments. Then we consider a multi-robot path planning problem. A collection of square robots need to move on the integer grid, from their given starting points to their target points, and without collision between robots, or between robots and a set of input obstacles. We designed and implemented three algorithms for this problem. First, we computed a feasible solution by placing middle-points outside of the minimum bounding box of the starting positions, the target positions and the obstacles, and moving each robot from its starting point to its target point through a middle-point. Second, we applied a simple local search approach where we repeatedly delete and insert again a random robot through an optimal path. It improves the quality of the solution, as the robots no longer need to go through the middle-points. Finally, we used simulated annealing to further improve this feasible solution.ope