No abstract

Let P be a polygonal curve in R d of length n, and S be a point-set of size k. The Curve/Point Set Matching problem consists of finding a polygonal curve Q on S such that the Fréchet distance from P is less than a given ε. We consider eight variations of the problem based on the distance metric used and the omittability or repeatability of the points. We provide closure to a recent series of complexity results for the case where S consists of precise points. More importantly, we formulate a more realistic version of the problem that takes into account measurement errors. This new problem is posed as the matching of a given curve to a set of imprecise points. We prove that all three variations of the problem that are in P when S consists of precise points become NP-complete when S consists of imprecise points. We also discuss approximation results.

Smart pixels" are sensor pixels which have been imbued with a limited amount of processing power. Consider a model of computation in which a grid of these smart pixels can perform simple arithmetic, store a small amount of data, and communicate with their direct neighbors. In this video, we review recent algorithms for well-known geometric problems developed within such a model. Specifically, given a black-and-white image on a w by h grid of smart pixels, we show how the sizes, orientations, centers of gravity, and convex hulls of multiple objects on the grid can be computed simultaneously in time O(w + h).