Convex Hulls under Uncertainty

TL;DR

This paper explores the problem of computing convex hulls in the presence of data uncertainty, proposing algorithms and new concepts to handle probabilistic input data in various applications.

Contribution

It introduces both exact and approximation algorithms for probabilistic convex hulls, and proposes a novel $ ext{some}$-hull concept for uncertain data representation.

Findings

Developed algorithms for probability of point inclusion in uncertain convex hulls

Established time-space tradeoffs for membership queries

Connected Tukey depth with membership query techniques

Abstract

We study the convex-hull problem in a probabilistic setting, motivated by the need to handle data uncertainty inherent in many applications, including sensor databases, location-based services and computer vision. In our framework, the uncertainty of each input site is described by a probability distribution over a finite number of possible locations including a \emph{null} location to account for non-existence of the point. Our results include both exact and approximation algorithms for computing the probability of a query point lying inside the convex hull of the input, time-space tradeoffs for the membership queries, a connection between Tukey depth and membership queries, as well as a new notion of $\some$-hull that may be a useful representation of uncertain hulls.