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Convex Hull for Probabilistic Points F Bryn Mawr College Scholarship, Research, and Creative Work at Bryn Mawr College Computer Science Faculty Research and Computer Science Scholarship 2016 Convex Hull for Probabilistic Points F. Betul Atalay Sorelle A. Friedler Dianna Xu Bryn Mawr College, [email protected] Let us know how access to this document benefits ouy . Follow this and additional works at: http://repository.brynmawr.edu/compsci_pubs Part of the Computer Sciences Commons Custom Citation F. B. Atalay, S. A. Friedler and D. Xu, "Convex Hull for Probabilistic Points," 2016 29th SIBGRAPI Conference on Graphics, Patterns and Images (SIBGRAPI), Sao Paulo, 2016, pp. 48-55. This paper is posted at Scholarship, Research, and Creative Work at Bryn Mawr College. http://repository.brynmawr.edu/compsci_pubs/62 For more information, please contact [email protected]. 2016 29th SIBGRAPI Conference on Graphics, Patterns and Images Convex Hull for Probabilistic Points F. Betul Atalay Sorelle A. Friedler Dianna Xu Dept. of Computer Engineering Dept. of Computer Science Dept. of Computer Science TOBB University of Economics Haverford College Bryn Mawr College and Technology 370 W. Lancaster Avenue 101 N. Merion Avenue Sogutozu, Ankara, Turkey Haverford, PA 19041, USA Bryn Mawr, PA 19010, USA [email protected] [email protected] [email protected] Abstract—We analyze the correctness of an O(n log n) time convex hull without requiring extensive modification to existing divide-and-conquer algorithm for the convex hull problem when algorithms. each input point is a location determined by a normal dis- We will show that the divide-and-conquer convex hull tribution. We show that the algorithm finds the convex hull of such probabilistic points to precision within some expected algorithm still produces an approximately correct convex hull correctness determined by a user-given confidence value φ.In even when its input point locations aren’t known exactly. This order to precisely explain how correct the resulting structure will require some modifications to the algorithm as well as an is, we introduce a new certificate error model for calculating introduction of a new error model in order to define what we and understanding approximate geometric error based on the mean by an approximately correct convex hull. We will build fundamental properties of a geometric structure. We show that this new error model implies correctness under a robust statistical this new approximate notion on boolean functions that certify error model, in which each point lies within the hull with geometric properties necessary to a correct calculation of the probability at least φ, for the convex hull problem. convex hull. These functions are borrowed from the study of kinetic data structures [5], and so some of this work will find application to other problems studied within that boolean I. INTRODUCTION certification framework (as well as allowing future work to The Convex Hull Problem is the problem of determining a extend these results to hold on moving points). A careful minimum convex bounding polygon that covers n points in analysis will show how potential errors in these certifications the Euclidean plane. propagate to the overall structure being calculated. The convex hull will be approximate in the sense that only a given percent of the points will be expected to lie within it. This matches the desires of some applications - for example, when determining the home range of an animal from (noisy) location observations, the goal is to compute a boundary containing some percentage of such observations [6]. A. Related Work Fig. 1. A point set and its convex hull Approximate correctness of a geometric structure has been considered under a number of different models, including This is a classic problem in computational geometry, with interpretations where the structure is considered to be fully well known solutions including Graham’s scan and divide-and- correct some percentage of time or where it is considered to be conquer (both take O(n log n) time) [8], [15]. The convex hull partially correct every time the algorithm is run. We are most is a fundamental primitive for many graphics problems, such as interested in this second interpretation, within which partial calculation of basic shape representations (e.g., bounding boxes) correctness has been considered under the absolute error model [17] and collision detection [13]. In application domains, point [7], the relative error model [3], and the robust error model [16]. locations are often the result of a machine learning algorithm Within the absolute error model a structure is considered to be that outputs a probability distribution for each point’s location correct up to some given fixed error bound ε that is constant (for a survey, see [11]). For example, in augmented reality, the for any set of points [7]. Under the relative error model a markerless tracking problem that aims to track the position and structure is considered to be correct up to some percentage orientation of a camera in a scene without using markers may based on the geometric structure [3]. The robust error model take a hybrid approach that relies on both computer vision is a per-point error model under which a structure is correct techniques and probabilistic GPS location information of the based on the percentage of points which are correct [16]. We type generated by such machine learning algorithms [18]. In will compare the error model we introduce to the robust error this paper, we are interested in examining what happens when model. the expected values of such probabilistic points are given as While classical computational geometry assumes exact input to the divide-and-conquer convex hull algorithm, with the knowledge of point location, goals of relaxing such assumptions goal of guaranteeing approximate correctness of the resulting have spurred several recent papers. Loeffler and Kreveld [14] 2377-5416/16 $31.00 © 2016 IEEE 48 DOI 10.1109/SIBGRAPI.2016.15 have considered approximate convex hulls under an imprecise point setting, where exact point location is unknown within a region but guaranteed not to be outside of it. They consider the convex hull under multiple variants of the relative error model and achieve running times that range from O(n log n) to O(n13). When considering approximate nearest neighbor searching, a model where points are described as probability distributions over their possible locations has also been con- sidered [1]. This latter model of point location, commonly used in application domains, is the same as the one we use here (and is described in more detail in Section II-A). The convex hull problem has been considered within the discrete version of this point location model (where the distributions are discrete) by Agarwal et al. [2]. Their results give a running 3 time of O(m log m), where m is the number of possible point locations in their discrete distributions. The robust error model in Argarwal et al implies the one we compute. While we solve a weaker version of the problem, we improve the running time O n n n to ( log ), where is the number of points. Additionally, Fig. 2. A Google Maps screenshot showing a probabilistic point pj under a ours is the first solution to hold on continuous distributions. normal distribution Dj where the central blue dot is vj and the lighter blue β (φ, D ) We use an O(n log n) divide-and-conquer algorithm to circle is its associated j j . compute the convex hull on a set of probabilistic points under normal distributions. Our solution is approximately correct d and vj ∈ R is an expected value for the point pj given under a robust error model with the correctness taken in distribution Nj. We are given a set P of n probabilistic points. expectation over all possible point locations, so that each point d pdf Dj = {x ∈ R |N (x) > 0} is the positive region of the has at least φ probability of being in the hull, for a parameter j N pdf Rd →{y ∈ R|y ≥ } φ. Ours is the first solution to hold for probabilistic points probability density function j : 0 .We D β φ, D ⊂ Rd with a continuous location probability distribution. assume that the region j is bounded. Let j( j) be p We achieve these results not by introducing a new algorithm, the boundary region of point j defined as the minimum-area p but by introducing a new error model and associated analysis convex set such that j is within the region with probability φ N pdf x dx φ φ ∈ , of the standard divide-and-conquer algorithm for calculating , i.e., j ( ) = . [0 1] is a user-given x∈βj (φ,Dj ) the convex hull via its upper envelope in the dual space [15]. confidence value, and Φ = 100 · φ is φ in percent form. We We introduce a certificate error model in which a structure assume that βj(φ, Dj) can be calculated in O(1) time. For is considered φ-correct if each Boolean certificate used to example, Figure 2 shows βj(φ, Dj) as the truncated Gaussian. φ calculate the structure is correct with probability at least .We For the remainder of this paper we will refer to these will show that approximate correctness under the certificate probabilistic points pj =(Dj,vj) simply as points. Within error model implies approximate correctness under the robust a machine learning context, these points would be generated error model for the convex hull. by a model M(j, E) → pj that, when given a point identity B. Contributions j and environmental data E, would return the probabilistic The rest of this paper shows the following results: point pj. More details about such models can be found in 1) We introduce a certificate error model guaranteeing that a survey of location models [11].
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