Convex Hull

The problem addressed is source localization from time differences of arrival (TDOA). This problem is also referred to as hyperbolic localization and it is non-convex in general. Traditional solutions proposed in the literature have generally poor robustness to errors in the TDOA estimates. More recent methods, which relax the non-convex problem to a convex optimization by applying a semi-definite relaxation (SDR) method, were found to be more robust to TDOA errors than the traditional methods. However, the SDR methods are not optimal in general. In this paper, three convex optimization methods with different computational costs are proposed to improve the hyperbolic localization accuracy. The first method takes an SDR approach to relax the hyperbolic localization to a convex optimization. The second method follows a linearized formulation of the problem and seeks for a biased estimate of improved accuracy. The first two methods perform comparably when the source is inside the convex hull of the sensors. When the source is located outside, the second approach performs better, at the cost of higher computation. A third method is proposed by exploiting the source sparsity. With this, the hyperbolic localization is formulated as an ℓ1-regularization problem, where the ℓ1-norm is used as source sparsity constraint. Computer simulations show that the ℓ1-regularization can offer further improved accuracy, but at the cost of additional computational effort.

We present a framework for multi-core implementations of divide and conquer algorithms and show its efficiency and ease of use by applying it to the fundamental geo- metric problem of computing the convex hull of a point set. We concentrate on the Quickhull algorithm intro- duced in (2). In general the framework can easily be used for any D&C-algorithm. It

In this paper we introduce the notion of approximate da2a siruclures, in which a small amount of error is tolerated in the output. Approximate data structures trade error of approximation for faster operation, leading to theoretical and practical speedups for a wide variety of algorithms. We give approximate variants of the van Emde Boas data structure, which support the same dynamic operations as the standard van Emde Boas data structure [28, 201, except that answers to queries are approximate. The variants support all operations in constant time provided the error of approximation is l/polylog(n), and in O(loglog n) time provided the error is l/polynomial(n), for n elements in the data structure. We consider the tolerance of prototypical algorithms to approximate data structures. We study in particular Prim’s minimumspanning tree algorithm, Dijkstra’s single-source shortest paths algorithm, and an on-line variant of Graham’s convex hull algorithm. To obtain output which approximat...

Several important factors must be taken into consideration in order to maximize the efficiency of rapid prototyping (RP) processes. The ability to select the optimal orientation of the build direction is one of the most critical factors in using RP processes, since it affects part quality, build time, and part cost. This study aims to determine the optimal build-up direction when a part is built with the variable layer thickness for different RP systems. The average weighted surface roughness (AWSR) that is generated from the stair stepping effect, the build time, and the part cost using the variable layer thickness are all considered in the process. Using the multi-attribute decision-making method, the best orientation is determined among the orientation candidates chosen from the convex hull of a model. The validity of the algorithm is illustrated by an example. The algorithm can help RP users select the best build-up direction of the part and create an efficient process planning.

In this paper, a new algorithm to find convex hull is presented which keeps the points into a form of nested binary tree. There are number of methods available for finding the convex hull but all of them are time consuming because they are based on comparing distances between the points and rotation of angles, whereas the proposed technique is based on binary tree which is efficient and more accurate than the existing algorithms as it decreases the computational cost and finds precise boundary of the object. Another main advantage of the proposed technique is the efficient memory usage because of binary tree structure.

Notice of Violation of IEEE Publication Principles"Towards a Unified Framework for Complexity Measurement in Aspect-Oriented Systems,"by A. Kumar, R. Kumar, P.S. Groverin the Proceedings of the 2008 International Conference on Computer Science and Software Engineering, vol.2, pp.98-103, Dec. 2008After careful and considered review of the content and authorship of this paper by a duly constituted expert committee, this paper has been found to be in violation of IEEE's Publication Principles.This paper contains significant portions of original text from the paper cited below. The original text was copied without attribution (including appropriate references to the original author(s) and/or paper title) and without permission.Due to the nature of this violation, reasonable effort should be made to remove all past references to this paper, and future references should be made to the following article:"Towards a Unified Coupling Framework for Measuring Aspect-oriented Programs"by T.T. Bartolomei, A. Garcia, C. Sant'Anna, C., E. Figueiredoin the Proceedings of the 3rd International Workshop on Software Quality Assurance (Portland, Oregon, November 06 - 06, 2006). SOQUA '06. ACMAspect-oriented programming (AOP) is an emerging technique that provides a mechanism to clearly encapsulate and implement concerns that crosscut other modules. It is claimed that this technique improves code modularization and therefore reduce complexity of object-oriented programs (OOP). Most of the proposed complexity measurement frameworks and metrics for AOP are for AspectJ programming language. In this paper we have defined a generic complexity measurement framework that takes into account three, the most well known families of available AOP languages, AspectJ, CaesarJ and Hyper/J. The proposed unified framework contributes to better understanding of complexity in AOP, which in turn help to (i) define new complexity metrics whic- h permit the analysis and comparison of Java, AspectJ, CaesarJ and Hyper/J implementations, and (ii) integrating different existing measures and examine same concepts in different ways.

Let V be a flnite set and M a flnite collection of subsets of V. Then M is an alignment of V if and only if M is closed under taking intersections and contains both V and the empty set. If M is an alignment of V , then the elements of M are called convex sets and the pair (V;M) is called an aligned space or a convexity space. If S µ V , then the convex hull of S, denoted by CH(S), is the smallest convex set that contains S. Suppose X 2 M. Then x 2 X is an extreme point for X if X ¡ fxg 2 M. A convex geometry on a flnite set is an aligned space with the additional property that every convex set is the convex hull of its extreme points. This property is referred to as the Minkowski-Krein-Milman property. The distance between a pair of vertices u;v of G is the number of edges in a u ¡ v geodesic (i.e., a shortest u ¡ v path) in G and is denoted by dG(u;v) or d(u;v). The interval between a pair u;v of vertices in a graph G, denoted by IG(u;v) or I(u;v), is the collection of all vertices...

Convexity predicates and the convex hull operator continue to play an important role in theories of spatial representation and reasoning, yet their first-order axiomatization is still a matter of controversy. In this paper, we present a new approach to adding convexity to mereotopological theory with boundary elements by specifying first-order axioms for a binary segment operator s. We show that our axioms yields a convex hull operator h that supports, not only the basic properties of convex regions, but also complex properties concerning region alignment. We also argue that h is stronger than convex hull operators from existing axiomatizations and show how to derive the latter from our axioms for s.

If two sets are linearly separable (LS), then there exists a single layer perceptron feedforward neural network that classifies them. We propose three methods for testing linear separability. The first method is based on the notion of convex hull, the second on a halting upper bound for the perceptron learning algorithm, and the third one, called the class of linear

The solution of the Chinese postman problem using matching theory is given. The convex hull of integer solutions is described as a linear programming polyhedron. This polyhedron is used to show that a good algorithm gives an optimum solution. The algorithm is a specialization of the more generalb-matching blossom algorithm. Algorithms for finding Euler tours and related problems are also discussed.