Preconditioning 2D Integer Data for Fast Convex Hull Computations

In order to accelerate computing the convex hull on a set of n points, a heuristic procedure is often applied to reduce the number of points to a set of s points, s ≤ n, which also contains the same hull. We present an algorithm to precondition 2D data with integer coordinates bounded by a box of size p × q before building a 2D convex hull, with three distinct advantages. First, we prove that under the condition min(p, q) ≤ n the algorithm executes in time within O(n); second, no explicit sorting of data is required; and third, the reduced set of s points forms a simple polygonal chain and thus can be directly pipelined into an O(n) time convex hull algorithm. This paper empirically evaluates and quantifies the speed up gained by preconditioning a set of points by a method based on the proposed algorithm before using common convex hull algorithms to build the final hull. A speedup factor of at least four is consistently found from experiments on various datasets when the condition min(p, q) ≤ n holds; the smaller the ratio min(p, q)/n is in the dataset, the greater the speedup factor achieved.

A new blind equalization algorithm based on convex combination

The convex combination is a mathematic approach to keep the advantages of its component algorithms for better performance. In this paper, we employ convex combination in the blind equalization to achieve better blind equalization. By combining the blind constant modulus algorithm (CMA) and decision directed algorithm, the combinative blind equalization (CBE) algorithm can retain the advantages from both. Furthermore, the convergence speed of the CBE algorithm is faster than both of its component equalizers. Simulation results are also given to verify the proposed algorithm.

Preconditioning 2D integer data for fast convex hull computations

In order to accelerate computing the convex hull on a set of n points, a heuristic procedure is often applied to reduce the number of points to a set of s points, s ≤ n, which also contains the same hull. We present an algorithm to precondition 2D data with integer coordinates bounded by a box of size p × q before building a 2D convex hull, with three distinct advantages. First, we prove that under the condition min(p, q) ≤ n the algorithm executes in time within O(n); second, no explicit sorting of data is required; and third, the reduced set of s points forms a simple polygonal chain and thus can be directly pipelined into an O(n) time convex hull algorithm. This paper empirically evaluates and quantifies the speed up gained by preconditioning a set of points by a method based on the proposed algorithm before using common convex hull algorithms to build the final hull. A speedup factor of at least four is consistently found from experiments on various datasets when the condition min(p, q) ≤ n holds; the smaller the ratio min(p, q)/n is in the dataset, the greater the speedup factor achieved.

Orthogonal packing of identical rectangles within isotropic convex regions

A mixed integer continuous nonlinear model and a solution method for the problem of orthogonally packing identical rectangles within an arbitrary convex region are introduced in the present work. The convex region is assumed to be made of an isotropic material in such a way that arbitrary rotations of the items, preserving the orthogonality constraint, are allowed. The solution method is based on a combination of branch and bound and active-set strategies for bound-constrained minimization of smooth functions. Numerical results show the reliability of the presented approach. (C) 2010 Elsevier Ltd. All rights reserved.

Fuzzy clustering of non-convex patterns using global optimization

This paper discusses various extensions of the classical within-group sum of squared errors functional, routinely used as the clustering criterion. Fuzzy c-means algorithm is extended to the case when clusters have irregular shapes, by representing the clusters with more than one prototype. The resulting minimization problem is non-convex and non-smooth. A recently developed cutting angle method of global optimization is applied to this difficult problem

Convex hulls as an hypothesis language bias

Classification learning is dominated by systems which induce large numbers of small axis-orthogonal decision surfaces which biases such systems towards particular hypothesis types. However, there is reason to believe that many domains have underlying concepts which do not involve axis orthogonal surfaces. Further, the multiplicity of small decision regions mitigates against any holistic appreciation of the theories produced by these systems, notwithstanding the fact that many of the small regions are individually comprehensible. We propose the use of less strongly biased hypothesis languages which might be expected to model' concepts using a number of structures close to the number of actual structures in the domain. An instantiation of such a language, a convex hull based classifier, CHI, has been implemented to investigate modeling concepts as a small number of large geometric structures in n-dimensional space. A comparison of the number of regions induced is made against other well-known systems on a representative selection of largely or wholly continuous valued machine learning tasks. The convex hull system is shown to produce a number of induced regions about an order of magnitude less than well-known systems and very close to the number of actual concepts. This representation, as convex hulls, allows the possibility of extraction of higher level mathematical descriptions of the induced concepts, using the techniques of computational geometry.

Fuzzy c-means density based clustering using data induced metric

We propose a new data induced metric to perform un supervised data classification (clustering). Our goal is to automatically recognize clusters of non-convex shape. We present a new version of fuzzy c-means al gorithm, based on the data induced metric, which is capable to identify non-convex d-dimensional clusters.

Efficacy of linear and convex transducers for ultrasound-guided transvaginal follicle aspiration

A new device was developed to hold linear transducers for transvaginal follicle aspiration. Efficacy of follicle aspiration was compared using a linear 6 MHz and a convex 5 MHz transducer. Fifty-five cows were submitted to follicle aspiration at random days of the estrous cycle. Aspirations were conducted with linear (n = 28) and convex (n = 38) transducers with 18 G needles at a negative pressure corresponding to 13 ml H2O/min. A greater number of follicles were aspirated using convex than to linear probe (12.4 versus 7.8, respectively, P < 0.05). Mean number of oocytes and recovery rates were similar for convex (5.4 and 48.6%) and linear (4.6 and 59.3%) transducers. Limited space between the linear transducer and needle guide restricted access to some portions of the ovary, reducing the number of follicles aspirated using a linear transducer. The newly developed adaptor allowed greater stability, holding the ovaries firmly against the linear transducer. This diminished mobility permitted a similar number of oocytes to be recovered with both transducers. In conclusion, this new adaptor provided a low cost alternative for routine follicle aspiration and oocyte recovery in cattle. (C) 2002 Elsevier B.V. All rights reserved.

Projections Onto Convex Sets through Particle Swarm Optimization and its application for remote sensing image restoration

Image restoration attempts to enhance images corrupted by noise and blurring effects. Iterative approaches can better control the restoration algorithm in order to find a compromise of restoring high details in smoothed regions without increasing the noise. Techniques based on Projections Onto Convex Sets (POCS) have been extensively used in the context of image restoration by projecting the solution onto hyperspaces until some convergence criteria be reached. It is expected that an enhanced image can be obtained at the final of an unknown number of projections. The number of convex sets and its combinations allow designing several image restoration algorithms based on POCS. Here, we address two convex sets: Row-Action Projections (RAP) and Limited Amplitude (LA). Although RAP and LA have already been used in image restoration domain, the former has a relaxation parameter (A) that strongly depends on the characteristics of the image that will be restored, i.e., wrong values of A can lead to poorly restoration results. In this paper, we proposed a hybrid Particle Swarm Optimization (PS0)-POCS image restoration algorithm, in which the A value is obtained by PSO to be further used to restore images by POCS approach. Results showed that the proposed PSO-based restoration algorithm outperformed the widely used Wiener and Richardson-Lucy image restoration algorithms. (C) 2010 Elsevier B.V. All rights reserved.

On a regular convex solver potential for an elastic-damage constitutive model: a theoretical analysis

This work is related with the proposition of a so-called regular or convex solver potential to be used in numerical simulations involving a certain class of constitutive elastic-damage models. All the mathematical aspects involved are based on convex analysis, which is employed aiming a consistent variational formulation of the potential and its conjugate one. It is shown that the constitutive relations for the class of damage models here considered can be derived from the solver potentials by means of sub-differentials sets. The optimality conditions of the resulting minimisation problem represent in particular a linear complementarity problem. Finally, a simple example is present in order to illustrate the possible integration errors that can be generated when finite step analysis is performed. (C) 2003 Elsevier Ltd. All rights reserved.

Trade-off study on deorbiting S/C in near polar orbit

Usual long, flexible, ED tethers kept vertical by the gravity gradient might be less efficient for deorbiting S/C in near-polar orbits than conventional (Hall, Ion) electrical thrusters. A trade-off study on this application is here presented for tethers kept horizontal and perpendicular to the orbital plane. A tether thus oriented must be rigid and short for structural reasons, requiring a non-convex cross section and a power supply as in the case of electrical thrusters. Very recent developments on bare-tether collection theory allow predicting the current collected by an arbitrary cross section. For the horizontal tether, structural considerations on length play the role of ohmic effects in vertical tethers, in determining the optimal contribution of tether mass to the overall deorbiting system. For a given deorbiting-mission impulse, tether-system mass is minimal at some optimal length that increases weakly with the impulse. The horizontal-tether system may beat both the vertical tether and the electrical thruster as regards mass requirements for a narrow length range centered at about 100 m, allowing, however, for a broad mission-impulse range.

Planar point sets with large minimum convex decompositions

We show the existence of sets with n points (n ? 4) for which every convex decomposition contains more than (35/32)n?(3/2) polygons,which refutes the conjecture that for every set of n points there is a convex decomposition with at most n+C polygons. For sets having exactly three extreme pointswe show that more than n+sqr(2(n ? 3))?4 polygons may be necessary to form a convex decomposition.

Motzkin decomposition of closed convex sets via truncation

A nonempty set F is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set C with a closed convex cone D. In that case, the sets C and D are called compact and conic components of F. This paper provides new characterizations of the Motzkin decomposable sets involving truncations of F (i.e., intersections of FF with closed halfspaces), when F contains no lines, and truncations of the intersection F̂ of F with the orthogonal complement of the lineality of F, otherwise. In particular, it is shown that a nonempty closed convex set F is Motzkin decomposable if and only if there exists a hyperplane H parallel to the lineality of F such that one of the truncations of F̂ induced by H is compact whereas the other one is a union of closed halflines emanating from H. Thus, any Motzkin decomposable set F can be expressed as F=C+D, where the compact component C is a truncation of F̂. These Motzkin decompositions are said to be of type T when F contains no lines, i.e., when C is a truncation of F. The minimality of this type of decompositions is also discussed.

On the stability of the Motzkin representation of closed convex sets

A set is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set with a closed convex cone. This paper analyzes the continuity properties of the set-valued mapping associating to each couple (C,D) formed by a compact convex set C and a closed convex cone D its Minkowski sum C + D. The continuity properties of other related mappings are also analyzed.

Perceived importance of zones surrounding a vehicle and learning to use a convex mirror effectively. Final report.

National Highway Traffic Safety Administration, Washington, D.C.