Ordered Set Of Points Research Articles

Solving the multi-objective path planning problem for mobile robot using an improved NSGA-II algorithm

Path planning is one of the most concerned problems in mobile robotics. Given the NP-hard nature of path planning problems, the non-dominated sorting genetic algorithm (NSGA-II), as one of the outstanding evolutionary algorithms with robust optimization capabilities, is a good candidate to deal with them. In this study, an improved NSGA-II (INSGA-II) is proposed to solve the multi-objective path planning problems, in which path length, path safety and path smoothness are optimized simultaneously. First, a grid-based environment model is created, and each feasible solution is encoded as a discrete ordered point set. Second, a hybrid initialization strategy is employed to enrich the diversity of initial population. Third, besides the usual selection, crossover and mutation operators, problem-specific evolutionary operators are developed. In addition, an adaptive variable neighborhood local search strategy is designed to increase the exploitation ability of the algorithm. Moreover, a hybrid global search strategy is applied to further improve the algorithm’s the exploration ability. Finally, the proposed INSGA-II is compared with six state-of-the-art algorithms on 16 instances of four representative maps. Simulation results indicate that INSGA-II can solve the multi-objective path planning problems with high efficiency.

Multisensor Data Fusion by Means of Voxelization: Application to a Construction Element of Historic Heritage

Point clouds are very common tools used in the work of documenting historic heritage buildings. These clouds usually comprise millions of unrelated points and are not presented in an efficient data structure, making them complicated to use. Furthermore, point clouds do not contain topological or semantic information on the elements they represent. Added to these difficulties is the fact that a variety of different kinds of sensors and measurement methods are used in study and documentation work: photogrammetry, LIDAR, etc. Each point cloud must be fused and integrated so that decisions can be taken based on the total information supplied by all the sensors used. A system must be devised to represent the discrete set of points in order to organise, structure and fuse the point clouds. In this work we propose the concept of multispectral voxels to fuse the point clouds, thus integrating multisensor information in an efficient data structure, and applied it to the real case of a building element in an archaeological context. The use of multispectral voxels for the fusion of point clouds integrates all the multisensor information in their structure. This allows the use of very powerful algorithms such as automatic learning and machine learning to interpret the elements studied.

The task of determining the matching of actual and planned operation plans in a dispatch control system of the road transport

The article deals with the task of determining the matching between actual and planned operation plans in a dispatch control system of the road transport. The actual route of movement along the transport network is recorded as navigation data in the form of an ordered point set which had been passed through: control points identifiers and time stamps for both arrival and departure times. A new approach is proposed to determining the matching between planned and actual movement routes, which is considered as required for the dispatch control system of the road transport. Determining the matching between actual and planned movement routes is proposed to consider as the task of finding the longest common subsequence of two ordered sets: points of the planned and actual movement routes. In this way, the problem under consideration is scaled down to the classical dynamic programming problem — finding the longest common subsequence of several rows. Based on that the article presents a developed algorithm for solving the problem of finding the longest common subsequence of planned and actual movement route points.

Synthesis of a robust controller with reduced dimension by the Loop Shaping Design Procedure and decomposition based on Laguerre functions

In this paper, we present the synthesis of a robust controller for uncertain discrete systems. The synthesis method of such a robust controller is the generalization of the Loop Shaping Design Procedure (LSDP) approach of McFarlane and Glover in the discrete case based on the work of Gu et al. We exploit the bilinear transform known as Tustin’s method in order to formulate the discrete loop shaping technique. A discrete weighting filter and a shaped discrete plant result from this technique. By taking into account the coprime factor uncertainty representation for the resulting shaped plant and by applying the small gain theorem, we define the concept of the robust stabilization of the discrete LSDP approach. This concept is based on the resolution of an optimization problem characterized by the maximum stability margin for the synthesis of the robust controller. To calculate the robust controller we transform this problem to a standard robust [Formula: see text] controller design based on the resolution of the Riccati equations. Also, we present the gap metric theory to characterize the controller’s robustness. We note that the resulting final controller is the combination of the discrete weighting filter and the robust controller. We propose then to exploit the recent work of Bouzrara et al. in order to develop a reduced robust controller by expanding the final controller on two independent Laguerre orthonormal bases. The discrete LSDP and the reduced controller approaches were validated on a Continuous Stirred Tank Reactor chemical reactor for a set of different equilibrium points in order to take into account the nonlinearities.

Cutting convex curves

We show that for any two convex curves C1 and C2 in Rd parametrized by [0,1] with opposite orientations, there exists a hyperplane H with the following property: For any t∈[0,1] the points C1(t) and C2(t) are never in the same open half space bounded by H. This will be deduced from a more general result on equipartitions of ordered point sets by hyperplanes.

New Bounds on the Size of Optimal Meshes

AbstractThe theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents a way to bound the feature size integral in terms of an easy to compute property of a suitable ordering of the point set. The key idea is to consider the pacing of an ordered point set, a measure of the rate of change in the feature size as points are added one at a time. In previous work, Miller et al. showed that if an ordered point set has pacing ϕ, then the number of vertices in an optimal mesh will be O(ϕdn), where d is the input dimension. We give a new analysis of this integral showing that the output size is only θ(n+nlogϕ). The new analysis tightens bounds from several previous results and provides matching lower bounds. Moreover, it precisely characterizes inputs that yield outputs of size O(n).

Rational surfaces with a large group of automorphisms

We classify rational surfaces for which the image of the automorphisms group in the group of linear transformations of the Picard group is the largest possible. This answers a question raised by Arthur Coble in 1928, and can be rephrased in terms of periodic orbits of a birational action of an infinite Coxeter group on ordered point sets in the projective plane modulo projective equivalence. We also outline the classification of non-rational surfaces with large automorphism groups.

Numerical model of wave-current interaction based on unstructured quadtree grid for nearshore coastal waters

According to the characteristics of complex topography, hydrology, sediment transport in coastal waters, the aim of this paper is to develop an implicit, depth-averaged 2-D numerical mode of coupling current and wave for coastal area. The hydrodynamic model is the two dimensional depth averaged non-linear shallow water equations with an unstructured non-staggered and multiple-level quadtree rectangular mesh with local refinement for important region. In this non-staggered system, primary variables u -, v -velocity, and water level are stored on the same set of grid points in order to simplify the program code, and fluxes at cell faces are determined using the Rhie and Chow’s momentum interpolation method to avoid spurious checkerboard oscillations. The discretized algebraic equations are solved iteratively using the GMRES method with ILUT preconditioning for speeding the computation. The Wave model is a two-dimensional spectral wave action model of taking into account the effect of wave breaking, shoaling, refraction, diffraction, wave-current interaction, bottom friction. The wave model readily provides the radiation stresses for the hydrodynamic model. The coupling model has been tested against measurement data for steady flow around a spur-dyke in a laboratory flume and tidal flows in Gironde Estuary, France and Grays Harbor, USA. The model reasonably well reproduces the temporal and spatial variations of water level, current speed and wave height observed in the measurements. It shows that the coupling current and wave model can simulate accurately the questions of tide and wave for complex domain, the modeling approach presented herein should be useful in engineering application and environmental protection for coastal area.

Constant time approximation scheme for largest well predicted subset

The largest well predicted subset problem is formulated for comparison of two predicted 3D protein structures from the same sequence. A 3D protein structure is represented by an ordered point set A={a 1,?,a n }, where each a i is a point in 3D space. Given two ordered point sets A={a 1,?,a n } and B={b 1,b 2,?b n } containing n points, and a threshold d, the largest well predicted subset problem is to find the rigid body transformation T for a largest subset B opt of B such that the distance between a i and T(b i ) is at most d for every b i in B opt . A meaningful prediction requires that the size of B opt is at least ?n for some constant ? (Li et al. in CPM 2008, 2008). We use LWPS(A,B,d,?) to denote the largest well predicted subset problem with meaningful prediction. An (1+? 1,1?? 2)-approximation for LWPS(A,B,d,?) is to find a transformation T to bring a subset B?⊆B of size at least (1?? 2)|B opt | such that for each b i ?B?, the Euclidean distance between the two points distance?(a i ,T(b i ))≤(1+? 1)d. We develop a constant time (1+? 1,1?? 2)-approximation algorithm for LWPS(A,B,d,?) for arbitrary positive constants ? 1 and ? 2. To our knowledge, this is the first constant time algorithm in this area. Li et al. (CPM 2008, 2008) showed an $O(n(\log n)^{2}/\delta_{1}^{5})$ time randomized (1+? 1)-distance approximation algorithm for the largest well predicted subset problem under meaningful prediction. We also study a closely related problem, the bottleneck distance problem, where we are given two ordered point sets A={a 1,?,a n } and B={b 1,b 2,?b n } containing n points and the problem is to find the smallest d opt such that there exists a rigid transformation T with distance(a i ,T(b i ))≤d opt for every point b i ?B. A (1+?)-approximation for the bottleneck distance problem is to find a transformation T, such that for each b i ?B, distance?(a i ,T(b i ))≤(1+?)d opt , where ? is a constant. For an arbitrary constant ?, we obtain a linear O(n/? 6) time (1+?)-algorithm for the bottleneck distance problem. The best known algorithms for both problems require super-linear time (Li et al. in CPM 2008, 2008).

Random generation of Q-convex sets

The problem of randomly generating Q-convex sets is considered. We present two generators. The first one uses the Q-convex hull of a set of random points in order to generate a Q-convex set included in the square [ 0 , n ) 2 . This generator is very simple, but is not uniform and its performance is quadratic in n. The second one exploits a coding of the salient points, which generalizes the coding of a border of polyominoes. It is uniform, and is based on the method by rejection. Experimentally, this algorithm works in linear time in the length of the word coding the salient points. Besides, concerning the enumeration problem, we determine an asymptotic formula for the number of Q-convex sets according to the size of the word coding the salient points in a special case, and in general only an experimental estimation.

Unsymmetric and symmetric meshless schemes for the unsteady convection–diffusion equation

In this paper we investigate the application of unsymmetric and symmetric meshless collocation techniques with radial basis functions for solving the unsteady convection–diffusion equation. We employ the method of lines approach to discretize the governing operator equation. The stability of both explicit and implicit time-stepping schemes are analyzed. Numerical results are presented for 1D and 2D problems to compare the performance of the unsymmetric and symmetric collocation techniques. We compare the performance of various globally supported radial basis functions such as multiquadric, inverse multiquadric, Gaussian, thin plate splines and quintics. Numerical studies suggest that symmetric collocation is only marginally better than the unsymmetric approach. Further, it appears that both collocation techniques require a very dense set of collocation points in order to achieve accurate results for high Péclet numbers.

Novel feature based image registration for automated patient positioning

In radiotherapy, current feature based image registration techniques (point or curve based) lack robustness and automation: curve based registrations require user input and can be affected by edge selection and noise, while point based techniques require equivalent and/or ordered point sets in the target and reference images. This limits their use for patient positioning using EPID imaging, and in optical based camera systems (bite-block intra-cranial positioning, external marker based body positioning and patient surface imaging).

Novel feature based image registration for automated patient positioning

In radiotherapy, current feature based image registration techniques (point or curve based) lack robustness and automation: curve based registrations require user input and can be affected by edge selection and noise, while point based techniques require equivalent and/or ordered point sets in the target and reference images. This limits their use for patient positioning using EPID imaging, and in optical based camera systems (bite-block intra-cranial positioning, external marker based body positioning and patient surface imaging).

Constructing iterative non-uniform B-spline curve and surface to fit data points

In this paper, based on the idea of profit and loss modification, we present the iterative non-uniform B-spline curve and surface to settle a key problem in computer aided geometric design and reverse engineering, that is, constructing the curve (surface) fitting (interpolating) a given ordered point set without solving a linear system. We start with a piece of initial non-uniform B-spline curve (surface) which takes the given point set as its control point set. Then by adjusting its control points gradually with iterative formula, we can get a group of non-uniform B-spline curves (surfaces) with gradually higher precision. In this paper, using modern matrix theory, we strictly prove that the limit curve (surface) of the iteration interpolates the given point set. The non-uniform B-spline curves (surfaces) generated with the iteration have many advantages, such as satisfying the NURBS standard, having explicit expression, gaining locality, and convexity preserving, etc.

On the Maximum Scatter Traveling Salesperson Problem

We study the problem of computing a Hamiltonian tour (cycle) or path on a set of points in order to maximize the minimum edge length in the tour or path. This maximum traveling salesperson problem (TSP) is closely related to the bottleneck TSP and is motivated by applications in manufacturing (e.g., sequencing of rivet operations) and medical imaging. In this paper, we give the first algorithmic study of these problems, including complexity results, approximation algorithms, and exact algorithms for special cases. In an attempt to model more accurately the real problems that arise in practice, we also generalize the basic problem to consider a more general measure of scatter in which points on a tour or path should be far not only from their immediate predecessor and successor, but also from other near-neighbors along the tour or path.

Approximate center points in dense point sets

The notion of center points is one way to generalize the median of a finite ordered point set to higher dimensions. A set P of n points in d-dimensional Euclidean space E d is δ- dense, if the ratio of the largest to the smallest distance between any two points of P is bounded by δn 1 d , with some constant δ. We describe a simple, yet efficient algorithm to compute an approximate center point for a δ-dense set P in time O( dn). The quality of the approximation depends on δ and exponentially on the dimension d, where the absolute value of the exponent is in Ω(d log d). We also present an iteration method with a linear rate of convergence, which computes an improved 1 (2(d − 1)(δ + 1) d) - center point for sufficiently large n.

A geometric description of modular lattices

Baer [1] observed that modular lattices of finite length (for example subgroup lattices of abelian groups) can be conceived as subspace lattices of a projective geometry structure on an ordered point set; the set of join irreducibles which in this case are the cyclic subgroups of prime power order. That modular lattices of finite length can be recaptured from the order on the points and, in addition, the incidence of points with ‘lines’, the joins of two points, or the blocks of collinear points has been elaborated by Kurinnoi [18] , Faigle and Herrmann [7], Benson and Conway [2] , and , in the general framework of the ‘core’ of a lattice, by Duquenne [5]. In [7] an axiomatization in terms of point-line incidence has been given. Here, we consider, more generally, modular lattices in which every element is the join of completely join irreducible ‘points’. We prove the isomorphy of an algebraic lattice of this kind and the associated subspace lattice and give a first order characterization of the associated ‘ordered spaces’ in terms of collinearity and order which appears more natural and powerful. The crucial axioms are a ‘triangle axiom’, which includes the degenerate cases, and a strengthened ‘line regularity axiom’, both derived from [7]. As a consequence, using Skolemization, we get that any variety of modular lattices is generated by subspace latices of countable spaces. The central concept, connecting the geometric structure and the lattice structure, is that of a line interval (p + q)/(p + q) where p and q are points and p, q their unique lower covers. Given a line interval, any choice of one incident point per atom of the line interval produces a line of the space.

An Interactive MOLP Procedure for the Extension of DEA to Effectiveness Analysis

This paper presents a new, interactive multi-objective linear-programming procedure to aid decisionmakers in setting up goals for desired outputs. The procedure relies on empirical production functions generated by the use of data envelopment analysis. It presents the decision-maker with a set of alternative efficient points in order either to compare different sets of inputs in terms of their effectiveness for goal achievement, or to set goals against which future management performance may be measured. With each iteration the new information provided by the decision-maker is used to adjust the procedure, leading to points which have greater effectiveness utility for the decision-maker. A numerical example is provided along with guidelines for future applications.

An Interactive MOLP Procedure for the Extension of DEA to Effectiveness Analysis

This paper presents a new, interactive multi-objective linear-programming procedure to aid decision-makers in setting up goals for desired outputs. The procedure relies on empirical production functions generated by the use of data envelopment analysis. It presents the decision-maker with a set of alternative efficient points in order either to compare different sets of inputs in terms of their effectiveness for goal achievement, or to set goals against which future management performance may be measured. With each iteration the new information provided by the decision-maker is used to adjust the procedure, leading to points which have greater effectiveness utility for the decision-maker. A numerical example is provided along with guidelines for future applications.

On the hermitean product of ordered point sets on the unit circle.

On the hermitean product of ordered point sets on the unit circle.