Two methods of characterizing the visibility of a moving point

Two methods are presented of determining the visibility (observability) of an object moving in space with an obstacle that hinders the motion and the perception of the object by an observer. The first method is based on taking into account the distance from the object to all possible observers. The second method uses not only the distance but also the size of the circular cone with the vertex at the observation point that contains a spherical neighborhood of the object. The directional differentiability of the functions characterizing the visibility of the object is established. The calculation of the derivatives is reduced to an extremal problem, for which “refinement” theorems are given.     

Object and observer: A tracking problem

The property of the visibility of an object for an observer is studied in the case when the motion takes place in a space with a fixed bodily set impeding the visibility. Directional derivatives of the visibility function are calculated. The problem of tracking the object by the observer is considered, when the object aims to minimize and the observer aims to maximize the maximum of the visibility function. An algorithm for constructing a trajectory that is best for the object is given for the case when the observer acts optimally.     

Object visibility for a group of observers with inaccurately given coordinates

A function is introduced that characterizes the relative visibility of a moving object for an observer with inaccurately given coordinates. The directional differentiability of this function is established and, for the case of a moving observer, a differentiation formula is given. The visibility characteristic of an object for a group of observers is defined and the problem of minimizing this characteristic is considered.     

Reconstruction of controls in hyperbolic systems by Tikhonov’s method with nonsmooth stabilizers

A function is introduced that characterizes the visibility of a moving object for an observer in a smooth strictly convex Banach space. The location of the observer is given inaccurately. The differentiability property of this function is investigated; in the case of a finite-dimensional space, an expression is found for the derivative in any direction. The visibility characteristic for a group of observers is studied.     

Moving Object in ℝ<Superscript>2</Superscript> and a Group of Observers

We formulate an extremal problem of constructing a trajectory of a moving object that is farthest from a group of observers with fixed visibility cones. Under some constraints on the arrangement of the observers, we give a characterization and a method of construction of an optimal trajectory.     

Gromov–Wasserstein Distances and the Metric Approach to Object Matching

This paper discusses certain modifications of the ideas concerning the Gromov–Hausdorff distance which have the goal of modeling and tackling the practical problems of object matching and comparison. Objects are viewed as metric measure spaces, and based on ideas from mass transportation, a Gromov–Wasserstein type of distance between objects is defined. This reformulation yields a distance between objects which is more amenable to practical computations but retains all the desirable theoretical underpinnings. The theoretical properties of this new notion of distance are studied, and it is established that it provides a strict metric on the collection of isomorphism classes of metric measure spaces. Furthermore, the topology generated by this metric is studied, and sufficient conditions for the pre-compactness of families of metric measure spaces are identified. A second goal of this paper is to establish links to several other practical methods proposed in the literature for comparing/matching shapes in precise terms. This is done by proving explicit lower bounds for the proposed distance that involve many of the invariants previously reported by researchers. These lower bounds can be computed in polynomial time. The numerical implementations of the ideas are discussed and computational examples are presented.     

Evolution equations for shapes and geometries

The first object of this paper is to introduce a new evolution equation for the characteristic function of the boundary Γ of a Lipschitzian domain Ω in the N-dimensional Euclidean space under the influence of a smooth time-dependent velocity field. The originality of this equation is that the evolution takes place in an L^p-space with respect to the (N − 1)-Hausdorff measure. A second more speculative objective is to discuss how that equation can be relaxed to rougher velocity fields via some weak formulation. A candidate is presented and some of the technical difficulties and open issues are discussed. Continuity results in several metric topologies are also presented. The paper also specializes the results on the evolution of the oriented distance function to initial sets with zero N-dimensional Lebesgue measure.     

Visibility metrics on the infinity boundary of the complex hyperbolic plane

Limiting spherical and horospherical metrics an the infinity boundary of the complex hyperbolic plane are constructed. It is proved that the limiting spherical metric, which automatically is the Carnot–Carathéodory metric, is also a visibility metric, i.e., it belongs to a canonical class of metrics on the infinity boundary. Bibliography: 6 titles.     

Informational Sets in a Model Problem of Homing

A linear pursuit problem in the plane under incomplete pursuer information about the evader is investigated. At discrete time instants, the pursuer measures with errors the angle of vision to the evader, the angular velocity of the line of sight, and the relative distance. Other combinations of measurable parameters are possible (for example, angle of vision and relative distance or angle of vision only). The measurements errors obey certain geometric constraints. The initial uncertainties on the evader coordinates and velocities are given in advance. Having a resource of impulse control, the pursuer tries to minimize the miss distance. The evader control is bounded in modulus.The problem is formulated as an auxiliary differential game. Here, the notion of informational set is central. The informational set is the totality of pointwise phase states consistent with the history of the observation-control process. The informational set depends on the current measurements; it changes in time and plays the role of a generalized state, which is used for constructing the pursuer control.A control method designed for the linear pursuit problem is used in the planar problem of a vehicle homing toward a dangerous space object. The nonlinear dynamics is described by the Kepler equations. Nonlinear terms of the equations in relative coordinates are small and are replaced by an uncertain vector parameter, which is bounded in modulus and is regarded as an evader control. As a result, we obtain the mentioned control problem in the plane.The final part of the paper is devoted to the simulation of a space vehicle homing toward a dangerous space object. In testing the control method developed, two variants are considered: random measurement errors and game method of constructing the measurements; the latter is also described in the paper.     

A blind image watermarking using multiresolution visibility map

A perceptual pyramid watermarking method is proposed. The key idea is to use the contrast sensitivity of the human visual system (HVS) to determine “invisible” regions where watermark energy can be adjusted providing an invisible and robust watermark. These invisibles regions are obtained by computing a “visibility map” at each level of the Gaussian pyramid (GP). The watermark is weighted by the local contrast and a global scaling factor. The embedding process is performed by modifying the values in some levels of the Laplacian Pyramid (LP) using the spread spectrum technique. Afterwards, the watermarked image can be constructed from the levels of the LP. For watermark detection, a blind detection scheme using the threshold-correlation based technique is proposed. Finally, the performances of the watermarking method are evaluated in terms of invisibility and robustness using some quality metrics and different attacks of Stirmark such as Gaussian noise, low-pass filtering, Jpeg compression and cropping. This evaluation is performed for the choice of some parameters of the watermarking system depending on performances such as invisibility and robustness. The design of our watermarking technique can finally be formulated as an optimisation problem where the objective is to guarantee a trade-off between invisibility and robustness.     

Universal Strict General Actors and Actors in Categories of Interest

For any category of interest ℂ we define a general category of groups with operations \mathbbC_G, \mathbbC\hookrightarrow\mathbbC_G\mathbb{C_G}, \mathbb{C}\hookrightarrow\mathbb{C_G}, and a universal strict general actor USGA(A) of an object A in ℂ, which is an object of \mathbbC_G\mathbb{C_G}. The notion of actor is equivalent to the one of split extension classifier defined for an object in more general settings of semi-abelian categories. It is proved that there exists an actor of A in ℂ if and only if the semidirect product \textUSGA(A)\ltimes A{\text{USGA}}(A)\ltimes A is an object of ℂ and if it is the case, then USGA(A) is an actor of A. We give a construction of a universal strict general actor for any A ∈ ℂ, which helps to detect more properties of this object. The cases of groups, Lie, Leibniz, associative, commutative associative, alternative algebras, crossed and precrossed modules are considered. The examples of algebras are given, for which always exist actors.     

On the use of divergence distance in fuzzy clustering

Clustering algorithms divide up a dataset into a set of classes/clusters, where similar data objects are assigned to the same cluster. When the boundary between clusters is ill defined, which yields situations where the same data object belongs to more than one class, the notion of fuzzy clustering becomes relevant. In this course, each datum belongs to a given class with some membership grade, between 0 and 1. The most prominent fuzzy clustering algorithm is the fuzzy c-means introduced by Bezdek (Pattern recognition with fuzzy objective function algorithms, 1981), a fuzzification of the k-means or ISODATA algorithm. On the other hand, several research issues have been raised regarding both the objective function to be minimized and the optimization constraints, which help to identify proper cluster shape (Jain et al., ACM Computing Survey 31(3):264–323, 1999). This paper addresses the issue of clustering by evaluating the distance of fuzzy sets in a feature space. Especially, the fuzzy clustering optimization problem is reformulated when the distance is rather given in terms of divergence distance, which builds a bridge to the notion of probabilistic distance. This leads to a modified fuzzy clustering, which implicitly involves the variance–covariance of input terms. The solution of the underlying optimization problem in terms of optimal solution is determined while the existence and uniqueness of the solution are demonstrated. The performances of the algorithm are assessed through two numerical applications. The former involves clustering of Gaussian membership functions and the latter tackles the well-known Iris dataset. Comparisons with standard fuzzy c-means (FCM) are evaluated and discussed.     

Visibilité dans le modèle Booléen

The aim of this Note is to give an estimate on the tail probability of the visibility function in a Boolean model: this function is defined as the length of the longest ray starting at the origin that does not intersect an obstacle. Convergence results are added, when the size of obstacles goes to zero and when the distance between the origin and the closest obstacle goes to infinity. To cite this article: P. Calka et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

A fuzzy closeness approach to fuzzy multi-attribute decision making

The aim of this paper is to develop a new fuzzy closeness (FC) methodology for multi-attribute decision making (MADM) in fuzzy environments, which is an important research field in decision science and operations research. The TOPSIS method based on an aggregating function representing “closeness to the ideal solution” is one of the well-known MADM methods. However, while the highest ranked alternative by the TOPSIS method is the best in terms of its ranking index, this does not mean that it is always the closest to the ideal solution. Furthermore, the TOPSIS method presumes crisp data while fuzziness is inherent in decision data and decision making processes, so that fuzzy ratings using linguistic variables are better suited for assessing decision alternatives. In this paper, a new FC method for MADM under fuzzy environments is developed by introducing a multi-attribute ranking index based on the particular measure of closeness to the ideal solution, which is developed from the fuzzy weighted Minkowski distance used as an aggregating function in a compromise programming method. The FC method of compromise ranking determines a compromise solution, providing a maximum “group utility” for the “majority” and a minimum individual regret for the “opponent”. A real example of a personnel selection problem is examined to demonstrate the implementation process of the method proposed in this paper.     

Observers and a moving object in ℝ3

Doklady Mathematics - Suppose that an object t moves within a given corridor Y in the presence of a groups S of hostile observers S ∉ Y, each having a fixed visibility cone K(S). The problem...     

Extension of the LINMAP for multiattribute decision making under Atanassov’s intuitionistic fuzzy environment

The aim of this article is further extending the linear programming techniques for multidimensional analysis of preference (LINMAP) to develop a new methodology for solving multiattribute decision making (MADM) problems under Atanassov’s intuitionistic fuzzy (IF) environments. The LINMAP only can deal with MADM problems in crisp environments. However, fuzziness is inherent in decision data and decision making processes. In this methodology, Atanassov’s IF sets are used to describe fuzziness in decision information and decision making processes by means of an Atanassov’s IF decision matrix. A Euclidean distance is proposed to measure the difference between Atanassov’s IF sets. Consistency and inconsistency indices are defined on the basis of preferences between alternatives given by the decision maker. Each alternative is assessed on the basis of its distance to an Atanassov’s IF positive ideal solution (IFPIS) which is unknown a prior. The Atanassov’s IFPIS and the weights of attributes are then estimated using a new linear programming model based upon the consistency and inconsistency indices defined. Finally, the distance of each alternative to the Atanassov’s IFPIS can be calculated to determine the ranking order of all alternatives. A numerical example is examined to demonstrate the implementation process of this methodology. Also it has been proved that the methodology proposed in this article can deal with MADM problems under not only Atanassov’s IF environments but also both fuzzy and crisp environments.     

Topologically sweeping visibility complexes via pseudotriangulations

This paper describes a new algorithm for constructing the set of free bitangents of a collection ofn disjoint convex obstacles of constant complexity. The algorithm runs in timeO(n logn + k), where,k is the output size, and uses,O(n) space. While earlier algorithms achieve the same optimal running time, this is the first optimal algorithm that uses only linear space. The visibility graph or the visibility complex can be computed in the same time and space. The only complicated data structure used by the algorithm is a splittable queue, which can be implemented easily using red-black trees. The algorithm is conceptually very simple, and should therefore be easy to implement and quite fast in practice. The algorithm relies on greedy pseudotriangulations, which are subgraphs of the visibility graph with many nice combinatorial properties. These properties, and thus the correctness of the algorithm, are partially derived from properties of a certain partial order on the faces of the visibility complex. A preliminary version of this work appeared in theProceedings of the 11th Annual ACM Symposium on Computational Geometry, Vancouver, June 1995, pages 248–257.     

A climbing girl's reflections about angles

The main research question in this paper is whether a climbing discourse can be a resource for a school-geometry discourse. The text is based on a 12-year old girl's story from an exciting climbing trip during her summer holiday. The girl uncovers some of her knowledge that had been invisible to her; she is guided to see some relations between her climbing and her understanding of angles. In the beginning, this girl believes her story does not concern angles at all. The tools for uncovering angles in her story are based on different levels of visibility and objects of the climbing discourse combined with different conceptions of space. The girl develops her consciousness about angles as natural elements in her climbing activity and she is guided to see the angle as an object of her climbing discourse.     

Advanced Genetic Programming Based Machine Learning

A Genetic Programming based approach for solving classification problems is presented in this paper. Classification is understood as the act of placing an object into a set of categories, based on the object’s properties; classification algorithms are designed to learn a function which maps a vector of object features into one of several classes. This is done by analyzing a set of input-output examples (“training samples”) of the function. Here we present a method based on the theory of Genetic Algorithms and Genetic Programming that interprets classification problems as optimization problems: Each presented instance of the classification problem is interpreted as an instance of an optimization problem, and a solution is found by a heuristic optimization algorithm. The major new aspects presented in this paper are advanced algorithmic concepts as well as suitable genetic operators for this problem class (mainly the creation of new hypotheses by merging already existing ones and their detailed evaluation). The experimental part of the paper documents the results produced using new hybrid variants of Genetic Algorithms as well as investigated parameter settings. Graphical analysis is done using a novel multiclass classifier analysis concept based on the theory of Receiver Operating Characteristic curves. The work described in this paper was done within the Translational Research Project L282 “GP-Based Techniques for the Design of Virtual Sensors” sponsored by the Austrian Science Fund (FWF).