Large deviations for heavy-tailed random elements in convex cones

We prove large deviation results for sums of heavy-tailed random elements in rather general convex cones being semigroups equipped with a rescaling operation by positive real numbers. In difference to previous results for the cone of convex sets, our technique does not use the embedding of cones in linear spaces. Examples include the cone of convex sets with the Minkowski addition, positive half-line with maximum operation and the family of square integrable functions with arithmetic addition and argument rescaling.

Voronoi cells via linear inequality systems

The theory and methods of linear algebra are a useful alternative to those of convex geometry in the framework of Voronoi cells and diagrams, which constitute basic tools of computational geometry. As shown by Voigt and Weis in 2010, the Voronoi cells of a given set of sites T, which provide a tesselation of the space called Voronoi diagram when T is finite, are solution sets of linear inequality systems indexed by T. This paper exploits systematically this fact in order to obtain geometrical information on Voronoi cells from sets associated with T (convex and conical hulls, tangent cones and the characteristic cones of their linear representations). The particular cases of T being a curve, a closed convex set and a discrete set are analyzed in detail. We also include conclusions on Voronoi diagrams of arbitrary sets.

Spline Subdivision Schemes for Compact Sets. A Survey

Dedicated to the memory of our colleague Vasil Popov January 14, 1942 – May 31, 1990 * Partially supported by ISF-Center of Excellence, and by The Hermann Minkowski Center for Geometry at Tel Aviv University, Israel

Some Fixed Point Theorems for Kannan Mappings

2000 Mathematics Subject Classification: Primary: 47H10; Secondary: 54H25.

Reconstruction from incomplete data in cone-beam tomography

A method for reconstruction of an object f(x) x=(x,y,z) from a limited set of cone-beam projection data has been developed. This method uses a modified form of convolution back-projection and projection onto convex sets (POCS) for handling the limited (or incomplete) data problem. In cone-beam tomography, one needs to have a complete geometry to completely reconstruct the original three-dimensional object. While complete geometries do exist, they are of little use in practical implementations. The most common trajectory used in practical scanners is circular, which is incomplete. It is, however, possible to recover some of the information of the original signal f(x) based on a priori knowledge of the nature of f(x). If this knowledge can be posed in a convex set framework, then POCS can be utilized. In this report, we utilize this a priori knowledge as convex set constraints to reconstruct f(x) using POCS. While we demonstrate the effectiveness of our algorithm for circular trajectories, it is essentially geometry independent and will be useful in any limited-view cone-beam reconstruction.

Contributions to the analysis and segmentation of remote sensing hyperspectral images

142 p.

Consensus under general convexity

A method is proposed to characterize contraction of a set through orthogonal projections. For discrete-time multi-agent systems, quantitative estimates of convergence (to a consensus) rate are provided by means of contracting convex sets. Required convexity for the sets that should include the values that the transition maps of agents take is considered in a more general sense than that of Euclidean geometry. © 2007 IEEE.

Quantitative convergence analysis of multi-agent systems

We introduce a characterization of contraction for bounded convex sets. For discrete-time multi-agent systems we provide an explicit upperbound on the rate of convergence to a consensus under the assumptions of contractiveness and (weak) connectedness (across an interval.) Convergence is shown to be exponential when either the system or the function characterizing the contraction is linear. Copyright © 2007 IFAC.

POCS-based reconstruction of multiplexed sensitivity encoded MRI (POCSMUSE): A general algorithm for reducing motion-related artifacts.

PURPOSE: A projection onto convex sets reconstruction of multiplexed sensitivity encoded MRI (POCSMUSE) is developed to reduce motion-related artifacts, including respiration artifacts in abdominal imaging and aliasing artifacts in interleaved diffusion-weighted imaging. THEORY: Images with reduced artifacts are reconstructed with an iterative projection onto convex sets (POCS) procedure that uses the coil sensitivity profile as a constraint. This method can be applied to data obtained with different pulse sequences and k-space trajectories. In addition, various constraints can be incorporated to stabilize the reconstruction of ill-conditioned matrices. METHODS: The POCSMUSE technique was applied to abdominal fast spin-echo imaging data, and its effectiveness in respiratory-triggered scans was evaluated. The POCSMUSE method was also applied to reduce aliasing artifacts due to shot-to-shot phase variations in interleaved diffusion-weighted imaging data corresponding to different k-space trajectories and matrix condition numbers. RESULTS: Experimental results show that the POCSMUSE technique can effectively reduce motion-related artifacts in data obtained with different pulse sequences, k-space trajectories and contrasts. CONCLUSION: POCSMUSE is a general post-processing algorithm for reduction of motion-related artifacts. It is compatible with different pulse sequences, and can also be used to further reduce residual artifacts in data produced by existing motion artifact reduction methods.

Convergence of bimodules over maximal abelian selfadjoint algebras

We prove a continuity result for the map sending a masa-bimodule to its support. We characterise the convergence of a net of weakly closed convex hulls of bilattices in terms of the convergence of the corresponding supports, and establish a lower-semicontinuity result for the map sending a support to the corresponding masa-bimodule.

Risk-informed decision-making in the presence of epistemic uncertainty

An important issue in risk analysis is the distinction between epistemic and aleatory uncertainties. In this paper, the use of distinct representation formats for aleatory and epistemic uncertainties is advocated, the latter being modelled by sets of possible values. Modern uncertainty theories based on convex sets of probabilities are known to be instrumental for hybrid representations where aleatory and epistemic components of uncertainty remain distinct. Simple uncertainty representation techniques based on fuzzy intervals and p-boxes are used in practice. This paper outlines a risk analysis methodology from elicitation of knowledge about parameters to decision. It proposes an elicitation methodology where the chosen representation format depends on the nature and the amount of available information. Uncertainty propagation methods then blend Monte Carlo simulation and interval analysis techniques. Nevertheless, results provided by these techniques, often in terms of probability intervals, may be too complex to interpret for a decision-maker and we, therefore, propose to compute a unique indicator of the likelihood of risk, called confidence index. It explicitly accounts for the decisionmaker’s attitude in the face of ambiguity. This step takes place at the end of the risk analysis process, when no further collection of evidence is possible that might reduce the ambiguity due to epistemic uncertainty. This last feature stands in contrast with the Bayesian methodology, where epistemic uncertainties on input parameters are modelled by single subjective probabilities at the beginning of the risk analysis process.

The basic principles of uncertain information fusion. An organised review of merging rules in different representation frameworks.

We propose and advocate basic principles for the fusion of incomplete or uncertain information items, that should apply regardless of the formalism adopted for representing pieces of information coming from several sources. This formalism can be based on sets, logic, partial orders, possibility theory, belief functions or imprecise probabilities. We propose a general notion of information item representing incomplete or uncertain information about the values of an entity of interest. It is supposed to rank such values in terms of relative plausibility, and explicitly point out impossible values. Basic issues affecting the results of the fusion process, such as relative information content and consistency of information items, as well as their mutual consistency, are discussed. For each representation setting, we present fusion rules that obey our principles, and compare them to postulates specific to the representation proposed in the past. In the crudest (Boolean) representation setting (using a set of possible values), we show that the understanding of the set in terms of most plausible values, or in terms of non-impossible ones matters for choosing a relevant fusion rule. Especially, in the latter case our principles justify the method of maximal consistent subsets, while the former is related to the fusion of logical bases. Then we consider several formal settings for incomplete or uncertain information items, where our postulates are instantiated: plausibility orderings, qualitative and quantitative possibility distributions, belief functions and convex sets of probabilities. The aim of this paper is to provide a unified picture of fusion rules across various uncertainty representation settings.

Some Problems In Algebra And Topology A Study Of Fuzzy Convexity With Special Reference To Separation Properties

This thesis is a study of abstract fuzzy convexity spaces and fuzzy topology fuzzy convexity spaces No attempt seems to have been made to develop a fuzzy convexity theoryin abstract situations. The purpose of this thesis is to introduce fuzzy convexity theory in abstract situations

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