FINDING PROJECTIONS ONTO THE INTERSECTION OF CONVEX SETS IN HILBERT SPACES. II.

We present a parallel iterative algorithm to find the shortest distance projection of a given point onto the intersection of a finite number of closed convex sets in a real Hilbert space ; the number of sets used at each iteration stept corresponding to the number of available processors, may be smaller than the total number of sets. The relaxation coefficient at each iteration step is determined by a geometrical condition in an associated Hilbert space, while for the weights mild conditions are given to assure norm convergence of the resulting sequence. These mild conditions leave enough flexibility to determine the weights more specifically in order to improve the speed of convergence.

Lifts of Non-Compact Convex Sets and Cone Factorizations

This paper generalizes the factorization theorem of Gouveia,Parrilo and Thomas to a broader class of convex sets.Given a general convex set,the authors define a slack operator associated to the set and its polar according to whether the convex set is full dimensional,whether it is a translated cone and whether it contains lines.The authors strengthen the condition of a cone lift by requiring not only the convex set is the image of an affine slice of a given closed convex cone,but also its recession cone is the image of the linear slice of the closed convex cone.The authors show that the generalized lift of a convex set can also be characterized by the cone factorization of a properly defined slack operator.

Convex Set Theory for Reliability Assessment of Steel Beam with Bounded Uncertainty

Probabilistic reliability model established by insufficient data is inaccessible. The convex model was applied to model the uncertainties of variables. A new non-probabilistic reliability model was proposed based on the robustness of system to uncertainty. The non-probabilistic reliability model,the infinite norm model,and the probabilistic model were used to assess the reliability of a steel beam,respectively. The results show that the resistance is allowed to couple with the action effect in the non-probabilistic reliability model. Additionally,the non-probabilistic reliability model becomes the same accurate as probabilistic model with the increase of the bounded uncertain information. The model is decided by the available data and information.

Iterative analytic extension in tomographic imaging

If a spatial-domain function has a finite support,its Fourier transform is an entire function.The Taylor series expansion of an entire function converges at every finite point in the complex plane.The analytic continuation theory suggests that a finite-sized object can be uniquely determined by its frequency components in a very small neighborhood.Trying to obtain such an exact Taylor expansion is difficult.This paper proposes an iterative algorithm to extend the measured frequency components to unmeasured regions.Computer simulations show that the proposed algorithm converges very slowly,indicating that the problem is too ill-posed to be practically solvable using available methods.

Optimization Processes of Tangible and Intangible Networks through the Laplace Problems for Regular Lattices with Multiple Obstacles along the Way

A systematic approach is proposed to the theme of safety,reliability and global quality of complex networks(material and immaterial)by means of special mathematical tools that allow an adequate geometric characterization and study of the operation,even in the presence of multiple obstacles along the path.To that end,applying the theory of graphs to the problem under study and using a special mathematical model based on stochastic geometry,in this article we consider some regular lattices in which it is possible to schematize the elements of the network,with the fundamental cell with six,eight or 2(n+2)obstacles,calculating the probability of Laplace.In this way it is possible to measure the“degree of impedance”exerted by the anomalies along the network by the obstacles examined.The method can be extended to other regular and/or irregular geometric figures,whose union together constitutes the examined network,allowing to optimize the functioning of the complex system considered.

A Comprehensive Model for Structural Non-Probabilistic Reliability and the Key Algorithms

It is very difficult to know the exact boundaries of the variable domain for problems with small sample size,and the traditional convex set model is no longer applicable.In view of this,a novel reliability model was proposed on the basis of the fuzzy convex set(FCS)model.This new reliability model can account for different relations between the structural failure region and variable domain.Key computational algorithms were studied in detail.First,the optimization strategy for robust reliability is improved.Second,Monte Carlo algorithms(i.e.,uniform sampling method)for hyper-ellipsoidal convex sets were studied in detail,and errors in previous reports were corrected.Finally,the Gauss-Legendre integral algorithm was used for calculation of the integral reliability index.Three numerical examples are presented here to illustrate the rationality and feasibility of the proposed model and its corresponding algorithms.

Lipschitz-continuity of Spherically Convex Functions

In this article,some basic and important properties of spherically convex functions,such as the Lipschitz-continuity,are investigated.It is shown that,under a weaker condition,every family of spherically convex functions is equi-Lipschitzian on each closed spherically convex subset contained in the relative interior of their common domain,and from which a powerful result is derived:the pointwise convergence of a sequence of spherically convex functions implies its uniform convergence on each closed spherically convex subset contained in the relative interior of their common domain.

Efficient Inverse Method for Structural Identification Considering Modeling and Response Uncertainties

The inverse problem analysis method provides an effective way for the structural parameter identification.However,uncertainties wildly exist in the practical engineering inverse problems.Due to the coupling of multi-source uncertainties in the measured responses and the modeling parameters,the traditional inverse method under the deterministic framework faces the challenges in solving mechanism and computing cost.In this paper,an uncertain inverse method based on convex model and dimension reduction decomposition is proposed to realize the interval identification of unknown structural parameters according to the uncertain measured responses and modeling parameters.Firstly,the polygonal convex set model is established to quantify the epistemic uncertainties of modeling parameters.Afterwards,a space collocation method based on dimension reduction decomposition is proposed to transform the inverse problem considering multi-source uncertainties into a few interval inverse problems considering response uncertainty.The transformed interval inverse problem involves the two-layer solving process including interval propagation and optimization updating.In order to solve the interval inverse problems considering response uncertainty,an efficient interval inverse method based on the high dimensional model representation and affine algorithm is further developed.Through the coupling of the above two strategies,the proposed uncertain inverse method avoids the time-consuming multi-layer nested calculation procedure,and then effectively realizes the uncertainty identification of unknown structural parameters.Finally,two engineering examples are provided to verify the effectiveness of the proposed uncertain inverse method.

Image enhancement and post-processing for low resolution compressed video

This research paper recommends the point spread function(PSF)forecasting technique based on the projection onto convex set(POCS)and regularization to acquire low resolution images.As the environment for the production of user created contents(UCC)videos(one of the contents on the Internet)becomes widespread,resolution reduction and image distortion occurs,failing to satisfy users who desire high quality images.Accordingly,this research neutralizes the coding artifact through POCS and regularization processes by:1)factoring the local characteristics of the image when it comes to the noise that results during the discrete cosine transform(DCT)and quantization process;and 2)removing the blocking and ring phenomena which are problems with the existing video compression.Moreover,this research forecasts the point spread function to obtain low resolution images using the above-mentioned methods.Thus,a method is suggested for minimizing the errors found among the forecasting interpolation pixels.Low-resolution image quality obtained through the experiment demonstrates that significant enhancement was made on the visual level compared to the original image.

SOME EQUIVALENT CONDITIONS OF PROXIMINALITY IN NONREFLEXIVE BANACH SPACES

In this paper,we discuss the relation between τ-strongly Chebyshev,approximatively τ-compact k-Chebyshev,approximatively τ-compact,τ-strongly proximinal and proximinal sets,where τ is the norm or the weak topology.We give some equivalent conditions regarding the above proximinality.Furthermore,we also propose the necessary and sufficient conditions that a half-space is τ-strongly proximinal,approximatively τ-compact and τ-strongly Chebyshev.

Super-resolution enhancement of UAV images based on fractional calculus and POCS

A super-resolution enhancement algorithm was proposed based on the combination of fractional calculus and Projection onto Convex Sets(POCS)for unmanned aerial vehicles(UAVs)images.The representative problems of UAV images including motion blur,fisheye effect distortion,overexposed,and so on can be improved by the proposed algorithm.The fractional calculus operator is used to enhance the high-resolution and low-resolution reference frames for POCS.The affine transformation parameters between low-resolution images and reference frame are calculated by Scale Invariant Feature Transform(SIFT)for matching.The point spread function of POCS is simulated by a fractional integral filter instead of Gaussian filter for more clarity of texture and detail.The objective indices and subjective effect are compared between the proposed and other methods.The experimental results indicate that the proposed method outperforms other algorithms in most cases,especially in the structure and detail clarity of the reconstructed images.

A Super-resolution Reconstruction Algorithm for Surveillance Video

Recent technological developments have resulted in surveillance video becoming a primary method of preserving public security.Many city crimes are observed in surveillance video.The most abundant evidence collected by the police is also acquired through surveillance video sources.Surveillance video footage offers very strong support for solving criminal cases,therefore,creating an effective policy and applying useful methods to the retrieval of additional evidence is becoming increasingly important.However,surveillance video has had its failings,namely,video footage being captured in low resolution(LR)and bad visual quality.In this paper,we discuss the characteristics of surveillance video and describe the manual feature registration-maximum a posteriori-projection onto convex sets to develop a super-resolution reconstruction method,which improves the quality of surveillance video.From this method,we can make optimal use of information contained in the LR video image,but we can also control the image edge clearly as well as the convergence of the algorithm.Finally,we make a suggestion on how to adjust the algorithm adaptability by analyzing the prior information of target image.