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On convergence rate of a rectangular partition based global optimization algorithm

JOURNAL OF GLOBAL OPTIMIZATION 2018年第1期71卷 165-191页

作者： Calvin, James Gimbutiene, Grazina Phillips, William O. Zilinskas, Antanas New Jersey Inst Technol Dept Comp Sci Newark NJ 07102 USA Vilnius Univ Inst Math & Informat Akad 6 LT-08663 Vilnius Lithuania

The convergence rate of a rectangular partition based algorithm is considered. A hyper-rectangle for the subdivision is selected at each step according to a criterion rooted in the statistical models based theory of global optimization;only the objective function values are used to compute the criterion of selection. The convergence rate is analyzed assuming that the objective functions are twice- continuously differentiable and defined on the unit cube in d-dimensional Euclidean space. An asymptotic bound on the convergence rate is established. The results of numerical experiments are included.

关键词： Global optimization Convergence rate rectangular partition Statistical models for global optimization Bayesian approach P-algorithm

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A Robust Image Watermarking Algorithm based on Non-Uniform rectangular partition and SVD

A Robust Image Watermarking Algorithm based on Non-Uniform R...

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Pacific-Asia Conference on Knowledge Engineering and Software Engineering

作者： U, KinTak Hu, ShengDun Qi, Dongxu Tang, Zesheng Macau Univ Sci & Technol Fac Informat Technol Macau Peoples R China

ISBN: (纸本)9780769539164

It is more likely to prevent the watermark from JPEG-compression attack, when it is embedded into maximum value of the SVD. However, embedding the watermark with the same intensity into them may distort the watermarked image to a large extent. To solve this problem, a novel robust image-watermark algorithm is proposed with the characteristics of selective embedding intensity based on the non-uniform partition. The SVD values of the scrambled watermark will be embedded into the maximum SVD value of each 8x8 block of the host image in order and repeatedly. Besides, in order to increase the anti-cropping function and the security of the algorithm, Arnold Scrambling Algorithm is applied before the watermark is embedded. The results of the experiments have showed that this algorithm is robust against noising attack, filtering attack and cropping attack, especially the JPEG-compression attack and no obvious distortion appeared in the watermarked image.

关键词： non-uniform partition of image rectangular partition image watermarking SVD

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A Robust Image Watermarking Algorithm based on Non-Uniform rectangular partition and DCT-SVD

A Robust Image Watermarking Algorithm based on Non-Uniform R...

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2010 International Conference on Measuring Technology and Mechatronics Automation(ICMTMA 2010)(2010年检测技术与机电自动化国际会议)

作者： KinTak U ShengDun Hu Dongxu Qi Zesheng Tang Faculty of Information Technology Macau University of Science and Technology Macau China

ISBN: (纸本)9781424450015;9780769539621

When embedding the watermark information into DC coefficient of each 8x8 DCT block of the host image, it is more likely to obtain a stronger anti-attack function against JPEG-compression attack. However, embedding the watermark with the same intensity into them may distort the watermarked image to a large extent. To solve this problem, a novel robust image-watermark algorithm is proposed with the characteristics of selective embedding intensity based on the non-uniform partition, in the meanwhile, Arnold Scrambling Algorithm is applied before the watermark is embedded to increase the security of the algorithm. The SVD values of the scrambled watermark will be embedded into the DC coefficient of each 8x8 DCT block of the host image in order and repeatedly and this may enhance the various antiattack ability of the whole algorithm. The experimental results have showed that this algorithm is robust against cropping attack, filtering attack and noising attack, especially the JPEGcompression attack and no obvious distortion appeared in the watermarked image.

关键词： rectangular partition non-uniform partition of image image watermarking DCT SVD

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rectangular partitionING FOR INTRA PREDICTION IN HEVC

RECTANGULAR PARTITIONING FOR INTRA PREDICTION IN HEVC

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IEEE Visual Communications and Image Processing (VCIP)

作者： Liu, Shan Zhang, Ximin Lei, Shawmin MediaTek USA Inc San Jose CA 95134 USA

ISBN: (纸本)9781467344050

This paper presents a mechanism of using rectangular prediction blocks for Intra prediction in the emerging High Efficiency Video Coding (HEVC) standard. In the previous and current video coding standards, Intra predictions have been processed on square blocks, e. g. 16x16, 8x8 and 4x4 pixel blocks H.264/MPEG4 AVC, and 2Nx2N or NxN square prediction units (PU) in the current HEVC. In this paper, it is proposed to include 2NxN and Nx2N prediction block sizes (or PU types) in a 2Nx2N coding unit (CU) for the Intra prediction, as in the Inter prediction. Experimental results show that, together with other tools (e. g. NSQT) significant coding gain, i.e. average 1.8% BD-rate reduction with the use of 2-point prediction and transforms, or 1.4% BD-rate reduction without the use of 2-point prediction and transforms (for "All Intra" configuration) can be achieved, compared with HM5.0 anchor.

关键词： High Efficiency Video Coding (HEVC) standards coding unit prediction unit rectangular partition

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Minimum stabbing rectangular partitions of rectilinear polygons

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COMPUTERS & OPERATIONS RESEARCH 2017年第Apr.期80卷 184-197页

作者： Piva, Breno de Souza, Cid C. Univ Estadual Campinas Inst Comp UNICAMP Campinas SP Brazil Univ Fed Sergipe Dept Comp Sao Cristovao SE Brazil Univ Estadual Campinas Inst Comp Ave Albert Einstein 1251Cidade Univ BR-13083852 Campinas SP Brazil

We study integer programming (IP) models for the problem of finding a rectangular partition of a rectilinear polygon with minimum stabbing number. Strong valid inequalities are introduced for an existing formulation and a new model is proposed. We compare the dual bounds yielded by the relaxations of the two models and prove that the new one is stronger than the old one. Computational experiments with the problem are reported for the first time in which polygons with thousands of vertices are solved to optimality. The (IP) branch-and-bound algorithm based on the new model is faster and more robust than those relying on the previous formulation. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： Stabbing number rectangular partition Rectilinear polygons Integer programming

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Exact solutions of rectangular partitions via integer programming

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INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS 2000年第5期10卷 477-522页

作者： De Meneses, CN De Souza, CC Univ Estadual Campinas Inst Comp BR-13083970 Campinas SP Brazil

Given a rectangle R in the plane and a finite set P of points in its interior, consider the partitions of the surface of R into smeller rectangles. A partition is feasible with respect to P if each point in P lie on the boundary of some rectangle of the partition. The length of a partition is computed as the sum of the lengths of the line segments defining the boundary of its rectangles. The goal is to find a feasible partition with minimum length. This problem, denoted by RGP, belongs to NP-hard and has application in VLSI design. In this paper we investigate how to obtain exact solutions for the RGP. We introduce two different Integer Programming formulations and carry out a theoretical study to evaluate and compare the strength of their bounds. Computational experiments are reported for Branch-and-Cut and Branch-and-Price algorithms we have implemented for the first and the second formulation, respectively. Randomly generated instances with \P\ less than or equal to 200 are solved exactly. The tests indicate that the size of the instances solved with our algorithms decrease by an order of magnitude in the absence of corectilinear points in P, a special case of RGP whose complexity is still open.

关键词： rectangular partition polyhedral combinatorics branch-and-cut branch-and-price

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rectangular partitions of arectilinear polygon

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COMPUTATIONAL GEOMETRY-THEORY AND APPLICATIONS 2023年 110卷

作者： Kim, Hwi Lee, Jaegun Ahn, Hee-Kap Pohang Univ Sci & Technol Dept Comp Sci & Engn Pohang South Korea Pohang Univ Sci & Technol Grad Sch Artificial Intelligence Dept Comp Sci & Engn Pohang South Korea

We investigate the problem of partitioning a rectilinear polygon P with n vertices and no holes into rectangles using disjoint line segments drawn inside P under two optimality criteria. In the minimum ink partition, the total length of the line segments drawn inside P is minimized. We present an O(n(3))-time algorithm using O(n(2)) space that returns a minimum ink partition of P. In the thick partition, the minimum side length over all resulting rectangles is maximized. We present an O(n(3)log(2)n)-time algorithm using O(n(3)) space that returns a thick partition using line segments incident to vertices of P, and an O(n(6)log(2)n)-time algorithm using O(n(6)) space that returns a thick partition using line segments incident to the boundary of P. We also show that if the input rectilinear polygon has holes, the corresponding decision problem for the thick partition problem using line segments incident to vertices of the polygon is NP-complete. We also present an O(m(3))-time 3-approximation algorithm for the minimum ink partition for a rectangle containing mpoint holes. (c) 2022 Elsevier B.V. All rights reserved.

关键词： rectangular partition Rectilinear polygon Minimum ink

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Optimal matrix-segmentation by rectangles

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DISCRETE APPLIED MATHEMATICS 2009年第9期157卷 2015-2030页

作者： Engel, Konrad Univ Rostock Inst Math D-18051 Rostock Germany

We study the problem of decomposing a nonnegative matrix into a nonnegative combination of 0-1-matrices whose ones form a rectangle such that the sum of the coefficients is minimal. We present for the case of two rows an easy algorithm that provides an optimal solution which is integral if the given matrix is integral. An additional integrality constraint makes the problem more difficult if the matrix has more than two rows. An algorithm that is based on the revised simplex method and uses only very few Gomory cuts yields exact integral solutions for integral matrices of reasonable size in a short time. For matrices of large dimension we propose a special greedy algorithm that provides sufficiently good results in numerical experiments. (C) 2008 Elsevier B.V. All rights reserved.

关键词： Network flow Intensity-modulated radiation therapy (IMRT) Segmentation algorithm Linear programming rectangular partition Matrix-decomposition Gomory cut Revised simplex method

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L^∞-error estimates of rectangular mixed finite element methods for bilinear optimal control problem

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APPLIED MATHEMATICS AND COMPUTATION 2017年 300卷 79-94页

作者： Lu, Zuliang Zhang, Shuhua Chongqing Three Gorges Univ Key Lab Signal & Informat Proc Key Lab Nonlinear Sci & Syst Struct Chongqing 404000 Peoples R China Tianjin Univ Finance & Econ Coordinated Innovat Ctr Computable Modeling Manag Tianjin 300222 Peoples R China

In this paper, we investigate L-infinity-error estimates of the bilinear elliptic optimal control problem by rectangular Raviart-Thomas mixed finite element methods. The control variable enters the state equation as a coefficient. The state and the co-state variables are approximated by the Raviart-Thomas mixed finite elements of order k = 1, and the control variable is approximated by piecewise linear functions. The L-infinity-error estimates are obtained for the control variable and coupled state variable, and the convergence rates of orders O(h(2)) and O(h3/2 vertical bar In h vertical bar 1/2) are also gained for the control and state variables and the flux of the state and co-state variables, respectively. In addition, the performance of the error estimates is assessed by two numerical examples. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Bilinear optimal control problem Raviart-Thomas mixed finite element methods rectangular partition L-infinity-error estimates

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Superconvergence of rectangular Mixed Finite Element Methods for Constrained Optimal Control Problem

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Advances in Applied Mathematics and Mechanics 2010年第1期2卷 56-75页

作者： Yanping Chen Li Dai Zuliang Lu School of Mathematical Sciences South China Normal UniversityGuangzhouGuangdongChina Mathematics Teaching and Research Group Xiguan Foreign Language SchoolGuangzhouGuangdongChina Hunan Key Laboratory for Computation and Simulation in Science and Engineering Department of MathematicsXiangtan UniversityXiangtanHunanChina

We investigate the superconvergence properties of the constrained quadratic elliptic optimal control problem which is solved by using rectangular mixed finite element *** use the lowest order Raviart-Thomas mixed finite element spaces to approximate the state and co-state variables and use piecewise constant functions to approximate the control *** obtain the superconvergence of O(h^(1+s))(0

关键词： Constrained optimal control problem linear elliptic equation mixed finite element methods rectangular partition superconvergence properties

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