Non-local Functionals Related to the Total Variation and Connections with Image Processing
Non-local Functionals Related to the Total Variation and Connections with Image Processing
复制标题
与总变分相关的非局部泛函及其与图像处理的联系
作者：
H. Brezis;Hoai
我们给出了关于总变分的逼近的新结果，∫Ω|∇u|DocumentClass[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{matrsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$INT_{Omega}|
Λδ(U)=∫Ω∫Ωδφ(|u(X)-u(Y)|/δ)|x-y|d+1dxdy，Documentclass[12pt]{minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{mathsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}egin{文档}$$egin{对准}lambda_Delta(U)=int_{ommega}int_{omegga}frac{Delta varphi ig(|u(X)-u(Y)|/ig)}{|x-y|^{d+1}，dX，dy，lambda_Delta(U)End{Aligned}$$End{Document}Asδ→0DocumentClass[12pt]{Minimal}Usepackage{amsath}usepackage{wa ysym}usepackage{amsFonts}usepackage{amssymb}usepackage{amsbsy}usepackage{mathsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$$Delta
Ightarrow 0$$End{Document}，其中ΩDocumentClass[12pt]{Minimum}Usepackage{amsath}Usepackage{amsFonts}Usepackage{amssymb}Usepackage{amsbsy}Usepackage{mathsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}Document{Document}$$Omega$$End{Document}中的域名是RdDocumentClass[12pt]{}Usepackage{amsmax}usepackage{amssymsm}usepackage{amssymssy}usepackage{mathsfs}usepackage{setgreupek}{setoddsidemin{Document}$69pt}中的一个域，{最小usepackage{amsath}usesepackage{wassymsb}usepackage{amsymssy}usepackage{mathssy}usepackage{-setupackage{setdsidemin{setdsidemin}Document{69pt}{}usepackage{amsath}usesepackage{amssymssym}usepackage{amesepackage{amsymsbsy}usepackage{mathssy}usepackage{mathsfks}usepackage{-}Usgepackage$$End{Document}中的域名[$Omega$$End{Document}，{}UsPack{amsmax}Usepackage{WASSYSYM}UsPACKAGE{amsPACKGE{WASSIPACKS}USIPACKGE{amssymsb}UsPackage{amSamssy y}Usepackage{mathamssy ssy ssyssy}UsPackage{+∞)DocumentClass[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{matrsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$varphi：[0，+inty]
Ightarrow[0，+infty)$$end{Document}是满足某些适当条件的非减函数。趋同的模式极其微妙，许多问题仍然悬而未决。De Giorgi的Γ文档类的概念[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{matrsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}egin{Document}$$Gamma$$End{Document}-收敛说明了这种情况，但也引入了神秘的新奇事物。我们工作的最初动机来自于图像处理。
We present new results concerning the approximation of the total variation, ∫Ω|∇u|documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$int _{Omega } |
abla u|$$end{document}, of a function u by non-local, non-convex functionals of the form Λδ(u)=∫Ω∫Ωδφ(|u(x)-u(y)|/δ)|x-y|d+1dxdy,documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$egin{aligned} Lambda _delta (u) = int _{Omega } int _{Omega } frac{delta varphi ig ( |u(x) - u(y)|/ delta ig )}{|x - y|^{d+1}} , dx , dy, end{aligned}$$end{document}as δ→0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$delta
ightarrow 0$$end{document}, where Ωdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$Omega $$end{document} is a domain in Rddocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$mathbb {R}^d$$end{document} and φ:[0,+∞)→[0,+∞)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$varphi : [0, + infty )
ightarrow [0, + infty )$$end{document} is a non-decreasing function satisfying some appropriate conditions. The mode of convergence is extremely delicate and numerous problems remain open. De Giorgi’s concept of Γdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$Gamma $$end{document}-convergence illuminates the situation, but also introduces mysterious novelties. The original motivation of our work comes from Image Processing.