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Nonlocal operators with applications to image processing. (English) Zbl 1181.35006

Summary: We propose the use of nonlocal operators to define new types of flows and functionals for image processing and elsewhere. A main advantage over classical PDE-based algorithms is the ability to handle better textures and repetitive structures. This topic can be viewed as an extension of spectral graph theory and the diffusion geometry framework to functional analysis and PDE-like evolutions. Some possible applications and numerical examples are given, as is a general framework for approximating Hamilton-Jacobi equations on arbitrary grids in high dimensions, e.g., for control theory.

MSC:

35A15 Variational methods applied to PDEs
68U10 Computing methodologies for image processing
70H20 Hamilton-Jacobi equations in mechanics
65D25 Numerical differentiation
35S05 Pseudodifferential operators as generalizations of partial differential operators
68R10 Graph theory (including graph drawing) in computer science
35F21 Hamilton-Jacobi equations
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