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Uniform a priori estimates for discrete solution of nonlinear tensor diffusion equation in image processing. (English) Zbl 1140.35362
Summary: This paper concerns with the finite volume scheme for nonlinear tensor diffusion in image processing. First we provide some basic information on this type of diffusion including a construction of its diffusion tensor. Then we derive a semi-implicit scheme with the help of the so-called diamond-cell method. Further, we prove existence and uniqueness of a discrete solution given by our scheme. The proof is based on a gradient bound in the tangential direction by a gradient in normal direction. Moreover, the proofs of \(L^2(\Omega)\)-a priori estimates for our discrete solution are given. Finally we present our computational results.
MSC:
35B45 A priori estimates in context of PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
74S10 Finite volume methods applied to problems in solid mechanics
68U10 Computing methodologies for image processing
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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References:
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