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finite volume method; diamond-cell method; image processing; nonlinear parabolic equation; tensor diffusion
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 so-called diamond-cell method (see [Coirier1] and [Coirier2]). 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.
[1] Catté F., Lions P. L., Morel J. M., Coll T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 129 (1991), 182–193 MR 1149092
[2] Coirier W. J.: An a Adaptively-Refined, Cartesian, Cell-Based Scheme for the Euler and Navier-Stokes Equations. PhD Thesis, Michigan Univ. NASA Lewis Research Center, 1994
[3] Coirier W. J., Powell K. G.: A cartesian, cell-based approach for adaptive-refined solutions of the Euler and Navier–Stokes equations. AIAA 1995
[4] Coudiere Y., Vila J. P., Villedieu P.: Convergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem. M2AN Math. Model. Numer. Anal. 33 (1999), 493–516 DOI 10.1051/m2an:1999149 | MR 1713235 | Zbl 0937.65116
[6] Eymard R., Gallouët, T., Herbin R.: Finite Volume Methods. In: Handbook for Numerical Analysis, Vol. 7 (Ph. Ciarlet, J. L. Lions, eds.), Elsevier, Amsterdam 2000 MR 1804748 | Zbl 1191.65142
[7] Guichard F., Morel J. M.: Image Analysis and P. D.E.s. IPAM GBM Tutorials, 2001
[8] Handlovičová A., Mikula, K., Sgallari F.: Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution. Numer. Math. 93 (2003), 675–695 DOI 10.1007/s002110100374 | MR 1961884 | Zbl 1065.65105
[9] Mikula K., Ramarosy N.: Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing. Numer. Math. 89 (2001), 561–590 DOI 10.1007/PL00005479 | MR 1864431 | Zbl 1013.65094
[10] Weickert J.: Coherence-enhancing diffusion filtering. Internat. J. Comput. Vision 31 (1999), 111–127 DOI 10.1023/A:1008009714131
[11] Weickert J., Scharr H.: A scheme for coherence-enhancing diffusion filtering with optimized rotation invariance. J. Visual Comm. and Image Repres. 13 (2002), 1–2, 103–118 DOI 10.1006/jvci.2001.0495
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