Uniform a priori estimates for discrete solution of nonlinear tensor diffusion equation in image processing

Olga Drblíková

Kybernetika (2007)

  • Volume: 43, Issue: 6, page 777-788
  • ISSN: 0023-5954

Abstract

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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 ( Ω ) – a priori estimates for our discrete solution are given. Finally we present our computational results.

How to cite

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Drblíková, Olga. "Uniform a priori estimates for discrete solution of nonlinear tensor diffusion equation in image processing." Kybernetika 43.6 (2007): 777-788. <http://eudml.org/doc/33895>.

@article{Drblíková2007,
abstract = {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.},
author = {Drblíková, Olga},
journal = {Kybernetika},
keywords = {finite volume method; diamond-cell method; image processing; nonlinear parabolic equation; tensor diffusion; finite volume scheme; diamond-cell method; gradient bound},
language = {eng},
number = {6},
pages = {777-788},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Uniform a priori estimates for discrete solution of nonlinear tensor diffusion equation in image processing},
url = {http://eudml.org/doc/33895},
volume = {43},
year = {2007},
}

TY - JOUR
AU - Drblíková, Olga
TI - Uniform a priori estimates for discrete solution of nonlinear tensor diffusion equation in image processing
JO - Kybernetika
PY - 2007
PB - Institute of Information Theory and Automation AS CR
VL - 43
IS - 6
SP - 777
EP - 788
AB - 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.
LA - eng
KW - finite volume method; diamond-cell method; image processing; nonlinear parabolic equation; tensor diffusion; finite volume scheme; diamond-cell method; gradient bound
UR - http://eudml.org/doc/33895
ER -

References

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  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 
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  5. 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 Zbl1191.65142MR1804748
  6. Guichard F., Morel J. M., Image Analysis and P, D.E.s. IPAM GBM Tutorials, 2001 
  7. Handlovičová A., Mikula, K., Sgallari F., 10.1007/s002110100374, Numer. Math. 93 (2003), 675–695 Zbl1065.65105MR1961884DOI10.1007/s002110100374
  8. Mikula K., Ramarosy N., 10.1007/PL00005479, Numer. Math. 89 (2001), 561–590 Zbl1013.65094MR1864431DOI10.1007/PL00005479
  9. Weickert J., 10.1023/A:1008009714131, Internat. J. Comput. Vision 31 (1999), 111–127 (1999) DOI10.1023/A:1008009714131
  10. Weickert J., Scharr H., 10.1006/jvci.2001.0495, J. Visual Comm. and Image Repres. 13 (2002), 1–2, 103–118 DOI10.1006/jvci.2001.0495

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