| Title:
|
Uniform a priori estimates for discrete solution of nonlinear tensor diffusion equation in image processing (English) |
| Author:
|
Drblíková, Olga |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
43 |
| Issue:
|
6 |
| Year:
|
2007 |
| Pages:
|
777-788 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 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. (English) |
| Keyword:
|
finite volume method |
| Keyword:
|
diamond-cell method |
| Keyword:
|
image processing |
| Keyword:
|
nonlinear parabolic equation |
| Keyword:
|
tensor diffusion |
| MSC:
|
35B45 |
| MSC:
|
35K57 |
| MSC:
|
35K60 |
| MSC:
|
65M60 |
| MSC:
|
68U10 |
| MSC:
|
74S10 |
| MSC:
|
94A08 |
| idZBL:
|
Zbl 1140.35362 |
| idMR:
|
MR2388392 |
| . |
| Date available:
|
2009-09-24T20:29:22Z |
| Last updated:
|
2013-09-21 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135814 |
| . |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[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 Zbl 0937.65116, MR 1713235, 10.1051/m2an:1999149 |
| Reference:
|
[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 Zbl 1191.65142, MR 1804748 |
| Reference:
|
[7] Guichard F., Morel J. M.: Image Analysis and P.D.E.s. IPAM GBM Tutorials, 2001 |
| Reference:
|
[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 Zbl 1065.65105, MR 1961884, 10.1007/s002110100374 |
| Reference:
|
[9] Mikula K., Ramarosy N.: Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing.Numer. Math. 89 (2001), 561–590 Zbl 1013.65094, MR 1864431, 10.1007/PL00005479 |
| Reference:
|
[10] Weickert J.: Coherence-enhancing diffusion filtering.Internat. J. Comput. Vision 31 (1999), 111–127 10.1023/A:1008009714131 |
| Reference:
|
[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 10.1006/jvci.2001.0495 |
| . |
