| Title:
|
Nonlinear Tensor Diffusion in Image Processing (English) |
| Author:
|
Stašová, Olga |
| Author:
|
Mikula, Karol |
| Author:
|
Handlovičová, Angela |
| Author:
|
Peyriéras, Nadine |
| Language:
|
English |
| Journal:
|
Proceedings of Equadiff 14 |
| Volume:
|
Conference on Differential Equations and Their Applications, Bratislava, July 24-28, 2017 |
| Issue:
|
2017 |
| Year:
|
|
| Pages:
|
377-386 |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper presents and summarize our results concerning the nonlinear tensor diffusion which enhances image structure coherence. The core of the paper comes from [3, 2, 4, 5]. First we briefly describe the diffusion model and provide its basic properties. Further we build a semi-implicit finite volume scheme for the above mentioned model with the help of a co-volume mesh. This strategy is well-known as diamond-cell method owing to the choice of co-volume as a diamondshaped polygon, see [1]. We present here 2D as well as 3D case of a numerical scheme, see [3, 4]. Then the convergence and error estimate analysis for 2D scheme is presented, see [3, 2]. Last part is devoted to results of computational experiments. They confirm the usefulness this diffusion type not just for an image improvement but also as a pre-processed algorithm. Numerical techniques which require a good coherence of image structures (like edge detection and segmentation) achieve much better results when we use images pre-processed by such a filtration. Let us note that this diffusion technique was successfully applied within the framework of EU projects. It was used to pre-process images for the structure segmentation in zebrafish embryogenesis, see [5]. (English) |
| Keyword:
|
Image processing, nonlinear tensor diffusion, coherence enhancing diffusion, numerical solution, semi-implicit scheme, diamond-cell finite volume method, convergence, error estimate, structure segmentation |
| MSC:
|
35B45 |
| MSC:
|
35K55 |
| MSC:
|
65M08 |
| MSC:
|
65M12 |
| MSC:
|
68U10 |
| . |
| Date available:
|
2019-09-27T08:23:28Z |
| Last updated:
|
2019-09-27 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/703032 |
| . |
| Reference:
|
[1] Coudiere, Y., Vila, J. P., Villedieu, P.: Convergence rate of a finite volume scheme fora two-dimensional convection-diffusion problem., M2AN Math. Model. Numer. Anal., 33,(1999), pp. 493–516. MR 1713235, 10.1051/m2an:1999149 |
| Reference:
|
[2] Drblíková, O., Handlovičová, A., Mikula, K.: Error Estimates of the Finite Volume Scheme for the Nonlinear Tensor Anisotropic Diffusion., Applied Numerical Mathematics 59(10) (2009), pp. 2548–2570. MR 2553154, 10.1016/j.apnum.2009.05.010 |
| Reference:
|
[3] Drblíková, O., Mikula, K.: Convergence Analysis of Finite Volume Scheme for Nonlinear Tensor Anisotropic Diffusion in Image Processing., SIAM J. Numer. Anal., 46(1) (2007), pp. 37–60. MR 2377254, 10.1137/070685038 |
| Reference:
|
[4] Drblíková, O., Mikula, K.: Semi-implicit Diamond-cell Finite Volume Scheme for 3DNonlinear Tensor Diffusion in Coherence Enhancing Image Filtering., Finite Volumes for Complex Applications, Proceedings of the 5th International Symposium on Finite Volumes for Complex Applications (FVCA5). Published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley & Sons, Inc., ISBN 978-1-84821-035-6, pp. 343–350. MR 2451426 |
| Reference:
|
[5] Drblíková, O., Mikula, K., Peyriéras, N.: The Nonlinear Tensor Diffusion in Segmentation of Meaningful Biological Structures from Image Sequences of Zebrafish Embryogenesis., Scale Space and Variational Methods in Computer Vision, Proceedings. Springer Berlin Heidelberg (2009), pp. 63–74. |
| Reference:
|
[6] Eymard, R., Gallou\"et, T., Herbin, R.: Finite Volume Methods., in: Handbook for Numerical Analysis, Vol. 7 (Ph. Ciarlet, J. L. Lions, eds.), Elsevier (2000). MR 1804748 |
| Reference:
|
[7] Meijering, E., Niessen, W., Weickert, J., Viergever, M.: Diffusion-Enhanced Visualization and Quantification of Vascular Anomalies in Three-Dimensional Rotational Angiography., Results of an In-Vitro Evaluation. Medical Image Analysis, 6(3) (2002), pp. 217–235. 10.1016/S1361-8415(02)00081-6 |
| Reference:
|
[8] Mikula, K., Ramarosy, N.: Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing., Numer. Math. 89 (3), (2001) pp. 561–590. MR 1864431, 10.1007/PL00005479 |
| Reference:
|
[9] Mikula, K., Sarti, A., Sgallari, F.: Co-volume level set method in subjective surface basedmedical image segmentation., in: Handbook of Medical Image Analysis: Segmentation and Registration Models (J. Suri et al., Eds.), Springer, New York, (2005) pp. 583–626. MR 2707803 |
| Reference:
|
[10] Sarti, A., Malladi, R., Sethian, J.A.: Subjective Surfaces: A Method for Completing Missing Boundaries., Proceedings of the National Academy of Sciences of the United States of America, 12 (97), (2000) pp. 6258–6263. MR 1760935 |
| Reference:
|
[11] Weickert, J.: Coherence-enhancing diffusion filtering., Int. J. Comput. Vision, Vol. 31, (1999) pp. 111–127. 10.1023/A:1008009714131 |
| . |
