A new nonlocal nonlinear diffusion equation for data analysis. (English) Zbl 1441.94037
Summary: In this paper we introduce and study a new feature-preserving nonlinear nonlocal diffusion equation for denoising signals. The proposed partial differential equation is based on a novel diffusivity coefficient that uses a nonlocal automatically detected parameter related to the local bounded variation and the local oscillating pattern of the noisy input signal. We provide a mathematical analysis of the existence of the solution in the two dimensional case, but easily extensible to the one-dimensional model. Finally, we show some numerical experiments, which demonstrate the effectiveness of the new approach.
MSC:
| 94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
| 35Q94 | PDEs in connection with information and communication |
| 35K57 | Reaction-diffusion equations |
References:
| [1] | Aletti, G.; Lonardoni, D.; Naldi, G.; Nieus, T., From dynamics to links: a sparse reconstruction of the topology of a neural network, Commun. Appl. Ind. Math., 10, 2, 2-11 (2019) · Zbl 1423.92009 |
| [2] | Alvarez, L.; Guichard, F.; Lions, P.-L.; Morel, J.-M., Axioms and fundamental equations of image processing, Arch. Ration. Mech. Anal., 123, 199-257 (1993) · Zbl 0788.68153 · doi:10.1007/BF00375127 |
| [3] | Angenent, S.; Pichon, E.; Tannenbaum, A., Mathematical methods in medical image processing, Bull. Amer. Math. Soc. (N.S.), 43, 365-396 (2006) · Zbl 1096.92027 · doi:10.1090/S0273-0979-06-01104-9 |
| [4] | Aubert, G.; Kornprobst, P., Mathematical Problems in Image Processing (2006), New York: Springer, New York · Zbl 1019.94002 |
| [5] | Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations (2011), New York: Springer, New York · Zbl 1220.46002 |
| [6] | Buades, A.; Coll, B.; Morel, J. M., A non-local algorithm for image denoising, Proc. IEEE CVPR, 60-65 (2005) · Zbl 1108.94004 |
| [7] | Buades, A.; Coll, B.; Morel, J. M., Neighborhood filters and PDE’s, Numer. Math., 105, 1-34 (2006) · Zbl 1151.94310 · doi:10.1007/s00211-006-0029-y |
| [8] | Catté, F.; Lions, P.-L.; Morel, J.-M.; Coll, T., Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29, 182-193 (1992) · Zbl 0746.65091 · doi:10.1137/0729012 |
| [9] | Chambolle, A.; Darbon, J., On total variation minimization and surface evolution using parametric maximum flows, Int. J. Comput. Vis., 84, 288-307 (2009) · Zbl 1371.94073 · doi:10.1007/s11263-009-0238-9 |
| [10] | Hummel, R. A., Representations based on zero-crossings in scale-space, Proceedings IEEE CVPR ’86, 204-209 (1986) |
| [11] | Kačur, J., Method of Rothe in Evolution Equations (1985) · Zbl 0582.65084 |
| [12] | Koenderink, J., The structure of images, Biol. Cybern., 50, 363-370 (1984) · Zbl 0537.92011 · doi:10.1007/BF00336961 |
| [13] | McOwen, R., Partial Differential Equations: Methods and Applications (2003), New York: Prentice Hall, New York |
| [14] | Meyer, Y., Oscillating Patterns in Image Processing and in Some Nonlinear Evolution Equations. The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures (2001), Providence: AMS, Providence · Zbl 0987.35003 |
| [15] | Palazzolo, G.; Moroni, M.; Soloperto, A.; Aletti, G.; Naldi, G.; Vassalli, M.; Nieus, T.; Difato, F., Fast wide-volume functional imaging of engineered in vitro brain tissues, Sci. Rep., 7 (2017) · doi:10.1038/s41598-017-08979-8 |
| [16] | Perona, P.; Malik, J., Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12, 629-639 (1990) · doi:10.1109/34.56205 |
| [17] | Rabier, P.; Thomas, J.-M., Exercices d’analyse numérique des équations aux dérivées partielles (1985), Paris: Masson, Paris · Zbl 0576.65090 |
| [18] | Sapiro, G., Geometric Partial Differential Equations and Image Analysis (2006), Cambridge: Cambridge University Press, Cambridge · Zbl 1141.35002 |
| [19] | Weickert, J., Anisotropic Diffusion in Image Processing (1998), Stuttgart: B. G. Teubner, Stuttgart · Zbl 0886.68131 |
| [20] | Witkin, A. P., Scale-space filtering, Proceedings of the 8th International Joint Conference on Artificial Intelligence, 1019-1022 (1983), Los Altos: William Kaufmann Inc., Los Altos |
| [21] | Ziemer, W. P., Weakly Differentiable Functions (1989), New York: Springer, New York · Zbl 0692.46022 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
