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Barbu, Tudor

Variational image denoising approach with diffusion porous media flow. (English) Zbl 1425.94007

Abstr. Appl. Anal. 2013, Article ID 856876, 8 p. (2013).
Summary: A novel PDE-based image denoising approach is proposed in this paper. One designs here a nonlinear filter for image noise reduction based on the diffusion flow generated by the porous media equation \(\partial u/\partial t = \Delta \beta (u)\), where \(\beta\) is a nonlinear continuous function of the form \(\beta(u) = \lambda u^m, 0 < m < 1\). With respect to standard 2D Gaussian smoothing and some nonlinear PDE-based filters, this one is more efficient to remove noise from degraded images and also to reduce “staircasing” effects and preserve the image edges.

Cited in 2 Documents

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
76S05 Flows in porous media; filtration; seepage
76M27 Visualization algorithms applied to problems in fluid mechanics
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References:

[1] Jain, A. K., Fundamentals of Digital Image Processing (1989), Englewood Cliffs, NJ, USA: Prentice Hall, Englewood Cliffs, NJ, USA · Zbl 0744.68134
[2] Gonzalez, R.; Woods, R., Digital Image Processing (2001), New York, NY, USA: Prentice Hall, New York, NY, USA
[3] Nallaperumal, K.; Selvakumar, R. K.; Arumugaperumal, S.; Annam, S.; Vishwanath, N., Hansa filter: a fuzzy adaptive nonlinear Gaussian filter: an adaptive non local algorithm, International Journal of Imaging and Engineering, 2, 1 (2008)
[4] Guichard, F.; Moisan, L.; Morel, J.-M., A review of P.D.E. models in image processing and image analysis, Journal de Physique, 4, 137-154 (2001)
[5] Weickert, J., Anisotropic Diffusion in Image Processing. Anisotropic Diffusion in Image Processing, European Consortium for Mathematics in Industry, xii+170 (1998), Stuttgart, Germany: B. G. Teubner, Stuttgart, Germany · Zbl 0886.68131
[6] Alvarez, L.; Berger, M.-O.; Deriche, R.; Herlin, I.; Jaffré, J.; Morel, J.-M., Images and PDE’s, ICAOS ’96 (Paris, 1996). ICAOS ’96 (Paris, 1996), Lecture Notes in Control and Information Sciences, 219, 3-14 (1996), London, UK: Springer, London, UK · Zbl 0851.68115 · doi:10.1007/3-540-76076-8_112
[7] Barbu, T.; Barbu, V., A PDE approach to image restoration problem with observation on a meager domain, Nonlinear Analysis: Real World Applications, 13, 3, 1206-1215 (2012) · Zbl 1239.94005 · doi:10.1016/j.nonrwa.2011.09.014
[8] Barbu, T.; Barbu, V.; Biga, V.; Coca, D., A PDE variational approach to image denoising and restoration, Nonlinear Analysis: Real World Applications, 10, 3, 1351-1361 (2009) · Zbl 1169.35341 · doi:10.1016/j.nonrwa.2008.01.017
[9] Barbu, T., A PDE based model for sonar image and video denoising, AnAlele Stiintifice Ale Universitatii Ovidius, Constanta, Seria Matematica, 19, 3, 51-58 (2011) · Zbl 1274.35394
[10] Perona, P.; Malik, J., Scale-space and edge detection using anisotropic diffusion, Proceedings of IEEE Computer Society Workshop on Computer Vision
[11] Catte, F.; Lions, P. L.; Morel, J.-M.; Coll, T., Image selective smoothing and edge detection by nonlinear diffusion, SIAM Journal on Numerical Analysis, 29, 1, 182-193 (1992) · Zbl 0746.65091
[12] Kačur, J.; Mikula, K., Slow and fast diffusion effects in image processing, Computing and Visualization in Science, 13, 4, 185-195 (2001) · Zbl 0971.65085 · doi:10.1007/s007910000047
[13] Barbu, V., Nonlinear Differential Equations of Montone Type in Banach Spaces (2010), New York, NY, USA: Springer, New York, NY, USA · Zbl 1197.35002
[14] Brezis, H.; Zarantonello, E., Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, Contributions to Nonlinear Functional Analysis (1971), New York, NY, USA: Acdemic Press, New York, NY, USA · Zbl 0278.47033
[15] Calder, J.; Mansouri, A.; Yezzi, A., New possibilities in image diffusion and sharpening via high-order Sobolev gradient flows, Journal of Mathematical Imaging and Vision, 40, 3, 248-258 (2011) · Zbl 1255.68215 · doi:10.1007/s10851-010-0250-2
[16] Calder, J.; Mansouri, A.; Yezzi, A., Image sharpening via Sobolev gradient flows, SIAM Journal on Imaging Sciences, 3, 4, 981-1014 (2010) · Zbl 1202.94019 · doi:10.1137/090771260
[17] Jin, Z.; Yang, X., Strong solutions for the generalized Perona-Malik equation for image restoration, Nonlinear Analysis: Theory, Methods and Applications, 73, 4, 1077-1084 (2010) · Zbl 1194.35503 · doi:10.1016/j.na.2010.04.039
[18] Black, M. J.; Sapiro, G.; Marimont, D. H.; Heeger, D., Robust anisotropic diffusion, IEEE Transactions on Image Processing, 7, 3, 421-432 (1998)
[19] Weickert, J.; ter Haar Romeny, B. M.; Viergever, M. A., Efficient and reliable schemes for nonlinear diffusion filtering, IEEE Transactions on Image Processing, 7, 3, 398-410 (1998)
[20] Kim, S.; Lim, H., A non-convex diffusion model for simultaneous image denoising and edge enhancement, Proceedings of the 6th Mississippi State-UBA Conference on Differential Equations and Computational Simulations, Southwest Texas State Univ. · Zbl 1112.35096
[21] Acton, S. T., Edge enhancement of infrared imagery by way of the anisotropic diffusion pyramid, Proceedings of IEEE International Conference on Image Processing (ICIP ’96)
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