# American Institute of Mathematical Sciences

June  2016, 5(2): 251-272. doi: 10.3934/eect.2016004

## On a parabolic-hyperbolic filter for multicolor image noise reduction

 1 Taras Shevchenko National University of Kyiv, Faculty of Cybernetics, 4D Glushkov Ave, 03680 Kyiv, Ukraine 2 Karlsruhe Institute of Technology, Department of Mathematics, Englerstrasse 2, 76131 Karlsruhe, Germany

Received  March 2016 Revised  May 2016 Published  June 2016

We propose a novel PDE-based anisotropic filter for noise reduction in multicolor images. It is a generalization of Nitzberg & Shiota's (1992) model being a hyperbolic relaxation of the well-known parabolic Perona & Malik's filter (1990). First, we consider a `spatial' mollifier-type regularization of our PDE system and exploit the maximal $L^{2}$-regularity theory for non-autonomous forms to prove a well-posedness result both in weak and strong settings. Again, using the maximal $L^{2}$-regularity theory and Schauder's fixed point theorem, respective solutions for the original quasilinear problem are obtained and the uniqueness of solutions with a bounded gradient is proved. Finally, the long-time behavior of our model is studied.
Citation: Valerii Maltsev, Michael Pokojovy. On a parabolic-hyperbolic filter for multicolor image noise reduction. Evolution Equations & Control Theory, 2016, 5 (2) : 251-272. doi: 10.3934/eect.2016004
##### References:
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El Ayyadi, A Volterra type model for image processing,, IEEE Transactions on Image Processing, 7 (1998), 292. doi: 10.1109/83.661179. Google Scholar [18] R. Dautray and J.-L. Lions, Evolution Problems, vol. 5 of Mathematical Analysis and Numerical Methods for Science and Technology,, Springer-Verlag, (1992). doi: 10.1007/978-3-642-58090-1. Google Scholar [19] D. Dier, Non-autonomous maximal regularity for forms of bounded variation,, Journal of Mathematical Analysis and Applications, 425 (2015), 33. doi: 10.1016/j.jmaa.2014.12.006. Google Scholar [20] M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds,, Archive for Rational Mechanics and Analysis, 31 (1968), 113. doi: 10.1007/BF00281373. Google Scholar [21] A. Handlovičová, K. Mikula and F. Sgallari, Variational numerical methods for solving nonlinear diffusion equations arising in image processing,, Journal of Visual Communication and Image Representation, 13 (2002), 217. Google Scholar [22] M. Hieber and M. Murata, The $L^p$-approach to the fluid-rigid body interaction problem for compressible fluids,, Evolution Equations and Control Theory, 4 (2015), 69. doi: 10.3934/eect.2015.4.69. Google Scholar [23] M. Hochbruck, T. Jahnke and R. Schnaubelt, Convergence of an ADI splitting for Maxwell's equations,, Numerische Mathematik, 129 (2015), 535. doi: 10.1007/s00211-014-0642-0. Google Scholar [24] S. L. Keeling and R. Stollberger, Nonlinear anisotropic diffusion filtering for multiscale edge enhancement,, Inverse Problems, 18 (2002), 175. doi: 10.1088/0266-5611/18/1/312. Google Scholar [25] D. Marr and E. Hildreth, Theory of edge detection,, Proceedings of the Royal Society B, 207 (1980), 187. doi: 10.1098/rspb.1980.0020. Google Scholar [26] S. A. Morris, The Schauder-Tychonoff fixed point theorem and applications,, Matematický Časopis, 25 (1975), 165. Google Scholar [27] M. Nitzberg and T. Shiota, Nonlinear image filtering with edge and corner enhancement,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 14 (1992), 826. doi: 10.1109/34.149593. Google Scholar [28] T. Ohkubo, Regularity of solutions to hyperbolic mixed problems with uniformly characteristic boundary,, Hokkaido Mathematical Journal, 10 (1981), 93. doi: 10.14492/hokmj/1381758116. Google Scholar [29] P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion,, IEEE Trans. Pattern Anal. Machine Intell., 12 (1990), 629. doi: 10.1109/34.56205. Google Scholar [30] J. Prüss, Maximal regularity of linear vector-valued parabolic Volterra equations,, Journal of Integral Equations and Applications, 3 (1991), 63. doi: 10.1216/jiea/1181075601. Google Scholar [31] J. Prüss, Evolutionary Integral Equations and Applications, vol. 87 of Monographs in Mathematics,, Birkhäuser Verlag, (1993). doi: 10.1007/978-3-0348-8570-6. Google Scholar [32] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D: Nonlinear Phenomena, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. Google Scholar [33] G. Savaré, Regularity results for elliptic equations in Lipschitz domains,, Journal of Functional Analysis, 152 (1998), 176. doi: 10.1006/jfan.1997.3158. Google Scholar [34] D. W. Scott, Multivariate Density Estimation: Theory, Practice, and Visualization,, 2nd edition, (). Google Scholar [35] P. Secchi, Well-posedness of characteristic symmetric hyperbolic systems,, Archive for Rational Mechanics and Analysis, 134 (1996), 155. doi: 10.1007/BF00379552. Google Scholar [36] K. Takezawa, Introduction to Nonparametric Regression,, Wiley Series in Probability and Mathematical Statistics, (2006). Google Scholar [37] J. Weickert, Anisotropic Diffusion in Image Processing,, B. G. Teubner, (1998). Google Scholar [38] A. P. Witkin, Scale-space filtering,, Readings in Computer Vision: Issues, (1987), 329. doi: 10.1016/B978-0-08-051581-6.50036-2. Google Scholar [39] R. Zacher, Maximal regularity of type $L_p$ for abstract parabolic Volterra equations,, Journal of Evolution Equations, 5 (2005), 79. doi: 10.1007/s00028-004-0161-z. Google Scholar

show all references

##### References:
 [1] L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel, Axioms and fundamental equations of image processing,, Archive for Rational Mechanics and Analysis, 123 (1993), 199. doi: 10.1007/BF00375127. Google Scholar [2] H. Amann, Compact embeddings of vector-valued Sobolev and Besov spaces,, Glasnik Matematički, 35 (2000), 161. Google Scholar [3] H. Amann, Non-local quasi-linear parabolic equations,, Russian Mathematical Surveys, 60 (2005), 1021. doi: 10.1070/RM2005v060n06ABEH004279. Google Scholar [4] H. Amann, Time-delayed Perona-Malik type problems,, Acta Mathematica Universitatis Comenianae, 76 (2007), 15. Google Scholar [5] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variational flow,, Differential and Integral Equations, 14 (2001), 321. Google Scholar [6] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Some qualitative properties for the total variation flow,, Journal of Functional Analysis, 188 (2002), 516. doi: 10.1006/jfan.2001.3829. Google Scholar [7] W. Arendt and R. Chill, Global existence for quasilinear diffusion equations in isotropic nondivergence form,, Annali della Scuola Normale Superiore di Pisa (5), 9 (2010), 523. Google Scholar [8] V. Barbu, Nonlinear Differential Equations Of Monotone Types in Banach Spaces,, Springer Monographs in Mathematics, (2010). doi: 10.1007/978-1-4419-5542-5. Google Scholar [9] A. Belahmidi, Équations Aux Dérivées Partielles Appliquées à la Restauration et à L'agrandissement des Images,, PhD thesis, (2003). Google Scholar [10] A. Belahmidi and A. Chambolle, Time-delay regularization of anisotropic diffusion and image processing,, ESAIM: Mathematical Modelling and Numerical Analysis, 39 (2005), 231. doi: 10.1051/m2an:2005010. Google Scholar [11] A. Belleni-Morante and A. C. McBride, Applied Nonlinear Semigroups: An Introduction,, Wiley Series in Mathematical Methods in Practice, (1998). Google Scholar [12] G. Bellettini, V. Caselles and M. Novaga, The total variation flow in $\mathbbR^N$,, Journal of Differential Equations, 184 (2002), 475. doi: 10.1006/jdeq.2001.4150. Google Scholar [13] M. Burger, A. C. G. Menucci, S. Osher and M. Rumpf (eds.), Level Set and PDE Based Reconstruction Methods in Imaging, vol. 2090 of Lecture Notes in Mathematics,, Springer International Publishing, (1992). Google Scholar [14] J. Canny, Finding Edges and Lines in Images,, Technical Report 720, (1983). Google Scholar [15] G. R. Cattaneo, Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée,, Comptes Rendus de l'Académie des Sciences, 247 (1958), 431. Google Scholar [16] F. Catté, P.-L. Lions, J.-M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion,, SIAM Journal on Numerical Analysis, 29 (1992), 182. doi: 10.1137/0729012. Google Scholar [17] G. H. Cottet and M. El Ayyadi, A Volterra type model for image processing,, IEEE Transactions on Image Processing, 7 (1998), 292. doi: 10.1109/83.661179. Google Scholar [18] R. Dautray and J.-L. Lions, Evolution Problems, vol. 5 of Mathematical Analysis and Numerical Methods for Science and Technology,, Springer-Verlag, (1992). doi: 10.1007/978-3-642-58090-1. Google Scholar [19] D. Dier, Non-autonomous maximal regularity for forms of bounded variation,, Journal of Mathematical Analysis and Applications, 425 (2015), 33. doi: 10.1016/j.jmaa.2014.12.006. Google Scholar [20] M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds,, Archive for Rational Mechanics and Analysis, 31 (1968), 113. doi: 10.1007/BF00281373. Google Scholar [21] A. Handlovičová, K. Mikula and F. Sgallari, Variational numerical methods for solving nonlinear diffusion equations arising in image processing,, Journal of Visual Communication and Image Representation, 13 (2002), 217. Google Scholar [22] M. Hieber and M. Murata, The $L^p$-approach to the fluid-rigid body interaction problem for compressible fluids,, Evolution Equations and Control Theory, 4 (2015), 69. doi: 10.3934/eect.2015.4.69. Google Scholar [23] M. Hochbruck, T. Jahnke and R. Schnaubelt, Convergence of an ADI splitting for Maxwell's equations,, Numerische Mathematik, 129 (2015), 535. doi: 10.1007/s00211-014-0642-0. Google Scholar [24] S. L. Keeling and R. Stollberger, Nonlinear anisotropic diffusion filtering for multiscale edge enhancement,, Inverse Problems, 18 (2002), 175. doi: 10.1088/0266-5611/18/1/312. Google Scholar [25] D. Marr and E. Hildreth, Theory of edge detection,, Proceedings of the Royal Society B, 207 (1980), 187. doi: 10.1098/rspb.1980.0020. Google Scholar [26] S. A. Morris, The Schauder-Tychonoff fixed point theorem and applications,, Matematický Časopis, 25 (1975), 165. Google Scholar [27] M. Nitzberg and T. Shiota, Nonlinear image filtering with edge and corner enhancement,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 14 (1992), 826. doi: 10.1109/34.149593. Google Scholar [28] T. Ohkubo, Regularity of solutions to hyperbolic mixed problems with uniformly characteristic boundary,, Hokkaido Mathematical Journal, 10 (1981), 93. doi: 10.14492/hokmj/1381758116. Google Scholar [29] P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion,, IEEE Trans. Pattern Anal. Machine Intell., 12 (1990), 629. doi: 10.1109/34.56205. Google Scholar [30] J. Prüss, Maximal regularity of linear vector-valued parabolic Volterra equations,, Journal of Integral Equations and Applications, 3 (1991), 63. doi: 10.1216/jiea/1181075601. Google Scholar [31] J. Prüss, Evolutionary Integral Equations and Applications, vol. 87 of Monographs in Mathematics,, Birkhäuser Verlag, (1993). doi: 10.1007/978-3-0348-8570-6. Google Scholar [32] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D: Nonlinear Phenomena, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. Google Scholar [33] G. Savaré, Regularity results for elliptic equations in Lipschitz domains,, Journal of Functional Analysis, 152 (1998), 176. doi: 10.1006/jfan.1997.3158. Google Scholar [34] D. W. Scott, Multivariate Density Estimation: Theory, Practice, and Visualization,, 2nd edition, (). Google Scholar [35] P. Secchi, Well-posedness of characteristic symmetric hyperbolic systems,, Archive for Rational Mechanics and Analysis, 134 (1996), 155. doi: 10.1007/BF00379552. Google Scholar [36] K. Takezawa, Introduction to Nonparametric Regression,, Wiley Series in Probability and Mathematical Statistics, (2006). Google Scholar [37] J. Weickert, Anisotropic Diffusion in Image Processing,, B. G. Teubner, (1998). Google Scholar [38] A. P. Witkin, Scale-space filtering,, Readings in Computer Vision: Issues, (1987), 329. doi: 10.1016/B978-0-08-051581-6.50036-2. Google Scholar [39] R. Zacher, Maximal regularity of type $L_p$ for abstract parabolic Volterra equations,, Journal of Evolution Equations, 5 (2005), 79. doi: 10.1007/s00028-004-0161-z. Google Scholar
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