# 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:
 [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), 199-257. doi: 10.1007/BF00375127.  Google Scholar [2] H. Amann, Compact embeddings of vector-valued Sobolev and Besov spaces, Glasnik Matematički, 35 (2000), 161-177.  Google Scholar [3] H. Amann, Non-local quasi-linear parabolic equations, Russian Mathematical Surveys, 60 (2005), 1021-1033. doi: 10.1070/RM2005v060n06ABEH004279.  Google Scholar [4] H. Amann, Time-delayed Perona-Malik type problems, Acta Mathematica Universitatis Comenianae, 76 (2007), 15-38.  Google Scholar [5] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variational flow, Differential and Integral Equations, 14 (2001), 321-360.  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), 516-547. 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), 523-539.  Google Scholar [8] V. Barbu, Nonlinear Differential Equations Of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer Science & Business Media, New York Dordrecht Heidelberg London, 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, Université de Paris-Dauphine, Paris, 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), 231-251. 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, John Wiley & Sons, Chichester, 1998.  Google Scholar [12] G. Bellettini, V. Caselles and M. Novaga, The total variation flow in $\mathbbR^N$, Journal of Differential Equations, 184 (2002), 475-525. 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, Switzerland, 1992. Google Scholar [14] J. Canny, Finding Edges and Lines in Images, Technical Report 720, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Boston, MA, 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), 431-433.  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), 182-193. 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), 292-303. 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, Berlin, 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), 33-54. 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), 113-126. 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), 217-237. 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), 69-87. 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), 535-561. 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), 175-190. 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), 187-217. doi: 10.1098/rspb.1980.0020.  Google Scholar [26] S. A. Morris, The Schauder-Tychonoff fixed point theorem and applications, Matematický Časopis, 25 (1975), 165-172.  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), 826-833. 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), 93-123. 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), 629-630. 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), 63-83. doi: 10.1216/jiea/1181075601.  Google Scholar [31] J. Prüss, Evolutionary Integral Equations and Applications, vol. 87 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 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), 259-268. 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), 176-201. 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), 155-197. doi: 10.1007/BF00379552.  Google Scholar [36] K. Takezawa, Introduction to Nonparametric Regression, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., Hoboken, New Jersey, 2006.  Google Scholar [37] J. Weickert, Anisotropic Diffusion in Image Processing, B. G. Teubner, Stuttgart, 1998.  Google Scholar [38] A. P. Witkin, Scale-space filtering, Readings in Computer Vision: Issues, Problem, Principles, and Paradigms, (1987), 329-332. 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), 79-103. 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), 199-257. doi: 10.1007/BF00375127.  Google Scholar [2] H. Amann, Compact embeddings of vector-valued Sobolev and Besov spaces, Glasnik Matematički, 35 (2000), 161-177.  Google Scholar [3] H. Amann, Non-local quasi-linear parabolic equations, Russian Mathematical Surveys, 60 (2005), 1021-1033. doi: 10.1070/RM2005v060n06ABEH004279.  Google Scholar [4] H. Amann, Time-delayed Perona-Malik type problems, Acta Mathematica Universitatis Comenianae, 76 (2007), 15-38.  Google Scholar [5] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variational flow, Differential and Integral Equations, 14 (2001), 321-360.  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), 516-547. 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), 523-539.  Google Scholar [8] V. Barbu, Nonlinear Differential Equations Of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer Science & Business Media, New York Dordrecht Heidelberg London, 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, Université de Paris-Dauphine, Paris, 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), 231-251. 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, John Wiley & Sons, Chichester, 1998.  Google Scholar [12] G. Bellettini, V. Caselles and M. Novaga, The total variation flow in $\mathbbR^N$, Journal of Differential Equations, 184 (2002), 475-525. 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, Switzerland, 1992. Google Scholar [14] J. Canny, Finding Edges and Lines in Images, Technical Report 720, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Boston, MA, 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), 431-433.  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), 182-193. 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), 292-303. 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, Berlin, 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), 33-54. 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), 113-126. 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), 217-237. 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), 69-87. 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), 535-561. 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), 175-190. 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), 187-217. doi: 10.1098/rspb.1980.0020.  Google Scholar [26] S. A. Morris, The Schauder-Tychonoff fixed point theorem and applications, Matematický Časopis, 25 (1975), 165-172.  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), 826-833. 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), 93-123. 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), 629-630. 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), 63-83. doi: 10.1216/jiea/1181075601.  Google Scholar [31] J. Prüss, Evolutionary Integral Equations and Applications, vol. 87 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 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), 259-268. 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), 176-201. 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), 155-197. doi: 10.1007/BF00379552.  Google Scholar [36] K. Takezawa, Introduction to Nonparametric Regression, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., Hoboken, New Jersey, 2006.  Google Scholar [37] J. Weickert, Anisotropic Diffusion in Image Processing, B. G. Teubner, Stuttgart, 1998.  Google Scholar [38] A. P. Witkin, Scale-space filtering, Readings in Computer Vision: Issues, Problem, Principles, and Paradigms, (1987), 329-332. 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), 79-103. doi: 10.1007/s00028-004-0161-z.  Google Scholar
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