American Institute of Mathematical Sciences

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

On a parabolic-hyperbolic filter for multicolor image noise reduction

Valerii Maltsev 1, and  Michael Pokojovy 2,

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.
Keywords: Image processing, weak solutions, strong solutions, maximal regularity., nonlinear partial differential equations.
Mathematics Subject Classification: Primary: 35G61, 35M33, 65J15; Secondary: 35B30, 35D30, 35D3.
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. 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

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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