# American Institute of Mathematical Sciences

March  2014, 34(3): 931-957. doi: 10.3934/dcds.2014.34.931

## ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing

 1 Cambridge Centre for Analysis, University of Cambridge, Wilberforce Road, CB3 0WA, Cambridge, United Kingdom 2 Department of Mathematics, University of Sussex, Pevensey II, BN1 9QH, Brighton, United Kingdom 3 Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge, Wilberforce Road, CB3 0WA, Cambridge, United Kingdom

Received  December 2012 Revised  May 2013 Published  August 2013

We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the $H^{-1}$-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation.
Citation: Luca Calatroni, Bertram Düring, Carola-Bibiane Schönlieb. ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 931-957. doi: 10.3934/dcds.2014.34.931
##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, Oxford, 2000.  Google Scholar [2] J.-F. Aujol and A. Chambolle, Dual norms and image decomposition models, International Journal of Computer Vision, 63 (2005), 85-104. doi: 10.1007/s11263-005-4948-3.  Google Scholar [3] J.-F. Aujol and G. Gilboa, Constrained and SNR-based solutions for TV-Hilbert space image denoising, Journal of Mathematical Imaging and Vision, 26 (2006), 217-237. doi: 10.1007/s10851-006-7801-6.  Google Scholar [4] D. Barash, M. Israeli and R. Kimmel, An accurate operator splitting scheme for nonlinear diffusion filtering, in "Scale-Space and Morphology in Computer Vision" (ed. M. Kerckhove), Lecture Notes in Computer Science 2106 (2006), 281-289. Google Scholar [5] M. Benning, "Singular Regularization of inverse Problems," PhD thesis, University of Münster, 2011. Google Scholar [6] M. Benning, L. Calatroni, B. Düring and C.-B. Schönlieb, A primal-dual approach for a total variation Wasserstein flow, to appear in GSI 2013 LNCS proceedings, Springer, (2013). Google Scholar [7] M. Burger, M. Franek and C.-B. Schönlieb, Regularised Regression and Density estimation based on Optimal Transport,, Appl. Math. Res. Express (AMRX), 2012 (): 209.   Google Scholar [8] M. Burger, L. He and C.-B. Schönlieb, Cahn-Hilliard inpainting and a generalization for grayvalue images, SIAM J. Imaging Sci., 2 (2009), 1129-1167. doi: 10.1137/080728548.  Google Scholar [9] T. F. Chan and J. J. Shen, "Image Processing and Analysis Variational, PDE, wavelet, and stochastic methods," SIAM, 2005. doi: 10.1137/1.9780898717877.  Google Scholar [10] A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vis., 20 (2004), 89-97. doi: 10.1023/B:JMIV.0000011321.19549.88.  Google Scholar [11] A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numer. Math. 76 (1997), 167-188. doi: 10.1007/s002110050258.  Google Scholar [12] T. F. Chan, S. H. Kang and J. Shen, Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., 63 (2002), 564-592. doi: 10.1137/S0036139901390088.  Google Scholar [13] T. F. Chan and P. Mulet, On the convergence of the lagged diffusivity fixed point method in total variation image restoration, SIAM Journal on Numerical Analysis, 36 (1999), 354-367. doi: 10.1137/S0036142997327075.  Google Scholar [14] T. F. Chan and J. Shen, Mathematical models for local non-texture inpaintings, SIAM J. Appl. Math., 62 (2001), 1019-1043. doi: 10.1137/S0036139900368844.  Google Scholar [15] T. F. Chan and J. Shen, Non-texture inpainting by curvature driven diffusions (CDD), J. Visual Comm. Image, Rep., 12 (2001), 436-449. Google Scholar [16] T. F. Chan and J. Shen, Variational image inpainting, Comm. Pure Applied Math, 58 (2005), 579-619. doi: 10.1002/cpa.20075.  Google Scholar [17] W. Chen and Z. Ditzian, Mixed and directional derivatives, Proceedings of the American Mathematical Society, 108 (1990), 177-185. doi: 10.1090/S0002-9939-1990-0994773-0.  Google Scholar [18] S. D. Conte and R. T. James, An alternating direction method for solving the biharmonic equation, Math. Tables Aids Comput., 12 (1958), 198-205. doi: 10.2307/2002021.  Google Scholar [19] S. Didas, J. Weickert and B. Burgeth, Stability and local feature enhancement of higher order nonlinear diffusion filtering, in "Pattern Recognition" (Eds. W. Kropatsch, R. Sablatnig and A. Hanbury), Lecture Notes in Computer Science, Springer, Berlin, 3663 (2005), 451-458. doi: 10.1007/11550518_56.  Google Scholar [20] B. Düring and C.-B. Schönlieb, A high-contrast fourth-order PDE from imaging: Numerical solution by ADI splitting, in "Multi-scale and High-Contrast Partial Differential Equations" (eds. H. Ammari et al.), 93-103, Contemporary Mathematics 577, American Mathematical Society, Providence, (2012). doi: 10.1090/conm/577/11465.  Google Scholar [21] B. Düring, D. Matthes and J.-P. Milisic, A gradient flow scheme for nonlinear fourth order equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 935-959. doi: 10.3934/dcdsb.2010.14.935.  Google Scholar [22] C. M. Elliott and D. A. French, Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math., 38 (1987), 97-128. doi: 10.1093/imamat/38.2.97.  Google Scholar [23] C. M. Elliott and S. A. Smitheman, Analysis of the TV regularization and H-1 fidelity model for decomposing an image into cartoon plus texture, Commun. on Pure and Appl. Anal., 6 (2007), 917-936. doi: 10.3934/cpaa.2007.6.917.  Google Scholar [24] C. M. Elliott and S. A. Smitheman, Numerical analysis of the TV regularization and $H^{-1}$ fidelity model for decomposing an image into cartoon plus texture, IMA Journal of Numerical Analysis, (2008), 1-39. doi: 10.1093/imanum/drn025.  Google Scholar [25] S. Esedoglu and J.-H. Shen, Digital inpainting based on the Mumford-Shah-Euler image model, Eur. J. Appl. Math., 13 (2002), 353-370. doi: 10.1017/S0956792502004904.  Google Scholar [26] T. Goldstein and S. Osher, The split Bregman method for $L^1$ regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891.  Google Scholar [27] L. Gosse and G. Toscani, Lagrangian numerical approximations to one-dimensional convolution-diffusion equations, SIAM J. Sci. Comput., 28 (2006), 1203-1227. doi: 10.1137/050628015.  Google Scholar [28] K. J. in 't Hout and B. D. Welfert, Stability of ADI schemes applied to convection-diffusion equations with mixed derivative terms, Appl. Numer. Math., 57 (2007), 19-35. doi: 10.1016/j.apnum.2005.11.011.  Google Scholar [29] K. J. in 't Hout and B. D. Welfert, Unconditional stability of second-order ADI schemes appliet to multi-dimensional diffusion equations with mixed derivative terms, Appl. Numer. Math., 59 (2009), 677-692. doi: 10.1016/j.apnum.2008.03.016.  Google Scholar [30] P. J. van der Houwen and J. G. Verwer, One-step splitting methods for semi-discrete parabolic equations, Computing, 22 (1979), 291-309. doi: 10.1007/BF02265311.  Google Scholar [31] W. Hundsdorfer and J. G. Verwer, "Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations," Berlin: Springer-Verlag, 2003.  Google Scholar [32] W. Hundsdorfer, Accuracy and stability of splitting with Stabilizing Corrections, Appl. Numer. Math., 42 (2002), 213-233. doi: 10.1016/S0168-9274(01)00152-0.  Google Scholar [33] L. Lieu and L. Vese, Image restoration and decompostion via bounded total variation and negative Hilbert-Sobolev spaces, Applied Mathematics & Optimization, 58 (2008), 167-193. doi: 10.1007/s00245-008-9047-8.  Google Scholar [34] T. Lu, P. Neittaanmälki and X.-C. Tai, A parallel splitting up method and its application to Navier-Stokes equations, Appl. Math. Lett., 4 (1991), 25-29. doi: 10.1016/0893-9659(91)90161-N.  Google Scholar [35] T. Lu, P. Neittaanmälki and X.-C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations, RAIRO Modél. Math. Anal. Numér., 26 (1992), 673-708.  Google Scholar [36] O. M. Lysaker and X.-C. Tai, Iterative image restoration combining total variation minimization and a second-order functional, International Journal of Computer Vision, 66 (2006), 5-18. doi: 10.1007/s11263-005-3219-7.  Google Scholar [37] S. Masnou and J. Morel, Level Lines based Disocclusion, 5th IEEE Int'l Conf. on Image Processing, Chicago, IL, (1998), 259-263. doi: 10.1109/ICIP.1998.999016.  Google Scholar [38] Y. Meyer, "Oscillating Patterns in Image Processing and Nonlinear Evolution Equations," Univ. Lecture Ser., 22, AMS, Providence, RI, 2002.  Google Scholar [39] J. Müller, "Parallel Total Variation Minimization," Diploma Thesis, University of Münster, WWU, 2008. Google Scholar [40] D. Mumford and B. Gidas, Stochastic models for generic images, Quart. Appl. Math., 59 (2001), 85-111.  Google Scholar [41] T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462. doi: 10.1137/S003614459529284X.  Google Scholar [42] A. Novick-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation, Phys. D, 10 (1984), 277-298. doi: 10.1016/0167-2789(84)90180-5.  Google Scholar [43] A. Oron, S. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films, Phys. Rev. Let., 85 (2000), 2108-2111. doi: 10.1103/RevModPhys.69.931.  Google Scholar [44] S. Osher, A. Sole and L. Vese, Image decomposition and restoration using total variation minimization and the $H^{-1}$ norm, Multiscale Modelling and Simulation: A SIAM Interdisciplinary Journal, 1 (2003), 349-370. doi: 10.1137/S1540345902416247.  Google Scholar [45] D. W. Peaceman and H. H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Industr. Appl. Math, 3 (1955), 28-41. doi: 10.1137/0103003.  Google Scholar [46] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.  Google Scholar [47] L. Rudin and S. Osher, Total variation based image restoration with free local constraints, Proc. 1st IEEE ICIP, 1 (1994), 31-35. doi: 10.1109/ICIP.1994.413269.  Google Scholar [48] C.-B. Schönlieb and A. Bertozzi, Unconditionally stable schemes for higher order inpainting, Commun. Math. Sci., 9 (2011), 413-457.  Google Scholar [49] C.-B. Schönlieb, A. Bertozzi, M. Burger and L. He, Image inpainting using a fourth-order total variation flow, Proc. Int. Conf. SampTA09, Marseilles, (2009). Google Scholar [50] C.-B. Schönlieb, Total variation minimization with an $H^{-1}$ constraint, CRM Series 9, Singularities in Nonlinear Evolution Phenomena and Applications Proceedings, Scuola Normale Superiore Pisa, (2009), 201-232.  Google Scholar [51] P. Smereka, Semi-implicit level set methods for curvature and surface diffusion motion, Special Issue in Honor of the Sixtieth Birthday of Stanley Osher, J. Sci. Comput., 19 (2003), 439-456. doi: 10.1023/A:1025324613450.  Google Scholar [52] L. Vese, A study in the BV space of a denoising-deblurring variational problem, Appl. Math. Optim., 44 (2001), 131-161. doi: 10.1007/s00245-001-0017-7.  Google Scholar [53] L. Vese and S. Osher, Image denoising and decomposition with total variation minimization and oscillatory functions, J. Math. Imaging Vision, 20 (2004), 7-18. doi: 10.1023/B:JMIV.0000011316.54027.6a.  Google Scholar [54] J. Weickert, B. M. ter Haar Romeny and M. A. Viergever, Efficient and reliable schemes for nonlinear diffusion filtering, IEEE Transactions on Image Processing, 7 (1998), 398-410. doi: 10.1109/83.661190.  Google Scholar [55] T. P. Witelski and M. Bowen, ADI schemes for higher-order nonlinear diffusion equations, Appl. Numer. Math., 45 (2003), 331-351. doi: 10.1016/S0168-9274(02)00194-0.  Google Scholar

show all references

##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, Oxford, 2000.  Google Scholar [2] J.-F. Aujol and A. Chambolle, Dual norms and image decomposition models, International Journal of Computer Vision, 63 (2005), 85-104. doi: 10.1007/s11263-005-4948-3.  Google Scholar [3] J.-F. Aujol and G. Gilboa, Constrained and SNR-based solutions for TV-Hilbert space image denoising, Journal of Mathematical Imaging and Vision, 26 (2006), 217-237. doi: 10.1007/s10851-006-7801-6.  Google Scholar [4] D. Barash, M. Israeli and R. Kimmel, An accurate operator splitting scheme for nonlinear diffusion filtering, in "Scale-Space and Morphology in Computer Vision" (ed. M. Kerckhove), Lecture Notes in Computer Science 2106 (2006), 281-289. Google Scholar [5] M. Benning, "Singular Regularization of inverse Problems," PhD thesis, University of Münster, 2011. Google Scholar [6] M. Benning, L. Calatroni, B. Düring and C.-B. Schönlieb, A primal-dual approach for a total variation Wasserstein flow, to appear in GSI 2013 LNCS proceedings, Springer, (2013). Google Scholar [7] M. Burger, M. Franek and C.-B. Schönlieb, Regularised Regression and Density estimation based on Optimal Transport,, Appl. Math. Res. Express (AMRX), 2012 (): 209.   Google Scholar [8] M. Burger, L. He and C.-B. Schönlieb, Cahn-Hilliard inpainting and a generalization for grayvalue images, SIAM J. Imaging Sci., 2 (2009), 1129-1167. doi: 10.1137/080728548.  Google Scholar [9] T. F. Chan and J. J. Shen, "Image Processing and Analysis Variational, PDE, wavelet, and stochastic methods," SIAM, 2005. doi: 10.1137/1.9780898717877.  Google Scholar [10] A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vis., 20 (2004), 89-97. doi: 10.1023/B:JMIV.0000011321.19549.88.  Google Scholar [11] A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numer. Math. 76 (1997), 167-188. doi: 10.1007/s002110050258.  Google Scholar [12] T. F. Chan, S. H. Kang and J. Shen, Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., 63 (2002), 564-592. doi: 10.1137/S0036139901390088.  Google Scholar [13] T. F. Chan and P. Mulet, On the convergence of the lagged diffusivity fixed point method in total variation image restoration, SIAM Journal on Numerical Analysis, 36 (1999), 354-367. doi: 10.1137/S0036142997327075.  Google Scholar [14] T. F. Chan and J. Shen, Mathematical models for local non-texture inpaintings, SIAM J. Appl. Math., 62 (2001), 1019-1043. doi: 10.1137/S0036139900368844.  Google Scholar [15] T. F. Chan and J. Shen, Non-texture inpainting by curvature driven diffusions (CDD), J. Visual Comm. Image, Rep., 12 (2001), 436-449. Google Scholar [16] T. F. Chan and J. Shen, Variational image inpainting, Comm. Pure Applied Math, 58 (2005), 579-619. doi: 10.1002/cpa.20075.  Google Scholar [17] W. Chen and Z. Ditzian, Mixed and directional derivatives, Proceedings of the American Mathematical Society, 108 (1990), 177-185. doi: 10.1090/S0002-9939-1990-0994773-0.  Google Scholar [18] S. D. Conte and R. T. James, An alternating direction method for solving the biharmonic equation, Math. Tables Aids Comput., 12 (1958), 198-205. doi: 10.2307/2002021.  Google Scholar [19] S. Didas, J. Weickert and B. Burgeth, Stability and local feature enhancement of higher order nonlinear diffusion filtering, in "Pattern Recognition" (Eds. W. Kropatsch, R. Sablatnig and A. Hanbury), Lecture Notes in Computer Science, Springer, Berlin, 3663 (2005), 451-458. doi: 10.1007/11550518_56.  Google Scholar [20] B. Düring and C.-B. Schönlieb, A high-contrast fourth-order PDE from imaging: Numerical solution by ADI splitting, in "Multi-scale and High-Contrast Partial Differential Equations" (eds. H. Ammari et al.), 93-103, Contemporary Mathematics 577, American Mathematical Society, Providence, (2012). doi: 10.1090/conm/577/11465.  Google Scholar [21] B. Düring, D. Matthes and J.-P. Milisic, A gradient flow scheme for nonlinear fourth order equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 935-959. doi: 10.3934/dcdsb.2010.14.935.  Google Scholar [22] C. M. Elliott and D. A. French, Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math., 38 (1987), 97-128. doi: 10.1093/imamat/38.2.97.  Google Scholar [23] C. M. Elliott and S. A. Smitheman, Analysis of the TV regularization and H-1 fidelity model for decomposing an image into cartoon plus texture, Commun. on Pure and Appl. Anal., 6 (2007), 917-936. doi: 10.3934/cpaa.2007.6.917.  Google Scholar [24] C. M. Elliott and S. A. Smitheman, Numerical analysis of the TV regularization and $H^{-1}$ fidelity model for decomposing an image into cartoon plus texture, IMA Journal of Numerical Analysis, (2008), 1-39. doi: 10.1093/imanum/drn025.  Google Scholar [25] S. Esedoglu and J.-H. Shen, Digital inpainting based on the Mumford-Shah-Euler image model, Eur. J. Appl. Math., 13 (2002), 353-370. doi: 10.1017/S0956792502004904.  Google Scholar [26] T. Goldstein and S. Osher, The split Bregman method for $L^1$ regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891.  Google Scholar [27] L. Gosse and G. Toscani, Lagrangian numerical approximations to one-dimensional convolution-diffusion equations, SIAM J. Sci. Comput., 28 (2006), 1203-1227. doi: 10.1137/050628015.  Google Scholar [28] K. J. in 't Hout and B. D. Welfert, Stability of ADI schemes applied to convection-diffusion equations with mixed derivative terms, Appl. Numer. Math., 57 (2007), 19-35. doi: 10.1016/j.apnum.2005.11.011.  Google Scholar [29] K. J. in 't Hout and B. D. Welfert, Unconditional stability of second-order ADI schemes appliet to multi-dimensional diffusion equations with mixed derivative terms, Appl. Numer. Math., 59 (2009), 677-692. doi: 10.1016/j.apnum.2008.03.016.  Google Scholar [30] P. J. van der Houwen and J. G. Verwer, One-step splitting methods for semi-discrete parabolic equations, Computing, 22 (1979), 291-309. doi: 10.1007/BF02265311.  Google Scholar [31] W. Hundsdorfer and J. G. Verwer, "Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations," Berlin: Springer-Verlag, 2003.  Google Scholar [32] W. Hundsdorfer, Accuracy and stability of splitting with Stabilizing Corrections, Appl. Numer. Math., 42 (2002), 213-233. doi: 10.1016/S0168-9274(01)00152-0.  Google Scholar [33] L. Lieu and L. Vese, Image restoration and decompostion via bounded total variation and negative Hilbert-Sobolev spaces, Applied Mathematics & Optimization, 58 (2008), 167-193. doi: 10.1007/s00245-008-9047-8.  Google Scholar [34] T. Lu, P. Neittaanmälki and X.-C. Tai, A parallel splitting up method and its application to Navier-Stokes equations, Appl. Math. Lett., 4 (1991), 25-29. doi: 10.1016/0893-9659(91)90161-N.  Google Scholar [35] T. Lu, P. Neittaanmälki and X.-C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations, RAIRO Modél. Math. Anal. Numér., 26 (1992), 673-708.  Google Scholar [36] O. M. Lysaker and X.-C. Tai, Iterative image restoration combining total variation minimization and a second-order functional, International Journal of Computer Vision, 66 (2006), 5-18. doi: 10.1007/s11263-005-3219-7.  Google Scholar [37] S. Masnou and J. Morel, Level Lines based Disocclusion, 5th IEEE Int'l Conf. on Image Processing, Chicago, IL, (1998), 259-263. doi: 10.1109/ICIP.1998.999016.  Google Scholar [38] Y. Meyer, "Oscillating Patterns in Image Processing and Nonlinear Evolution Equations," Univ. Lecture Ser., 22, AMS, Providence, RI, 2002.  Google Scholar [39] J. Müller, "Parallel Total Variation Minimization," Diploma Thesis, University of Münster, WWU, 2008. Google Scholar [40] D. Mumford and B. Gidas, Stochastic models for generic images, Quart. Appl. Math., 59 (2001), 85-111.  Google Scholar [41] T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462. doi: 10.1137/S003614459529284X.  Google Scholar [42] A. Novick-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation, Phys. D, 10 (1984), 277-298. doi: 10.1016/0167-2789(84)90180-5.  Google Scholar [43] A. Oron, S. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films, Phys. Rev. Let., 85 (2000), 2108-2111. doi: 10.1103/RevModPhys.69.931.  Google Scholar [44] S. Osher, A. Sole and L. Vese, Image decomposition and restoration using total variation minimization and the $H^{-1}$ norm, Multiscale Modelling and Simulation: A SIAM Interdisciplinary Journal, 1 (2003), 349-370. doi: 10.1137/S1540345902416247.  Google Scholar [45] D. W. Peaceman and H. H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Industr. Appl. Math, 3 (1955), 28-41. doi: 10.1137/0103003.  Google Scholar [46] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.  Google Scholar [47] L. Rudin and S. Osher, Total variation based image restoration with free local constraints, Proc. 1st IEEE ICIP, 1 (1994), 31-35. doi: 10.1109/ICIP.1994.413269.  Google Scholar [48] C.-B. Schönlieb and A. 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