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May  2018, 17(3): 959-985. doi: 10.3934/cpaa.2018047

Stability of traveling waves of models for image processing with non-convex nonlinearity

Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA

* Corresponding author: Tong Li

Received  September 2017 Revised  September 2017 Published  January 2018

We establish the existence and stability of smooth large-amplitude traveling waves to nonlinear conservation laws modeling image processing with general flux functions. We innovatively construct a weight function in the weighted energy estimates to overcome the difficulties caused by the absence of the convexity of fluxes in our model. Moreover, we prove that if the integral of the initial perturbation decays algebraically or exponentially in space, the solution converges to the traveling waves with rates in time, respectively. Furthermore, we are able to construct another new weight function to deal with the degeneracy of fluxes in establishing the stability.

Citation: Tong Li, Jeungeun Park. Stability of traveling waves of models for image processing with non-convex nonlinearity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 959-985. doi: 10.3934/cpaa.2018047
References:
[1]

M. Bertalmio, A. Bertozzi and G. Sapiro, Navier-stokes, fluid dynamics and image and video inpainting, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Kayai, HI, (2001), 355–362. Google Scholar

[2]

M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting, in Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, ACM Press/Addison-Wesley, New York, (2000), 417–424. Google Scholar

[3]

J. Canny, A computational approach to edge detection, IEEE Trans. Pattern Anal. Machine Intell., PAMI-8 (1986), 679-698.   Google Scholar

[4]

J. GoodmanA. Kurganov and P. Rosenau, Breakdown in Burgers-type equations with saturating dissipation fluxes, Nonlinearity, 12 (1999), 247-268.   Google Scholar

[5]

J. B. Greer and A. L. Bertozzi, Traveling wave solutions of fourth order PDEs for image processing, SIAM J. Math. Anal., 36 (2004), 38-68.   Google Scholar

[6]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127.   Google Scholar

[7]

S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Commun. Pure Appl. Math., 47 (1994), 1547-1569.   Google Scholar

[8]

J. J. Koenderink, The structure of image, Bio. Cybernet, 50 (1984), 363-370.   Google Scholar

[9]

A. KurganovD. Levy and P. Rosenau, On Burgers-Type equations with nonmonotonic dissipative fluxes, Commun. Pure Appl. Math., 51 (1998), 0443-0473.   Google Scholar

[10]

T-P. Liu and L-A Ying, Nonlinear stability of strong detonations for a viscous combustion model, SIAM J. Math. Anal., 26 (1995), 519-528.   Google Scholar

[11]

T. Li, Rigorous asymptotic stability of a Chapman-Jouguet detonation wave in the limit of small resolved heat release, Combust. Theory Model, 1 (1997), 259-270.   Google Scholar

[12]

T. Li, Stability of strong detonation waves and rates of convergence, Electron. J. Differential Equations, 1998 (1998), 1-17.   Google Scholar

[13]

T. Li, Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and diffusion, SIAM J. Math. Anal., 40 (2008), 1058-1075.   Google Scholar

[14]

T. Li and H. Liu, Stability of a traffic flow model with nonconvex relaxation, Commun. Math. Sci., 3 (2005), 101-118.   Google Scholar

[15]

T. Li and J. Park, Stability of traveling wave solutions of nonlinear conservation laws for image processing, Commun. Math. Sci., 15 (2017), 1073-1106.   Google Scholar

[16]

D. Marr and E. Hildreth, Theory of edge detection, Proc. Roy. Soc. London Ser. B, 207 (1980), 187-217.   Google Scholar

[17]

A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Commun. Math. Phys., 165 (1994), 83-96.   Google Scholar

[18]

M. Mei, Remark on stability of shock profile for nonconvex scalar viscous conservation laws, Bull. Inst. Math. Acad. Sin., 27 (1999), 213-226.   Google Scholar

[19]

M. Mei and T. Yang, Convergence rates to traveling waves for a nonconvex relaxation model, Proc. Roy. Soc. Edinburgh, 128A (1998), 1053-1068.   Google Scholar

[20]

T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. D'osay 78-02, Départment de Mathématique, Université de Paris-Sud, Orsay, France, 78 (1978).  Google Scholar

[21]

M. Nishikata, Convergence rate to the traveling wave for viscous conservation laws, Funkcialoj Ekvacioj, 41 (1998), 107-132.   Google Scholar

[22]

J. Pan and H. Liu, Convergence rates to traveling waves for viscous conservation laws with dispersion, J. Diff. Eqs., 187 (2003), 337-358.   Google Scholar

[23]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE. Trans. Pattern Anal. Machine Intell., 12 (1990), 629-639.   Google Scholar

[24]

J. Tumblin and G. Turk, A boundary hierarchy for detail-preserving contrast reduction, in Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, ACM Press/Addison-Wesley, New York, (1990), 83–90. Google Scholar

[25]

A. Witkin, Scale-space filtering, in Proceedings of the International Joint Conference on Artificial Intelligence, Karlsruhe, Germany, (1983), 1019–1021. Google Scholar

[26]

Y. Wu, The stability of traveling fronts for some quasilinear Burgers-type equations, Adv. Math. (China), 31 (2002), 363-371.   Google Scholar

[27]

Y.-L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process, 9 (2000), 1723-1730.   Google Scholar

show all references

References:
[1]

M. Bertalmio, A. Bertozzi and G. Sapiro, Navier-stokes, fluid dynamics and image and video inpainting, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Kayai, HI, (2001), 355–362. Google Scholar

[2]

M. Bertalmio, G. Sapiro, V. Caselles and C. Ballester, Image inpainting, in Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, ACM Press/Addison-Wesley, New York, (2000), 417–424. Google Scholar

[3]

J. Canny, A computational approach to edge detection, IEEE Trans. Pattern Anal. Machine Intell., PAMI-8 (1986), 679-698.   Google Scholar

[4]

J. GoodmanA. Kurganov and P. Rosenau, Breakdown in Burgers-type equations with saturating dissipation fluxes, Nonlinearity, 12 (1999), 247-268.   Google Scholar

[5]

J. B. Greer and A. L. Bertozzi, Traveling wave solutions of fourth order PDEs for image processing, SIAM J. Math. Anal., 36 (2004), 38-68.   Google Scholar

[6]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127.   Google Scholar

[7]

S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Commun. Pure Appl. Math., 47 (1994), 1547-1569.   Google Scholar

[8]

J. J. Koenderink, The structure of image, Bio. Cybernet, 50 (1984), 363-370.   Google Scholar

[9]

A. KurganovD. Levy and P. Rosenau, On Burgers-Type equations with nonmonotonic dissipative fluxes, Commun. Pure Appl. Math., 51 (1998), 0443-0473.   Google Scholar

[10]

T-P. Liu and L-A Ying, Nonlinear stability of strong detonations for a viscous combustion model, SIAM J. Math. Anal., 26 (1995), 519-528.   Google Scholar

[11]

T. Li, Rigorous asymptotic stability of a Chapman-Jouguet detonation wave in the limit of small resolved heat release, Combust. Theory Model, 1 (1997), 259-270.   Google Scholar

[12]

T. Li, Stability of strong detonation waves and rates of convergence, Electron. J. Differential Equations, 1998 (1998), 1-17.   Google Scholar

[13]

T. Li, Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and diffusion, SIAM J. Math. Anal., 40 (2008), 1058-1075.   Google Scholar

[14]

T. Li and H. Liu, Stability of a traffic flow model with nonconvex relaxation, Commun. Math. Sci., 3 (2005), 101-118.   Google Scholar

[15]

T. Li and J. Park, Stability of traveling wave solutions of nonlinear conservation laws for image processing, Commun. Math. Sci., 15 (2017), 1073-1106.   Google Scholar

[16]

D. Marr and E. Hildreth, Theory of edge detection, Proc. Roy. Soc. London Ser. B, 207 (1980), 187-217.   Google Scholar

[17]

A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Commun. Math. Phys., 165 (1994), 83-96.   Google Scholar

[18]

M. Mei, Remark on stability of shock profile for nonconvex scalar viscous conservation laws, Bull. Inst. Math. Acad. Sin., 27 (1999), 213-226.   Google Scholar

[19]

M. Mei and T. Yang, Convergence rates to traveling waves for a nonconvex relaxation model, Proc. Roy. Soc. Edinburgh, 128A (1998), 1053-1068.   Google Scholar

[20]

T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. D'osay 78-02, Départment de Mathématique, Université de Paris-Sud, Orsay, France, 78 (1978).  Google Scholar

[21]

M. Nishikata, Convergence rate to the traveling wave for viscous conservation laws, Funkcialoj Ekvacioj, 41 (1998), 107-132.   Google Scholar

[22]

J. Pan and H. Liu, Convergence rates to traveling waves for viscous conservation laws with dispersion, J. Diff. Eqs., 187 (2003), 337-358.   Google Scholar

[23]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE. Trans. Pattern Anal. Machine Intell., 12 (1990), 629-639.   Google Scholar

[24]

J. Tumblin and G. Turk, A boundary hierarchy for detail-preserving contrast reduction, in Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, ACM Press/Addison-Wesley, New York, (1990), 83–90. Google Scholar

[25]

A. Witkin, Scale-space filtering, in Proceedings of the International Joint Conference on Artificial Intelligence, Karlsruhe, Germany, (1983), 1019–1021. Google Scholar

[26]

Y. Wu, The stability of traveling fronts for some quasilinear Burgers-type equations, Adv. Math. (China), 31 (2002), 363-371.   Google Scholar

[27]

Y.-L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process, 9 (2000), 1723-1730.   Google Scholar

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