Analysis of total variation flow and its finite element approximations

Xiaobing Feng; Andreas Prohl

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 37, Issue: 3, page 533-556
  • ISSN: 0764-583X

Abstract

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We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter ε, see (1.7)) and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since our approach is constructive and variational, finite element methods can be naturally applied to approximate weak solutions of the limiting gradient flow problem. We propose a fully discrete finite element method and establish convergence to the regularized gradient flow problem as h,k → 0, and to the total variation gradient flow problem as h,k,ε → 0 in general cases. Provided that the regularized gradient flow problem possesses strong solutions, which is proved possible if the datum functions are regular enough, we establish practical a priori error estimates for the fully discrete finite element solution, in particular, by focusing on the dependence of the error bounds on the regularization parameter ε. Optimal order error bounds are derived for the numerical solution under the mesh relation k = O(h2). In particular, it is shown that all error bounds depend on 1 ε only in some lower polynomial order for small ε.

How to cite

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Feng, Xiaobing, and Prohl, Andreas. "Analysis of total variation flow and its finite element approximations." ESAIM: Mathematical Modelling and Numerical Analysis 37.3 (2010): 533-556. <http://eudml.org/doc/194177>.

@article{Feng2010,
abstract = { We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter ε, see (1.7)) and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since our approach is constructive and variational, finite element methods can be naturally applied to approximate weak solutions of the limiting gradient flow problem. We propose a fully discrete finite element method and establish convergence to the regularized gradient flow problem as h,k → 0, and to the total variation gradient flow problem as h,k,ε → 0 in general cases. Provided that the regularized gradient flow problem possesses strong solutions, which is proved possible if the datum functions are regular enough, we establish practical a priori error estimates for the fully discrete finite element solution, in particular, by focusing on the dependence of the error bounds on the regularization parameter ε. Optimal order error bounds are derived for the numerical solution under the mesh relation k = O(h2). In particular, it is shown that all error bounds depend on $\frac\{1\}\{\varepsilon\}$ only in some lower polynomial order for small ε. },
author = {Feng, Xiaobing, Prohl, Andreas},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Bounded variation; gradient flow; variational inequality; equations of prescribed mean curvature and minimal surface; fully discrete scheme; finite element method.; fully discrete scheme; finite element method},
language = {eng},
month = {3},
number = {3},
pages = {533-556},
publisher = {EDP Sciences},
title = {Analysis of total variation flow and its finite element approximations},
url = {http://eudml.org/doc/194177},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Feng, Xiaobing
AU - Prohl, Andreas
TI - Analysis of total variation flow and its finite element approximations
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 3
SP - 533
EP - 556
AB - We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter ε, see (1.7)) and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since our approach is constructive and variational, finite element methods can be naturally applied to approximate weak solutions of the limiting gradient flow problem. We propose a fully discrete finite element method and establish convergence to the regularized gradient flow problem as h,k → 0, and to the total variation gradient flow problem as h,k,ε → 0 in general cases. Provided that the regularized gradient flow problem possesses strong solutions, which is proved possible if the datum functions are regular enough, we establish practical a priori error estimates for the fully discrete finite element solution, in particular, by focusing on the dependence of the error bounds on the regularization parameter ε. Optimal order error bounds are derived for the numerical solution under the mesh relation k = O(h2). In particular, it is shown that all error bounds depend on $\frac{1}{\varepsilon}$ only in some lower polynomial order for small ε.
LA - eng
KW - Bounded variation; gradient flow; variational inequality; equations of prescribed mean curvature and minimal surface; fully discrete scheme; finite element method.; fully discrete scheme; finite element method
UR - http://eudml.org/doc/194177
ER -

References

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  1. L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. The Clarendon Press Oxford University Press, New York (2000).  
  2. F. Andreu, C. Ballester, V. Caselles and J.M. Mazón, The Dirichlet problem for the total variation flow. J. Funct. Anal.180 (2001) 347–403.  
  3. F. Andreu, C. Ballester, V. Caselles and J.M. Mazón, Minimizing total variation flow. Differential Integral Equations14 (2001) 321–360.  
  4. F. Andreu, V. Caselles, J.I. Díaz and J.M. Mazón, Some qualitative properties for the total variation flow. J. Funct. Anal.188 (2002) 516–547.  
  5. G. Bellettini and V. Caselles, The total variation flow in RN. J. Differential Equations (accepted).  
  6. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Springer-Verlag, New York, 2nd ed. (2002).  
  7. H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam, North-Holland Math. Stud., No. 5. Notas de Matemática (50) (1973).  
  8. E. Casas, K. Kunisch and C. Pola, Regularization by functions of bounded variation and applications to image enhancement. Appl. Math. Optim.40 (1999) 229–257.  
  9. A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems. Numer. Math.76 (1997) 167–188.  
  10. T. Chan and J. Shen, On the role of the BV image model in image restoration. Tech. Report CAM 02-14, Department of Mathematics, UCLA (2002).  
  11. T.F. Chan, G.H. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration. SIAM J. Sci. Comput.20 (1999) 1964–1977 (electronic).  
  12. P.G. Ciarlet, The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam, Stud. Math. Appl.4 (1978).  
  13. M.G. Crandall and T.M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces. Amer. J. Math.93 (1971) 265–298.  
  14. D.C. Dobson and C.R. Vogel, Convergence of an iterative method for total variation denoising. SIAM J. Numer. Anal.34 (1997) 1779–1791.  
  15. C. Gerhardt, Boundary value problems for surfaces of prescribed mean curvature. J. Math. Pures Appl.58 (1979) 75–109.  
  16. C. Gerhardt, Evolutionary surfaces of prescribed mean curvature. J. Differential Equations36 (1980) 139–172.  
  17. D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin (2001). Reprint of the 1998 ed.  
  18. E. Giusti, Minimal surfaces and functions of bounded variation. Birkhäuser Verlag, Basel (1984).  
  19. R. Hardt and X. Zhou, An evolution problem for linear growth functionals. Comm. Partial Differential Equations19 (1994) 1879–1907.  
  20. C. Johnson and V. Thomée, Error estimates for a finite element approximation of a minimal surface. Math. Comp.29 (1975) 343–349.  
  21. A. Lichnewsky and R. Temam, Pseudosolutions of the time-dependent minimal surface problem. J. Differential Equations30 (1978) 340–364.  
  22. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod (1969).  
  23. R. Rannacher, Some asymptotic error estimates for finite element approximation of minimal surfaces. RAIRO Anal. Numér.11 (1977) 181–196.  
  24. L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms. Phys. D60 (1992) 259–268.  
  25. J. Simon, Compact sets in the space Lp(0,T;B). Ann. Mat. Pura Appl.146 (1987) 65–96.  
  26. M. Struwe, Applications to nonlinear partial differential equations and Hamiltonian systems, in Variational methods. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics (Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics), Vol. 34. Springer-Verlag, Berlin, 3rd ed. (2000).  

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