A modified TV-Stokes model for image processing

SIAM Journal on Scientific Computing

Volumn 33, Issue 4, 2011, Pages 1574-1597

  Litvinov, William G ^a   Rahman, Talal ^b   Tai, Xue Cheng ^c  

^a UNIVERSITY OF AUGSBURG   (Germany)

^b BERGEN UNIVERSITY COLLEGE   (Norway)

^c UNIVERSITY OF BERGEN   (Norway)

Author keywords

Image inpainting; Image restoration; Total variation

Indexed keywords

RESTORATION;

DIVERGENCE FREE; GEOMETRIC INFORMATION; IMAGE INPAINTING; IMAGES PROCESSING; MINIMIZATION PROBLEMS; MODEL-BASED OPC; STOKES MODEL; TOTAL VARIATION MINIMIZATION; TOTAL-VARIATION; TWO-STEP MODEL;

IMAGE RECONSTRUCTION;

EID: 80052744038     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/080727506     Document Type: Article

References (33)

1. R. ACAR AND C. R. VOGEL, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10(1994), pp. 1217-1229.
2. L. AMBROSIO, N FUSCO, AND D. PALLARA, Functions of Bounded Variation and Free Discontinuity Problems, The Caledron Press, Oxford University Press, New York, 2000.
3. D. N. ARNOLD, F. BREZZI, AND M. FORTIN, A stable finite element for the Stokes equations, Calcolo, 21(1984), pp. 337-344.
4. G. AUBERT AND P. KORNPROBST, Mathematical problems in image processing: Partial differential equations and the calculus of variations, Appl. Math. Sci. 147, Springer-Verlag, New York, 2002.
5. M. S. BELONOSOV AND W. G. LITVINOV, Finite element method for nonlinearly viscous fluids, Z. Angew. Math. Mech., 76(1996), pp. 307-320.
6. M. BERTALMIO, A. L. BERTOZZI, AND G. SAPIRO, Navier-Stokes, fluid dynamics and image and video inpainting, in Proc. Conf. Comp. Vision Pattern Rec., 1(2001), pp. 355-362.
7. A. BUADES, B. COLL, AND J. M. MOREL, A review of image denoising algorithms, with a new one, Multiscale Model. Simul., 4(2005), pp. 490-530. (Pubitemid 43890582)
8. P. BURCHARD, T. TASDIZEN, R. WHITAKER, AND S. OSHER, Geometric surface processing via normal maps, Technical report 02-3, Los Angeles, CA, UCLA, 2002.
9. A. CHAMBOLLE, An algorithm for total variation minimization and applications, J. Math. Imaging. Vision, 20(2004), pp. 89-97.
10. A. CHAMBOLLE AND P. L. LIONS, Image recovery via total variation minimization and related problems, Numer. Math., 76(1997), pp. 167-188.
11. T. F. CHAN AND J. SHEN, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, Philadelphia, 2005.
12. T. F. CHAN, A. MARQUINA, AND P. MULLET, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22(2000), pp. 503-516.
13. T. F. CHAN, J. SHEN, AND H. M. ZHOU, Total variation wavelet inpainting, J. Math. Imaging Vision, 25(2006), pp. 107-125. (Pubitemid 44544743)
14. P. CIARLET, Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
15. F. DONG, Z. LIU, D. KONG, AND K. LIU, An improved LOT model for image restoration, J. Math. Imaging Vision, 34(2009), pp. 89-97.
16. V. DUVAL, J.-F. AUJOL, AND Y. GOUSSEAU, The TVL1 model: A geometric point of view, Multiscale Model. Simul., 8(2009), pp. 154-189.
17. C. ELO, A. MALYSHEV, AND T. RAHMAN, A dual formulation of the TV-Stokes algorithm for image denoising, Lecture Notes in Comput. Sci. 5567, Springer-Verlag, New York, 2009, pp. 307-318.
18. S. FUČIK, A. KRATOCHVIL, AND J. NEČAS, Kachanov-Galerkin method, Comment. Math. Univ. Carolin., 14(1973), pp. 651-659.
19. V. GIRAULT AND P. A. RAVIART, Finite Element Approximation of the Navier-Stokes Equations, Springer, New York, 1981.
20. E. GIUSTI, Minimal surfaces and functions of bounded variation, Monographs in Mathematics 80, A. Borel, J. Moser, and S.-T. Yau, eds., Birkhäuser, Boston, Cambridge, MA, 1984.
21. C. KENNY AND J. LANGAN, A New Image Processing Primitive: Reconstructing Images from Modified Flow Fields, University of California, Santa Barbara, CA, 1999, pp. 1-10.
22. J.-L. LIONS, Problèmes aux Limites non Hmogènes at Applications, Dunod, Paris, 1968.
23. W. G. LITVINOV, Optimization in elliptic problems with applications to mechanics of deformable bodies and fluid mechanics, Monographs in Mathematics, Birkhäuser, Boston, Cambridge, MA, 2000.
24. W. G. LITVINOV, Problem on stationary flow of electrorheological fluids at the generalized conditions of slip on the boundary, Commun. Pure Appl. Anal., 6(2007), pp. 349-370.
25. Q. LIU, Z. YAO, AND Y. KE, Entropy solutions for a fourth-order nonlinear degenerate problem for noise removal, Nonlinear Anal., 67(2007), pp. 1908-1918. (Pubitemid 46634851)
26. A. LUNDERVOLD, M. LYSAKER, AND X. C. TAI, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, Technical report 02-44, Los Angeles, CA, UCLA, 2002.
27. M. LYSAKER, S. OSHER, AND X. C. TAI, Noise removal using smoothed normals and surface fitting, IEEE Trans. Image Process., 13(2004), pp. 1345-1357.
28. S. OSHER, A. SOLE, AND L. VESE, Image decomposition and restoration using total variation minimization and the H-1 norm, Multiscale Model. Simul., 1(2003), pp. 1579-1590.
29. T. RAHMAN, X. C. TAI, AND S. OSHER, A TV-Stokes denoising algorithm, Lecture Notes in Comput. Sci. 4485, F. Sgallari et al., eds., Springer-Verlag, New York, 2007, pp. 473-483. (Pubitemid 350269819)
30. L. I. RUDIN, S. OSHER, AND E. FATEMI, Nonlinear total variation based noise removal algorithms, Phys. D., 60(1992), pp. 259-268.
31. C. SHANNON AND W. WEAVER, The Mathematical Theory of Communication, University of Illinois Press, Champaign, IL, 1998.
32. X. C. TAI, S. OSHER, AND R. HOLM, Image inpainting using TV-Stokes equation, in Image Processing Based on Partial Differential Equations, Springer, New York, 2006.
33. Y.-L. YOU, AND M. KAVEH, Fourth-order partial differential equation for noise removal, IEEE Trans. Image Process., 9(2000), pp. 1723-1730.