Total bounded variation regularization as a bilaterally constrained optimization problem

SIAM Journal on Applied Mathematics

Volumn 64, Issue 4, 2004, Pages 1311-1333

  Hintermuller M ^a   Kunisch, K ^b  

^a Rice University   (United States)

^b UNIVERSITY OF GRAZ   (Austria)

Author keywords

Box constraints; Image reconstruction; Predual; Semismooth Newton methods; Total bounded variation

Indexed keywords

BOX CONSTRAINTS; IMAGE CONSTRUCTIONS; PREDUAL; SEMISMOOTH NEWTON METHODS; TOTAL BOUNDED VARIATION;

BOUNDARY CONDITIONS; CONVERGENCE OF NUMERICAL METHODS; INTEGRATION; MATHEMATICAL OPERATORS; OPTIMIZATION; SET THEORY; TOPOLOGY;

CONSTRAINT THEORY;

EID: 4544249333     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139903422784     Document Type: Article

References (26)

1. R. ACAR AND C. VOGEL, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10 (1994). pp. 1217-1229.
2. G. AUBERT AND L. VESE, A variational method in image recovery, SIAM J. Numer. Anal., 34 (1997), pp. 1948-1979.
3. V. BARBU AND T. PRECUPANU, Convexity and Optimization in Banach Spaces, 2nd rev. and extended ed., Mathematics and Its Applications (East European Series) 10, translated from the Romanian, D. Reidel, Dordrecht/Boston/Lancaster, Editura Academiei, Bucuresti, 1986.
4. P. CARBONNIER, L. BLAUD-FÉRAND, G. AUBERT, AND M. BARLAUD, Deterministic edge-preserving regularisation in computed imaging. IEEE Trans. Image Process., 6 (1997), pp. 298-311.
5. E. CASAS, K. KUNISCH, AND C. POLA, Regularization by functions of bounded variation and applications to image enhancement, Appl. Math. Optim., 40 (1999), pp. 229-257.
6. F. CATTÉ, P.-L. LIONS, J.-M. MOREL, AND T. COLL, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29 (1992), pp. 182-193.
7. A. CHAMBOLLE AND P.-L. LIONS, Image recovery via total variation minimization and related problems. Numer. Math., 76 (1997), pp. 167-188.
8. 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), pp. 1964-1977.
9. G. CHAVENT AND K. KUNISCH, Regularization of linear least squares problems by total bounded variation, ESAIM Control Optim. Calc. Var., 2 (1997), pp. 359-376.
10. R. DAUTRAY AND J.-L. LIONS, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3: Spectral Theory and Applications, with the collaboration of Michel Artola and Michel Cessenat, translated from the French by John C. Amson, Springer-Verlag, Berlin, 2000.
11. P. L. DAVIES AND A. KOVAC, Local extremes, runs, strings and multiresolution, Ann. Statist., 29 (2001), pp. 1-65.
12. D. C. DOBSON AND F. SANTOSA, Recovery of blocky images from noisy and blurred data, SIAM J. Appl. Math., 56 (1996), pp. 1181-1198.
13. I. EKELAND AND R. TÉMAM, Convex Analysis and Variational Problems, (corrected republicaticn of the 1976 English original) Classics in Appl. Math. 28, SIAM, Philadelphia, 1999.
14. D. GEMAN AND C. YANG. Nonlinear image recovery with half-quadratic regularization, IEEE Trans. Image Process., 4 (1995), pp. 932-945.
15. V. GIRAULT AND P.-A. RAVIART, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Ser. Comput. Math. 5, Springer-Verlag, Berlin, 1986.
16. E. GIUSTI, Minimal Surfaces and Functions of Bounded Variation, Monogr. Math. 80, Birkhäuser, Boston, Basel, Stuttgart, 1984.
17. M. HINTERMÜLLER, K. ITO, AND K. KUNISCH, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim., 13 (2003), pp. 865-888.
18. K. ITO AND K. KUNISCH, An active set strategy based on the augmented Lagrangian formulation for image restoration, M2AN Math. Model. Numer. Anal., 33 (1999). pp. 1-21.
19. K. ITO AND K. KUNISCH, BV-type regularization methods for convoluted objects with edge, flat and grey scales, Inverse Problems, 16 (2000), pp. 909-928.
20. F. MALGOUYRES AND F. GUICHARD, Edge direction preserving image zooming: A mathematical and numerical analysis, SIAM J. Numer. Anal., 39 (2001), pp. 1-37.
21. E. MAMMEN AND S. VAN DE GEER, Locally adaptive regression splines, Ann. Statist., 25 (1997), pp. 387-413.
22. L. Q. QI AND J. SUN, A nonsmooth version of Newton's method, Math. Programming, 58 (1993), pp. 353-367.
23. L. I. RUDIN, S. OSHER, AND E. FATEMI, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259-268.
24. M. ULBRICH, Semismooth Newton methods for operator equations in function spaces, SIAM J. Optim., 13 (2003), pp. 805-841.
25. C. R. VOGEL, Computational Methods for Inverse Problems, Frontiers Appl. Math. 23, SIAM, Philadelphia, 2002.
26. C. R. VOGEL AND M. E. OMAN, Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE Trans. Image Process., 7 (1998), pp. 813-824.