zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Duality-based algorithms for total-variation-regularized image restoration. (English) Zbl 1208.90165
Summary: Image restoration models based on total variation (TV) have become popular since their introduction by {\it L. I. Rudin, S. Osher} and {\it E. Fatemi} [Physica D 60, No. 1--4, 259--268 (1992; Zbl 0780.49028)]. The dual formulation of this model has a quadratic objective with separable constraints making projections onto the feasible set easy to compute. This paper proposes the application of gradient projection (GP) algorithms to the dual formulation. We test variants of GP with different step length selection and line search strategies including techniques based on the Barzilai-Borwein method. Global convergence can in some cases be proved by appealing to the existing theory. We also propose a sequential quadratic programming (SQP) approach that takes account of the curvature of the boundary of the dual feasible set. Computational experiments show that the proposed approaches perform well in a wide range of applications and that some are significantly faster than previously proposed methods, particularly when only modest accuracy in the solution is required.

MSC:
90C30Nonlinear programming
90C52Methods of reduced gradient type
90C55Methods of successive quadratic programming type
Software:
CONV_QP; GPDT
WorldCat.org
Full Text: DOI
References:
[1] Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8, 141--148 (1988) · Zbl 0638.65055 · doi:10.1093/imanum/8.1.141
[2] Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Nashua (1999)
[3] Birgin, E.G., Martínez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10(4), 1196--1211 (2000) · Zbl 1047.90077 · doi:10.1137/S1052623497330963
[4] Carter, J.L.: Dual method for total variation-based image restoration. Report 02-13, UCLA CAM (2002)
[5] Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89--97 (2004) · Zbl 02060335 · doi:10.1023/B:JMIV.0000011320.81911.38
[6] Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 167--188 (1997) · Zbl 0874.68299 · doi:10.1007/s002110050258
[7] Chan, T.F., Zhu, M.: Fast algorithms for total variation-based image processing. In: Proceedings of the 4th ICCM. Hangzhuo, China (2007)
[8] Chan, T.F., Golub, G.H., Mulet, P.: A nonlinear primal-dual method for total variation based image restoration. SIAM J. Sci. Comput. 20, 1964--1977 (1999) · Zbl 0929.68118 · doi:10.1137/S1064827596299767
[9] Chan, T.F., Esedoglu, S., Park, F., Yip, A.: Total variation image restoration: Overview and recent developments. In: Paragios, N., Chen, Y., Faugeras, O. (eds.) Handbook of Mathematical Models in Computer Vision. Springer, Berlin (2005)
[10] Dai, Y.H., Fletcher, R.: Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming. Numer. Math. 100, 21--47 (2005) · Zbl 1068.65073 · doi:10.1007/s00211-004-0569-y
[11] Dai, Y.H., Hager, W.W., Schittkowski, K., Zhang, H.: The cyclic Barzilai-Borwein method for unconstrained optimization. IMA J. Numer. Anal. 26, 604--627 (2006) · Zbl 1147.65315 · doi:10.1093/imanum/drl006
[12] Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. SIAM Classics in Applied Mathematics. SIAM, Philadelphia (1999) · Zbl 0939.49002
[13] Goldfarb, D., Yin, W.: Second-order cone programming methods for total variation-based image restoration. SIAM J. Sci. Comput. 27, 622--645 (2005) · Zbl 1094.68108 · doi:10.1137/040608982
[14] Hintermüller, M., Stadler, G.: An infeasible primal-dual algorithm for TV-based inf-convolution-type image restoration. SIAM J. Sci. Comput. 28, 1--23 (2006) · Zbl 1136.94302 · doi:10.1137/040613263
[15] Hiriart-Urruty, J., Lemaréchal, C.: Convex Analysis and Minimization Algorithms, vol. I. Springer, Berlin (1993) · Zbl 0795.49001
[16] Osher, S., Marquina, A.: Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal. SIAM J. Sci. Comput. 22, 387--405 (2000) · Zbl 0969.65081 · doi:10.1137/S1064827599351751
[17] Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259--268 (1992) · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[18] Serafini, T., Zanghirati, G., Zanni, L.: Gradient projection methods for large quadratic programs and applications in training support vector machines. Optim. Methods Softw. 20(2--3), 353--378 (2004) · Zbl 1072.90026 · doi:10.1080/10556780512331318182
[19] Vogel, C.R., Oman, M.E.: Iterative methods for total variation denoising. SIAM J. Sci. Comput. 17, 227--238 (1996) · Zbl 0847.65083 · doi:10.1137/0917016
[20] Wang, Y., Ma, S.: Projected Barzilai-Borwein methods for large-scale nonnegative image restoration. Inverse Probl. Sci. Eng. 15(6), 559--583 (2007) · Zbl 1202.94077 · doi:10.1080/17415970600881897