The Split Bregman method is an effective method for solving the problem with L1 regularization. Chen and et al. improved the split Bregman method by adopting the linearization technique, variate and nonmonotone step size, and proposed a new split Bregman method with variate step size (BOSVS). The BOSVS can be used to solve image deblurring and denoising problem with Gaussian noise, and its numerical results are satisfactory. However, it can not be used to solve image deblurring and denoising problem with impulse noise. In this paper, we improve the BOSVS and propose a new split Bregman method with variate step size for solving image deblurring and denoising problem with impulse noise. On one hand, it retains the advantages of BOSVS such as linearization technique, variate and nonmonotone step size; on the other hand, by adding a L1 regularization term to the objective function, the model can deal with Gaussian noise as well as impulse noise. Therefore, the new method has wider range of application than the BOSVS. Preliminary experimental results show that, compared with an efficient method for the same problem, the new method can obtain a better result, and its computation time and algorithm efficiency are also competitive.
. A NEW SPLIT BREGMAN METHOD WITH VARIATE STEP FOR IMAGE DEBLURRING AND DENOISING WITH IMPULSE NOISE[J]. Journal of Numerical Methods and Computer Applicat, 2018, 39(1): 44-59.
Rudin L, Osher S, Fatemi E. Nolinear total variation based noise removal algorithms[J]. Physica D, 1992, 60(1-4): 259-268.
Chan T F, Mulet P. On the convergence of the lagged diffusivity fixed point method in total variation image restoration[J]. SIAM J. Numer. Anal., 1999, 36(2): 354-367.
Wang Y L, Yang J F, Yin W T, Zhang Y. A new alternating minimization algorithm for total variation image reconstruction[J]. SIAM J. Imaging Sci., 2008, 1(3): 284-272.
Yang J F, Zhang Y, Yin W T. An efficient TVL1 algorithm for deblurring multichannel images corrupted by implulsive noise[J]. SIAM J. Sci. Comput., 2009, 31(4): 2842-2865.
Tao M, Yang J F. Alernating direction algorithms for total variation deconvolution in image reconstruction[R]. Rice CAAM tech report TR0918, 2009.
Chambolle A, Pock T. A First-Order Primal-Dual algorithm for convex problems with applications to imaging[J]. J. Math. Imaging Vis., 2011, 40(1): 120-145.
Li C B, Yin W T, Zhang Y. Tv minimization by augmented lagrangian and alternating direction algorithms[R]. Technical report, Department of CAAM, Rice University, Houston, Texas, 77005, 2011. http://www.caam.rice.edu/optimization/L1/TVAL3/.
Osher S, Burger M, Goldfarb D, Xu J J, Yin W T. An iterative regularization method for total variation-based image restoration[J]. Multiscale Model. Sim., 2005, 4(2): 460-489.
Goldstein T, Osher S. The split Bregman method for L1-regularized problems[J]. SIAM J. Imaging Sci., 2009, 2(2): 323-343.
Chen Y M, Hager W W, Yashtini M, Ye X J, Zhang H C. Bregman operator splitting with variable stepsize for total variation image reconstruction[J]. Comput. Optim. Appl., 2013, 54(2): 317-342.
Zhang H C, Hager W W. A nonmonotone line search technique and its application to unconstrained optimization[J]. SIAM J. Optim., 2006, 14(4): 1043-1056.
Hestenes M R. Multiplier and gradient methods[J]. J. Optim. Theory Appl., 1969, 4(5): 303-320.
Barzilai J, Borwein J M. Two-point step size gradient methods[J]. Ima J. Numer. Anal., 1988, 8(1): 141-148.
Rockafellar R T. Convex Analysis[M]. Princeton University Press, 1970.
Cai J F, Chan R H, Nikolova M. Two phase methods for deblurring images corrupted by impulse plus Gaussian noise[J]. Inverse Probl. Imag., 2010, 2(2): 187-204.
Chan R H, Ho C W, Nikolova M. Salt-and-Pepper noise removal by median-type noise detectors and detail-preserving regularization[J]. IEEE T. Image Process., 2005, 14(10): 1479-1485.