# zbMATH — the first resource for mathematics

Bregman operator splitting with variable stepsize for total variation image reconstruction. (English) Zbl 1290.90071
Summary: This paper develops a Bregman operator splitting algorithm with variable stepsize (BOSVS) for solving problems of the form $$\min\{\phi(Bu) +1/2\|Au-f\|_{2}^{2}\}$$, where $$\phi$$ may be nonsmooth. The original Bregman operator splitting (BOS) algorithm employs a fixed stepsize, while BOSVS uses a line search to achieve better efficiency. These schemes are applicable to total variation (TV)-based image reconstruction. The stepsize rule starts with a Barzilai-Borwein (BB) step, and increases the nominal step until a termination condition is satisfied. The stepsize rule is related to the scheme used in SpaRSA (sparse reconstruction by separable approximation). Global convergence of the proposed BOSVS algorithm to a solution of the optimization problem is established. BOSVS is compared with other operator splitting schemes using partially parallel magnetic resonance image reconstruction problems. The experimental results indicate that the proposed BOSVS algorithm is more efficient than the BOS algorithm and another split Bregman Barzilai-Borwein algorithm known as SBB.

##### MSC:
 90C30 Nonlinear programming
RecPF
Full Text:
##### References:
 [1] Baraniuk, R., Steeghs, P.: Compressive radar imaging. In: 2007 IEEE Radar Conference, pp. 128–133 (2007) [2] 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 [3] Block, K., Uecker, M., Frahm, J.: Undersampled radial MRI with multiple coils: iterative image reconstruction using a total variation constraint. Magn. Reson. Med. 57, 1086–1098 (2007) · doi:10.1002/mrm.21236 [4] Chan, T.F., Golub, G.H., Mulet, P.: A nonlinear primal-dual method for total variationbased image restoration. SIAM J. Optim. 20, 1964–1977 (1999) · Zbl 0929.68118 [5] Chen, Y., Hager, W.W., Huang, F., Phan, D.T., Ye, X., Yin, W.: Fast algorithms for image reconstruction with application to partially parallel MR imaging. SIAM J. Imaging Sci. 5, 90–118 (2012) · Zbl 1247.90212 · doi:10.1137/100792688 [6] Darbon, J., Sigelle, M.: A fast and exact algorithm for total variation minimization. In: Lecture Notes in Comput. Sci. IbPRIA, vol. 3522, pp. 351–359. Springer, Berlin (2005) [7] Donoho, D.: Compressed sensing. IEEE Trans. Inf. Theory 52, 351–359 (2006) · Zbl 1288.94016 [8] Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrange Methods: Applications to the Solution of Boundary-Valued Problems, pp. 299–331. North Holland, Amsterdam (1983) [9] Goldstein, T., Osher, S.: The split Bregman method for L1 regularized problems. SIAM J. Imaging Sci. 2, 323–343 (2009) · Zbl 1177.65088 · doi:10.1137/080725891 [10] Hager, W.W., Phan, D.T., Zhang, H.: Gradient-based methods for sparse recovery. SIAM J. Imaging Sci. 4, 146–165 (2011) · Zbl 1209.90266 · doi:10.1137/090775063 [11] Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303–320 (1969) · Zbl 0174.20705 · doi:10.1007/BF00927673 [12] Huang, Y., Ng, M.K., Wen, Y.: A new total variation method for multiplicative noise removal. SIAM J. Imaging Sci. 2, 20–40 (2009) · Zbl 1187.68655 · doi:10.1137/080712593 [13] Li, Y., Santosa, F.: An affine scalling algorithm for minimizing total variation in image enhancement. Tech. Rep. TR94-1470, Cornell Theory Center, Cornell University, Ithaca, NY (1994) [14] Lustig, M., Donoho, D., Pauly, J.M.: Sparse MRI: the application of compressed sensing for rapid MR imaging. Magn. Reson. Med. 58, 1182–1195 (2007) · doi:10.1002/mrm.21391 [15] Pruessmann, K., Weiger, M., Bornert, P., Boesiger, P.: Advances in sensitivity encoding with arbitrary k-space trajectories. Magn. Reson. Med. 46, 638–651 (2001) · doi:10.1002/mrm.1241 [16] Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) · Zbl 0193.18401 [17] Rudin, L., Osher, S., Fatemi, E.: Non-linear total variation noise removal algorithm. Physica D 60, 259–268 (1992) · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F [18] Strong, D., Chan, T.: Edge preserving and scale-dependent properties of total variation regularization. Inverse Probl. 19, 165–187 (2003) · Zbl 1043.94512 [19] Vogel, C.R.: A multigrid method for total variation-based image denoising. In: Bowers, K., Lund, J. (eds.) Computation and Control IV. Progess in Systems and Control Theory, vol. 20, pp. 323–331. Birkhauser, Boston (1995) · Zbl 0831.93062 [20] 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 [21] Wang, Y., Yang, J., Yin, W., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci. 1, 248–272 (2008) · Zbl 1187.68665 · doi:10.1137/080724265 [22] Wright, S.J., Nowak, R.D., Figueiredo, M.A.T.: Sparse reconstruction by separable approximation. IEEE Trans. Signal Process. 57, 2479–2493 (2009) · Zbl 1391.94442 · doi:10.1109/TSP.2009.2016892 [23] Yang, J., Zhang, Y., Yin, W.: A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data. IEEE J. Sel. Top. Signal Process. 4, 288–297 (2010) · doi:10.1109/JSTSP.2010.2042333 [24] Ye, X., Chen, Y., Huang, F.: Computational acceleration for MR image reconstruction in partially parallel imaging. IEEE Trans. Med. Imaging 30, 1055–1063 (2011) · doi:10.1109/TMI.2010.2073717 [25] Zhang, X., Burger, M., Bresson, X., Osher, S.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. SIAM J. Imaging Sci. 3, 253–276 (2011) · Zbl 1191.94030 · doi:10.1137/090746379 [26] Zhang, X., Burger, M., Osher, S.: A unified primal-dual algorithm framework based on Bregman iteration. J. Sci. Comput. 46, 20–46 (2011) · Zbl 1227.65052 · doi:10.1007/s10915-010-9408-8 [27] Zhu, M., Chan, T.: An efficient primal-dual hybrid gradient algorithm for total variation image restoration. Tech. Rep. 08-34, CAM UCLA (2008) [28] Zhu, M., Wright, S., Chan, T.: Duality-based algorithms for total-variation-regularized image restoration. Comput. Optim. Appl. 47, 377–400 (2010) · Zbl 1208.90165 · doi:10.1007/s10589-008-9225-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.