Abstract
A qualitative comparison of total variation like penalties (total variation, Huber variant of total variation, total generalized variation, . . .) is made in the context of global seismic tomography. Both penalized and constrained formulations of seismic recovery problems are treated. A number of simple iterative recovery algorithms applicable to these problems are described. The convergence speed of these algorithms is compared numerically in this setting. For the constrained formulation a new algorithm is proposed and its convergence is proven.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beck A., Teboulle M.: A fast iterative shrinkage-threshold algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183–202 (2009). doi:10.1137/080716542
Bredies K., Kunisch K., Pock T.: Total generalized variation. SIAM J. Imaging Sci. 3, 492–526 (2010). doi:10.1137/090769521
Candes E., Demanet L., Donoho D., Ying L.: Fast discrete curvelet transforms. Multiscale Model. Simul. 5(3), 861–899 (2006). doi:10.1137/05064182X
Chambolle, A.: Total variation minimization and a class of binary MRF models. In: Energy Minimization Methods in Computer Vision and Pattern Recognition. Lecture Notes in Computer Science, vol. 3757, pp. 136–152. Springer, Berlin (2005)
Chambolle A., Lions P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 167–188 (1997). doi:10.1007/s002110050258
Chambolle A., Pock T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120–145 (2011). doi:10.1007/s10851-010-0251-1
Chan T.F., Marquina A., Mulet P.: Higher order total variation-based image restoration. SIAM J. Sci. Comput. 22, 503–516 (2000). doi:10.1137/S1064827598344169
Chen S.S., Donoho D.L., Saunders M.A.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20(1), 33–61 (1998). doi:10.1137/S1064827596304010
Cohen A., Daubechies I., Feauveau J.: Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45, 485–560 (1992). doi:10.1002/cpa.3160450502
Combettes, P.L., Pesquet, J.-C.: Proximal splitting methods in signal processing. In: Bauschke, H.H., Burachik, R.S., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 185–212. Springer, Berlin (2011)
Daubechies I., Defrise M., De Mol C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57(11), 1413–1457 (2004). doi:10.1002/cpa.20042
Esser E., Zhang X., Chan T.F.: A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM J. Imaging Sci. 3(4), 1015–1046 (2010). doi:10.1137/09076934X
Gholami A., Siahkoohi H.R.: Regularization of linear and non-linear geophysical ill-posed problems with joint sparsity constraints. Geophys. J. Int. 180(2), 871–882 (2010). doi:10.1111/j.1365-246X.2009.04453.x
Goldstein T., Osher S.: The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci. 2, 323–343 (2009). doi:10.1137/080725891
Hennenfent G., van Berg E., Friedlander M.P., Herrmann F.J.: New insights into one-norm solvers from the Pareto curve. Geophysics 73(4), A23–A26 (2008). doi:10.1190/1.2944169
Herrmann F.J., Hennenfent G.: Non-parametric seismic data recovery with curvelet frames. Geophys. J. Int. 173(1), 233–248 (2008). doi:10.1111/j.1365-246X.2007.03698.x
Huber P.J.: Robust estimation of a location parameter. Ann. Stat. 53, 73–101 (1964). doi:10.1214/aoms/1177703732
Labate, D., Lim, W., Kutyniok, G., Weiss, G.: Sparse multidimensional representation using shearlets. In: SPIE Proc. 5914, SPIE, Bellingham, WA, 2005. Wavelets XI (San Diego, CA, 2005), pp. 254–262. doi:10.1117/12.613494
Loris I., Verhoeven C.: On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty. Inverse Probl. 27, 125007 (2011). doi:10.1088/0266-5611/27/12/125007
Loris I., Nolet G., Daubechies I., Dahlen F.A.: Tomographic inversion using ℓ1-norm regularization of wavelet coefficients. Geophys. J. Int. 170(1), 359–370 (2007). doi:10.1111/j.1365-246X.2007.03409.x
Lukic T., Lindblad J., Sladoje N.: Regularized image denoising based on spectral gradient optimization. Inverse Probl. 27(8), 085010 (2011). doi:10.1088/0266-5611/27/8/085010
Mallat S.: A Wavelet Tour of Signal Processing: The Sparse Way, 3rd edn. Academic Press, New York (2009)
Nesterov Y.E.: A method for solving a convex programming problem with convergence rate \({\mathcal{O}(1/k^2)}\) . Soviet Math. Dokl. 27(2), 372–376 (1983)
Ronchi C., Iacono R., Paolucci P.: The “cubed sphere”: a new method for the solution of partial differential equations in spherical geometry. J. Comput. Phys. 124, 93–114 (1996). doi:10.1006/jcph.1996.0047
Rudin L.I., Osher S., Fatemi E.: Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena 60(1–4), 259–268 (1992). doi:10.1016/0167-2789(92)90242-F
Setzer S., Steidl G., Teuber T.: Infimal convolution regularizations with discrete ℓ1-type functionals. Commun. Math. Sci. 9(3), 797–827 (2011)
Simons F.J., Loris I., Nolet G., Daubechies I.C., Voronin S., Judd J.S., Vetter P.A., CharlTty J., Vonesch C.: Solving or resolving global tomographic models with spherical wavelets, and the scale and sparsity of seismic heterogeneity. Geophys. J. Int. 187(2), 969–988 (2011). doi:10.1111/j.1365-246X.2011.05190.x
Strong D., Chan T.: Edge-preserving and scale-dependent properties of total variation regularization. Inverse Probl. 19(6), S165 (2003). doi:10.1088/0266-5611/19/6/059
Trampert J., Woodhouse J.H.: Global phase-velocity maps of Love and Rayleigh-waves between 40 and 150 seconds. Geophys. J. Int. 122(2), 675–690 (1995). doi:10.1111/j.1365-246X.1995.tb07019.x
Trampert J., Woodhouse J.H.: High resolution global phase velocity distributions. Geophys. Res. Lett. 23(1), 21–24 (1996). doi:10.1029/95GL03391
Trampert J., Woodhouse J.H.: Assessment of global phase velocity models. Geophys. J. Int. 144(1), 165–174 (2001). doi:10.1046/j.1365-246x.2001.00307.x
van Berg E., Friedlander M.P.: Sparse optimization with least-squares constraints. SIAM J. Optim. 21(4), 1201–1229 (2011). doi:10.1137/100785028
Yamagishi M., Yamada I.: Over-relaxation of the fast iterative shrinkage-thresholding algorithm with variable stepsize. Inverse Probl. 27(10), 105008 (2011). doi:10.1088/0266-5611/27/10/105008
Zhang X., Burger M., Osher S.: A unified primal-dual algorithm framework based on Bregman iteration. J. Sci. Comput. 46, 20–46 (2011). doi:10.1007/s10915-010-9408-8
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Loris, I., Verhoeven, C. Iterative algorithms for total variation-like reconstructions in seismic tomography. Int J Geomath 3, 179–208 (2012). https://doi.org/10.1007/s13137-012-0036-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13137-012-0036-3