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A nonlinear multigrid method for total variation minimization from image restoration. (English) Zbl 1129.65046
Summary: Image restoration has been an active research topic and variational formulations are particularly effective in high quality recovery. Although there exist many modelling and theoretical results, available iterative solvers are not yet robust in solving such modeling equations. Recent attempts on developing optimisation multigrid methods have been based on first order conditions. Different from this idea, this paper proposes to use piecewise linear function spanned subspace correction to design a multilevel method for directly solving the total variation minimisation. Our method appears to be more robust than the primal-dual method by {\it T. F. Chan, G. H. Golub}, and {\it P. Mulet} [SIAM J. Sci. Comput. 20, No. 6, 1964--1977 (1999; Zbl 0929.68118)] previously found reliable. Supporting numerical results are presented.

65K10Optimization techniques (numerical methods)
49J20Optimal control problems with PDE (existence)
49M27Decomposition methods in calculus of variations
94A08Image processing (compression, reconstruction, etc.)
68U10Image processing (computing aspects)
65F10Iterative methods for linear systems
Full Text: DOI
[1] Acar, R., Vogel, C.R.: Analysis of total variation penalty method for ill-posed problems. Inverse Probl. 10, 1217--1229 (1994) · Zbl 0809.35151 · doi:10.1088/0266-5611/10/6/003
[2] Acton, S.T.: Multigrid anisotropic diffusion. IEEE Trans. Image Process. 3(3), 280--291 (1998) · doi:10.1109/83.661178
[3] Arian, E., Ta’asan, S.: Multigrid one-shot methods for optimal control problems. ICASE Technical Report No. 94-52, USA (1994)
[4] Blomgren, P., Chan, T.F., Mulet, P., Vese, L., Wan, W.L.: Variational PDE models and methods for image processing. In: Research Notes in Mathematics, vol. 420, pp. 43--67. Chapman & Hall/CRC (2000) · Zbl 0953.68621
[5] Brandt, A.: Multigrid solvers and multilevel optimization strategies. In: Cong, J., Shinnerl, J.R. (eds.) Multiscale Optimization and VLSI/CAD, pp. 1--68. Kluwer Academic, Boston (2000)
[6] Carter, J.L.: Dual method for total variation-based image restoration. CAM report 02-13, UCLA, USA; see http://www.math.ucla.edu/applied/cam/index.html . Ph.D. thesis, University of California, LA (2002)
[7] Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Image Vis. 20, 89--97 (2004) · Zbl 02060336 · doi:10.1023/B:JMIV.0000011321.19549.88
[8] Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167--188 (1997) · Zbl 0874.68299 · doi:10.1007/s002110050258
[9] Chan, R.H., Wong, C.K.: Sine transform based preconditioners for elliptic problems. Numer. Linear Algebra Appl. 4, 351--368 (1997) · Zbl 0889.65047 · doi:10.1002/(SICI)1099-1506(199709/10)4:5<351::AID-NLA103>3.0.CO;2-4
[10] Chan, R.H., Chan, T.F., Wan, W.L.: Multigrid for differential convolution problems arising from image processing. In: Chan R., Chan T.F., Golub G.H. (eds.) Proc. Sci. Comput. Workshop. Springer, see also CAM report 97-20, UCLA, USA (1997) · Zbl 0922.65097
[11] Chan, R.H., Chang, Q.S., Sun, H.W.: Multigrid method for ill-conditioned symmetric Toeplitz systems. SIAM J. Sci. Comput. 19, 516--529 (1998) · Zbl 0916.65029 · doi:10.1137/S1064827595293831
[12] Chan, R.H., Ho, C.W., Nikolova, M.: Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization. IEEE Trans. Image Process. 14, 1479--1485 (2005) · Zbl 05452810 · doi:10.1109/TIP.2005.852196
[13] Chan, T.F., Chen, K.: On a nonlinear multigrid algorithm with primal relaxation for the image total variation minimisation. Numer. Algorithms 41, 387--411 (2006) · Zbl 1096.94003 · doi:10.1007/s11075-006-9020-z
[14] Chan, T.F., Mulet, P.: Iterative methods for total variation restoration. CAM report 96-38, UCLA, USA; see http://www.math.ucla.edu/applied/cam/index.html (1996) · Zbl 0938.65155
[15] Chan, T.F., Tai, X.C.: Identification of discontinuous coefficient from elliptic problems using total variation regularization. SIAM J. Sci. Comput. 25(3), 881--904 (2003) · Zbl 1046.65090 · doi:10.1137/S1064827599326020
[16] Chan, T.F., Vese, L.: Image segmentation using level sets and the piecewise-constant Mumford-Shah model. UCLA CAM report CAM00-14, USA (2000)
[17] Chan, T.F., Golub, G.H., Mulet, P.: A nonlinear primal dual method for total variation based image restoration. SIAM J. Sci. Comput. 20(6), 1964--1977 (1999) · Zbl 0929.68118 · doi:10.1137/S1064827596299767
[18] Chang, Q.S., Chern, I.L.: Acceleration methods for total variation-based image denoising. SIAM J. Sci. Comput. 25, 982--994 (2003) · Zbl 1046.65048 · doi:10.1137/S106482750241534X
[19] Chen, K.: Matrix Preconditioning Techniques and Applications. Cambridge Monographs on Applied and Computational Mathematics, No. 19. Cambridge University Press, Cambridge (2005) · Zbl 1079.65057
[20] Frohn-Schauf, C., Henn, S., Witsch, K.: Nonlinear multigrid methods for total variation image denoising. Comput. Visual Sci. 7, 199--206 (2004) · Zbl 1071.65093
[21] Kelley, C.T.: Iterative Methods for Solving Linear and Nonlinear Equations. SIAM Publications (1995) · Zbl 0832.65046
[22] Kenigsberg, A., Kimmel, R., Yavneh, I.: A multigrid approach for fast geodesic active contours. CIS report 2004-06 (2004)
[23] Kimmel, R., Yavneh, I.: An algebraic multigrid approach for image analysis. SIAM J. Sci. Comput. 24(4), 1218--1231 (2003) · Zbl 1034.65049 · doi:10.1137/S1064827501389229
[24] Li, Y.Y., Santosa, F.: A computational algorithm for minimizing total variation in image restoration. IEEE Trans. Image Process 5(6), 987--995 (1996) · doi:10.1109/83.503914
[25] Lysaker, M., Lundervold, A., Tai, X.C.: Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Trans. Image Process. 12(12), 1579--1590 (2003) · Zbl 1286.94020 · doi:10.1109/TIP.2003.819229
[26] Malgouyres, F.: Minimizing the total variation under a general convex constraint for image restoration. IEEE Trans. Image Process. 11(12), 1450--1456 (2002) · Zbl 05453118 · doi:10.1109/TIP.2002.806241
[27] Nash, S.: A multigrid approach to discretized optimisation problems. J. Optim. Methods Softw. 14, 99--116 (2000) · Zbl 0988.90040 · doi:10.1080/10556780008805795
[28] Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2003) · Zbl 1026.76001
[29] 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(2), 387--405 (2000) · Zbl 0969.65081 · doi:10.1137/S1064827599351751
[30] Osher, S., Sethian, J.: Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12--49 (1988) · Zbl 0659.65132 · doi:10.1016/0021-9991(88)90002-2
[31] Rudin, L.I., 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
[32] Savage, J., Chen, K.: An improved and accelerated nonlinear multigrid method for total-variation denoising. Int. J. Comput. Math. 82(8), 1001--1015 (2005) · Zbl 1072.65096 · doi:10.1080/00207160500069904
[33] Stuben, K.: An introduction to algebraic multigrid. In: Trottenberg, U., Oosterlee, C.W., Schuller, A. (eds.) Multigrid (2000), Appendix A. Also appeared as GMD report 70 from http://www.gmd.de and http://publica.fhg.de/english/index.htm
[34] Ta’asan, S.: Multigrid one-shot methods and design strategy. Lecture Note 4 of Von-Karmen Institute Lectures, http://www.math.cmu.edu/$\sim$shlomo/VKI-Lectures/lecture4 (1997)
[35] Tai, X.C.: Rate of convergence for some constraint decomposition methods for nonlinear variational inequalities. Numer. Math. 93, 755--786 (2003). Available online at http://www.mi.uib.no/$\sim$tai · Zbl 1057.65040 · doi:10.1007/s002110200404
[36] Tai, X.C., Espedal, M.: Rate of convergence of some space decomposition methods for linear and nonlinear problems. SIAM J. Numer. Anal. 35, 1558--1570 (1998) · Zbl 0915.65063 · doi:10.1137/S0036142996297461
[37] Tai, X.C., Tseng, P.: Convergence rate analysis of an asynchronous space decomposition method for convex minimization. Math. Comput. 71, 1105--1135 (2001) · Zbl 0997.65088 · doi:10.1090/S0025-5718-01-01344-8
[38] Tai, X.C., Xu, J.C.: Global and uniform convergence of subspace correction methods for some convex optimization problems. Math. Comput. 71, 105--124 (2001) · Zbl 0985.65065 · doi:10.1090/S0025-5718-01-01311-4
[39] Trottenberg, U., Oosterlee, C.W., Schuller, A.: Multigrid. Academic, London (2001)
[40] Vogel, C.R.: A multigrid method for total variation-based image denoising. In: Bowers, K., Lund, J. (eds.) Computation and Control IV. Progress in Systems and Control Theory, vol. 20. Birkhäuser, Boston (1995) · Zbl 0831.93062
[41] Vogel, C.R.: Negative results for multilevel preconditioners in image deblurring. In: Nielson, M. (ed.) Scale-Space Theories in Computer Vision, pp. 292--304. Springer, New York (1999)
[42] Vogel, C.R.: Computational Methods for Inverse Problems. SIAM Publications (2002) · Zbl 1008.65103
[43] Vogel, C.R., Oman, M.E.: Iterative methods for total variation denoising. SIAM J. Sci. Stat. Comput. 17, 227--238 (1996) · Zbl 0847.65083 · doi:10.1137/0917016
[44] Vogel, C.R., Oman, M.E.: Fast, robust total variation-based reconstruction of noisy, blurred images. IEEE Trans. Image Process. 7, 813--824 (1998) · Zbl 0993.94519 · doi:10.1109/83.679423
[45] Wan, W.L., Chan, T.F., Smith, B.: An energy-minimizing interpolation for robust multigrid methods. SIAM J. Sci. Comput. 21(4), 1632--1649 (2000) · Zbl 0966.65098 · doi:10.1137/S1064827598334277
[46] Xu, J.C.: Iteration methods by space decomposition and subspace correction. SIAM Rev. 4, 581--613 (1992) · Zbl 0788.65037 · doi:10.1137/1034116
[47] Yip, A.M., Park, F.: Solution dynamics, causality, and critical behavior of the regularization parameter in total variation denoising problems. CAM report 03-59, UCLA, USA (on-line) (2003)