Infimal convolution regularisation functionals of BV and \(\mathrm{L}^p\) spaces. I: The finite \(p\) case. (English) Zbl 1342.49014
Summary: We study a general class of infimal convolution type regularization functionals suitable for applications in image processing. These functionals incorporate a combination of the total variation seminorm and \(\mathrm {L}^{p}\) norms. A unified well-posedness analysis is presented and a detailed study of the one-dimensional model is performed, by computing exact solutions for the corresponding denoising problem and the case \(p=2\). Furthermore, the dependence of the regularization properties of this infimal convolution approach on the choice of \(p\) is studied. It turns out that in the case \(p=2\) this regularizer is equivalent to the Huber-type variant of total variation regularization. We provide numerical examples for image decomposition as well as for image denoising. We show that our model is capable of eliminating the staircasing effect, a well-known disadvantage of total variation regularization. Moreover, as \(p\) increases we obtain almost piecewise affine reconstructions, leading also to a better preservation of hat-like structures.
MSC:
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 49M30 | Other numerical methods in calculus of variations (MSC2010) |
| 68U10 | Computing methodologies for image processing |
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
Keywords:
regularization functionals; infimal convolution; total variation distance; image denoising; image decomposition; staircasing; \(\mathrm {L}^{p}\)-normsSoftware:
CVXReferences:
| [1] | Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Science Publications, Oxford (2000) · Zbl 0957.49001 |
| [2] | Attouch, H., Brezis, H.: Duality for the sum of convex functions in general Banach spaces. North-Holland Mathematical Library 34, 125-133 (1986) · Zbl 0642.46004 · doi:10.1016/S0924-6509(09)70252-1 |
| [3] | Benning, M., Brune, C., Burger, M., Müller, J.: Higher-order TV methods—enhancement via Bregman iteration. J. Sci. Comput. 54(2-3), 269-310 (2013). doi:10.1007/s10915-012-9650-3 · Zbl 1308.94012 · doi:10.1007/s10915-012-9650-3 |
| [4] | Bergounioux, M., Piffet, L.: A second-order model for image denoising. Set-Valued Var. Anal. 18(3-4), 277-306 (2010). doi:10.1007/s11228-010-0156-6 · Zbl 1203.94006 · doi:10.1007/s11228-010-0156-6 |
| [5] | Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492-526 (2010). doi:10.1137/090769521 · Zbl 1195.49025 · doi:10.1137/090769521 |
| [6] | Bredies, K., Kunisch, K., Valkonen, T.: Properties of L \[^11\]-TGV \[^{\,2}2\]: the one-dimensional case. J. Math. Anal. Appl. 398(1), 438-454 (2013). doi:10.1016/j.jmaa.2012.08.053 · Zbl 1253.49024 · doi:10.1016/j.jmaa.2012.08.053 |
| [7] | Burger, M.; Osher, S., A guide to the TV zoo, 1-70 (2013), Berlin · Zbl 1342.94014 · doi:10.1007/978-3-319-01712-9_1 |
| [8] | Burger, M., Papafitsoros, K., Papoutsellis, E., Schönlieb, C.-B.: Infimal convolution regularisation functionals of \[{\rm BV}\] BV and \[{\rm L}^p\] Lp spaces. The case \[p=\infty\] p=∞. Submitted (2015) · Zbl 1342.49014 |
| [9] | Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numerische Mathematik 76, 167-188 (1997). doi:10.1007/s002110050258 · Zbl 0874.68299 · doi:10.1007/s002110050258 |
| [10] | Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120-145 (2011). doi:10.1007/s10851-010-0251-1 · Zbl 1255.68217 · doi:10.1007/s10851-010-0251-1 |
| [11] | Chan, T., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503-516 (2001). doi:10.1137/S1064827598344169 · Zbl 0968.68175 · doi:10.1137/S1064827598344169 |
| [12] | Chan, T.F., Esedoglu, S., Park, F.E.: Image decomposition combining staircase reduction and texture extraction. J. Vis. Commun. Image Represent. 18(6), 464-486 (2007). doi:10.1016/j.jvcir.2006.12.004 · doi:10.1016/j.jvcir.2006.12.004 |
| [13] | Chen, C., He, B., Ye, Y., Yuan, X.: The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent, Math. Program. 1-23 (2014) · Zbl 1332.90193 |
| [14] | Yuan, X., Dai, Y., Han, D., Zhang, W.: A sequential updating scheme of Lagrange multiplier for separable convex programming, Math. Comput., to appear (2013) · Zbl 1348.90520 |
| [15] | Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. SIAM, Philadelphia (1999) · Zbl 0939.49002 · doi:10.1137/1.9781611971088 |
| [16] | Elion, C., Vese, L.: An image decomposition model using the total variation and the infinity Laplacian. Proc. SPIE 6498, 64980W-64980W-10 (2007). doi:10.1117/12.716079 · doi:10.1117/12.716079 |
| [17] | Esser, E.: Applications of Lagrangian-based alternating direction methods and connections to split Bregman. CAM Rep. 9, 31 (2009) |
| [18] | Goldstein, T., Osher, S.: The split Bregman algorithm method for \[L_1\] L1-regularized problems. SIAM J. Imaging Sci. 2, 323-343 (2009). doi:10.1137/080725891 · Zbl 1177.65088 · doi:10.1137/080725891 |
| [19] | Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.1, (2014) · Zbl 1187.35280 |
| [20] | Hansen, P.: Discrete inverse problems. Soc. Ind. Appl. Math. (2010). doi:10.1137/1.9780898718836 · Zbl 1197.65054 |
| [21] | He, B., Hou, L., Yuan, X.: On full jacobian decomposition of the augmented lagrangian method for separable convex programming, preprint (2013) · Zbl 1327.90209 |
| [22] | He, B., Tao, M., Yuan, X.: Convergence rate and iteration complexity on the alternating direction method of multipliers with a substitution procedure for separable convex programming, Math. Oper. Res., under revision 2 (2012) |
| [23] | Hintermüller, M., Stadler, G.: An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration. SIAM J. Sci. Comput. 28(1), 1-23 (2006). doi:10.1137/040613263 · Zbl 1136.94302 · doi:10.1137/040613263 |
| [24] | Huber, P.J.: Robust regression: asymptotics, conjectures and monte carlo. Ann. Stat. 1, 799-821 (1973). http://www.jstor.org/stable/2958283 · Zbl 0289.62033 |
| [25] | Kim, Y., Vese, L.: Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability. Inverse Probl Imaging 3, 43-68 (2009). doi:10.3934/ipi.2009.3.43 · Zbl 1187.35280 · doi:10.3934/ipi.2009.3.43 |
| [26] | Kuijper, A.: P-Laplacian driven image processing. IEEE Int. Conf. Image Process. 5, 257-260 (2007). doi:10.1109/ICIP.2007.4379814 · doi:10.1109/ICIP.2007.4379814 |
| [27] | Lefkimmiatis, S., Bourquard, A., Unser, M.: Hessian-based norm regularization for image restoration with biomedical applications. IEEE Trans. Image Process. 21, 983-995 (2012). doi:10.1109/TIP.2011.2168232 · Zbl 1372.94145 · doi:10.1109/TIP.2011.2168232 |
| [28] | Lindqvist, P.: Notes on the \[p\] p-Laplace equation, University of Jyvaskyla (2006) · Zbl 1256.35017 |
| [29] | 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). doi:10.1109/TIP.2003.819229 · Zbl 1286.94020 · doi:10.1109/TIP.2003.819229 |
| [30] | Lysaker, M., Tai, X.C.: Iterative image restoration combining total variation minimization and a second-order functional. Int. J. Comput. Vis. 66(1), 5-18 (2006). doi:10.1007/s11263-005-3219-7 · Zbl 1286.94021 · doi:10.1007/s11263-005-3219-7 |
| [31] | Nien, H., Fessler, J.A.: A convergence proof of the split Bregman method for regularized least-squares problems, arXiv:1402.4371, preprint (2014) · Zbl 1195.49025 |
| [32] | Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation based image restoration. SIAM Multisc. Model. Simul. 4, 460-489 (2005). doi:10.1137/040605412 · Zbl 1090.94003 · doi:10.1137/040605412 |
| [33] | Papafitsoros, K., Bredies, K.: A study of the one dimensional total generalised variation regularisation problem. Inverse Probl. Imaging 9, 511-550 (2015). doi:10.3934/ipi.2015.9.511 · Zbl 1336.49044 · doi:10.3934/ipi.2015.9.511 |
| [34] | Papafitsoros, K., Schönlieb, C.B.: A combined first and second order variational approach for image reconstruction. J. Math. Imaging Vis. 48(2), 308-338 (2014). doi:10.1007/s10851-013-0445-4 · Zbl 1362.94009 · doi:10.1007/s10851-013-0445-4 |
| [35] | Papafitsoros, K., Valkonen, T.: Asymptotic behaviour of total generalised variation. Scale Space and Variational Methods in Computer Vision 9087, 702-714 (2015) · Zbl 1444.94020 |
| [36] | Papoutsellis, E.: First-order gradient regularisation methods for image restoration. reconstruction of tomographic images with thin structures and denoising piecewise affine images, Ph.D. thesis, University of Cambridge, Cambridge (2015) · Zbl 0847.65083 |
| [37] | Ring, W.: Structural properties of solutions to total variation regularisation problems. ESAIM Math. Model. Numer. Anal. 34, 799-810 (2000). doi:10.1051/m2an:2000104 · Zbl 1018.49021 · doi:10.1051/m2an:2000104 |
| [38] | Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D: Nonlinear Phenom. 60, 259-268 (1992). doi:10.1016/0167-2789(92)90242-F · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F |
| [39] | Vese, L.: A study in the BV space of a denoising-deblurring variational problem. Appl. Math. Optim. 44(2), 131-161 (2001) · Zbl 1003.35009 · doi:10.1007/s00245-001-0017-7 |
| [40] | Vogel, C., Oman, M.: Iterative methods for total variation denoising. SIAM J. Sci. Comput. 17(1), 227-238 (1996). doi:10.1137/0917016 · Zbl 0847.65083 · doi:10.1137/0917016 |
| [41] | Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13, 600-612 (2004). doi:10.1109/TIP.2003.819861 · doi:10.1109/TIP.2003.819861 |
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