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A dual algorithm for minimization of the LLT model. (English) Zbl 1170.94006
Summary: We apply the dual algorithm of Chambolle for the minimization of the LLT model. A convergence theorem is given for the proposed algorithm. The algorithm overcomes the numerical difficulties related to the non-differentiability of the LLT model. The dual algorithm is faster than the original gradient descent algorithm. Numerical experiments are supplied to demonstrate the efficiency of the algorithm.
MSC:
94A08Image processing (compression, reconstruction, etc.)
References:
[1]Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithm. Physica D 60, 259–268 (1992) · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[2]Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: partial differential equations and the calculus of variations. Applied Mathematical Science, vol. 147, 2nd edn. Springer (2006)
[3]Andreu, F., Ballester, C., Caselles, V., Mazon, J.M.: Minimizing total variation flow. CRAS I-Mathématique 331(11), 867–872 (2000)
[4]Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 167–188 (1997) · Zbl 0874.68299 · doi:10.1007/s002110050258
[5]Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problem. Inverse Probl. 10, 1217–1229 (1994) · Zbl 0809.35151 · doi:10.1088/0266-5611/10/6/003
[6]Vese, L.: A study in the BV space of a denosing–deblurring variationl problem. Appl. Math. Optim. 44, 131–161 (2001) · Zbl 1003.35009 · doi:10.1007/s00245-001-0017-7
[7]Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89–97 (2004) · Zbl 02060335 · doi:10.1023/B:JMIV.0000011320.81911.38
[8]Blomgern, P., Chan, T.F., Mulet, P., Wong, C.: Total variation image restoration: Numerical methods and extensions. In: Proceedings, IEEE International Conference on Image Processing, III, pp. 384–387 (1997)
[9]Nikolova, M.: Weakly constrained minimization: application to the estimation of images and signals involving constant regions. J. Math. Imaging Vis. 21, 155–175 (2004) · doi:10.1023/B:JMIV.0000035180.40477.bd
[10]Chan, T.F., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000) · Zbl 0968.68175 · doi:10.1137/S1064827598344169
[11]You, Y.-L., Kaveh, M.: Fourth-order partial differential equation for noise removal. IEEE Trans. Image Process. 9(10), 1723–1730 (2000) · Zbl 0962.94011 · doi:10.1109/83.869184
[12]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 05453109 · doi:10.1109/TIP.2003.819229
[13]Osher, S., Sole, A., Vese, L.: Image decomposition and restoration using total variation minimization and the H 1 norm. Multiscale Model. Simul. 1(3), 349–370 (2003) · Zbl 1051.49026 · doi:10.1137/S1540345902416247
[14]Didas, S., Burgeth, B., Imiya, A., Weickert, J.: Regularity and scale- space properties of fractional high order linear filtering. In: Kimmel, R., Sochen, N., Weickert, J. (eds.) Scale-space and PDE Methods in Computer Vision. Lecture Notes in Computer Science, vol. 3459. Springer, Berlin (2005)
[15]Didas, S., Weickert, J., Burgeth, B.: Stability and local feature enhancement of higher order nonlinear diffusion filtering. In: Kropatsch, W., Sablatnig, R., Hanbury, A. (eds.) Pattern Recognition. Lecture Notes in Computer Science, vol. 3663, pp. 451–458. Springer, Berlin (2005)
[16]Osher, S., Scherzer, O.: G-norm properties of bounded variation regularization. Comm. Math. Sci. 2(2), 237–254 (2004)
[17]Obereder, A., Osher, S., Scherzer, O.: On the use of dual norms in bounded variation type regularization. UCLA cam report, cam-04-35. Available from: http://www.math.ucla.edu/applied/cam/ (2004)
[18]Fang, L., Shen, C., Fan, J., Shen, C.: Image restoration combining a total variational filter and a fourth-order filter. J. Vis. Commun. Image Represent. 18(4), 322–330 (2007) · Zbl 05461584 · doi:10.1016/j.jvcir.2006.10.003
[19]Bertsekas, D.P.: Convex Analysis and Optimization. Tsinghua University Press (2006)
[20]Yuan, Y.X.: Numerical Method For Nonlinear Programming. Shanghai Scientific and Technical Publishers (1992)
[21]Liu, Q., Yao, Z.G., Ke, Y.: Entropy solutions for a fourth-order nonlinear degenerate problem for noise removal. Nonlinear Anal. Theory Methods Appl. 67(6), 1908–1918 (2007) · Zbl 1119.35031 · doi:10.1016/j.na.2006.08.016
[22]Liu, Q., Yao, Z.G., Ke, Y.: Solutions of fourth-order partial differential equations in a noise removal model. Electron. J. Diff. Equ. 2007(120), 11 (2007)
[23]Chan, T.F., Esedoglu, S., Park, F.: Image decomposition combining staircase reduction and texture extraction. J. Vis. Commun. Image Represent. 18(6), 464–486 (2007) · Zbl 05461772 · doi:10.1016/j.jvcir.2006.12.004