American Institute of Mathematical Sciences

Advanced
May  2014, 8(2): 507-535. doi: 10.3934/ipi.2014.8.507

A stable method solving the total variation dictionary model with $L^\infty$ constraints

Liyan Ma 1, , Lionel Moisan 2, , Jian Yu 3, and  Tieyong Zeng 4,

1. 

Institute of Microelectronics, Chinese Academy of Sciences, Beijing, China
2. 

MAP5, Université Paris Descartes, Paris, 75006, France
3. 

School of Computer and Information Technology, Beijing Jiaotong University, Beijing, China
4. 

Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China

Received  November 2011 Revised  May 2013 Published  May 2014

Image restoration plays an important role in image processing, and numerous approaches have been proposed to tackle this problem. This paper presents a modified model for image restoration, that is based on a combination of Total Variation and Dictionary approaches. Since the well-known TV regularization is non-differentiable, the proposed method utilizes its dual formulation instead of its approximation in order to exactly preserve its properties. The data-fidelity term combines the one commonly used in image restoration and a wavelet thresholding based term. Then, the resulting optimization problem is solved via a first-order primal-dual algorithm. Numerical experiments demonstrate the good performance of the proposed model. In a last variant, we replace the classical TV by the nonlocal TV regularization, which results in a much higher quality of restoration.
Keywords: nonlocal total variation, wavelet packet decomposition., Image restoration, total variation, proximal gradient method.
Mathematics Subject Classification: 52A41, 65F22, 65K10, 65K05, 68U10, 90C25, 90C4.
Citation: Liyan Ma, Lionel Moisan, Jian Yu, Tieyong Zeng. A stable method solving the total variation dictionary model with $L^\infty$ constraints. Inverse Problems & Imaging, 2014, 8 (2) : 507-535. doi: 10.3934/ipi.2014.8.507
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show all references

References:
[1]

IEEE Trans. Image Process., 19 (2010), 2345-2356. doi: 10.1109/TIP.2010.2047910.  Google Scholar

[2]

Oxford, U.K.: Oxford Univ. Press 2000.  Google Scholar

[3]

Applied Mathematical Sciences, 147. Springer, New York, 2006.  Google Scholar

[4]

SIAM J. Imag. Sci., 2 (2009), 183-202. doi: 10.1137/080716542.  Google Scholar

[5]

in Proc. SIGGRAPH, New York, (2000), 417-424. doi: 10.1145/344779.344972.  Google Scholar

[6]

Cambridge University Press, 2004. doi: 10.1017/CBO9780511804441.  Google Scholar

[7]

Lecture Notes in Statistics, 37 (1986), 28-47. doi: 10.1007/978-1-4613-9940-7_3.  Google Scholar

[8]

Technical Report, 2009. Google Scholar

[9]

Inverse Problems and Imaging, 2 (2008), 455-484. doi: 10.3934/ipi.2008.2.455.  Google Scholar

[10]

Multiscale Model. Simul., 4 (2005), 490-530. doi: 10.1137/040616024.  Google Scholar

[11]

IEEE Trans. Image Process., 19 (2010), 2634-2645. doi: 10.1109/TIP.2010.2049240.  Google Scholar

[12]

Signal Processing, 82 (2002), 1519-1543. Google Scholar

[13]

J. Math. Imag. Vis., 20 (2004), 89-97. doi: 10.1023/B:JMIV.0000011321.19549.88.  Google Scholar

[14]

IEEE Trans. Image Process., 7 (1998), 319-335. doi: 10.1109/83.661182.  Google Scholar

[15]

J. Math. Imaging Vis., 40 (2011), 120-145. doi: 10.1007/s10851-010-0251-1.  Google Scholar

[16]

SIAM J. Sci. Comp., 20 (1999), 1964-1977. doi: 10.1137/S1064827596299767.  Google Scholar

[17]

Int. J. of Imaging Systems and Technology, 15 (2005), 92-102. doi: 10.1002/ima.20041.  Google Scholar

[18]

Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. doi: 10.1137/1.9780898717877.  Google Scholar

[19]

SIAM J. Imag. Sci., 2 (2009), 730-762. doi: 10.1137/080727749.  Google Scholar

[20]

Multiscale Model. Simul., 4 (2005), 1168-1200. doi: 10.1137/050626090.  Google Scholar

[21]

SIAM Publ., Philadelphia, 1992. doi: 10.1137/1.9781611970104.  Google Scholar

[22]

Commun.Pure Appl. Math., 57 (2004), 1413-1457. doi: 10.1002/cpa.20042.  Google Scholar

[23]

Biometrika, 81 (1994), 425-455. doi: 10.1093/biomet/81.3.425.  Google Scholar

[24]

J. Amer. Statist. Assoc., 90 (1995), 1200-1224. doi: 10.1080/01621459.1995.10476626.  Google Scholar

[25]

Studies Math. Appl., American Elsevier, Amsterdam, New York, 1976.  Google Scholar

[26]

in Augmented Lagrangian Methods: Applications to the Solution of Boundary-Valued Problems, M. Fortin and R. Glowinski, eds., North-Holland, Amsterdam, 1983, 299-331. Google Scholar

[27]

Multiscale Model. Simul., 7 (2008), 1005-1028. doi: 10.1137/070698592.  Google Scholar

[28]

SIAM J. Imag. Sci., 2 (2009), 323-343. doi: 10.1137/080725891.  Google Scholar

[29]

Inverse. Probl., 20 (2004), 815-831. doi: 10.1088/0266-5611/20/3/010.  Google Scholar

[30]

IEEE Trans. on Image Process, 22 (2013), 1108-1120. doi: 10.1109/TIP.2012.2227766.  Google Scholar

[31]

Appl. and Comp. Harmonic Analysis, 12 (2002), 309-331. doi: 10.1006/acha.2002.0379.  Google Scholar

[32]

Journal of Information Processes, 2 (2002), 1-10. Google Scholar

[33]

IEEE Trans. on Image Process, 11 (2002), 1450-1456. doi: 10.1109/TIP.2002.806241.  Google Scholar

[34]

Int. Conf. on Image Processing, 3 (1998), 259-263. doi: 10.1109/ICIP.1998.999016.  Google Scholar

[35]

Inverse Probl., 27 (2011), 45009-45038. doi: 10.1088/0266-5611/27/4/045009.  Google Scholar

[36]

C.R. Acad. Sci. Paris Ser. A Math, 255 (1962), 2897-2899.  Google Scholar

[37]

Bull. Soc. Math. France, 93 (1965), 273-299.  Google Scholar

[38]

Dokl. Akad. Nauk SSSR, 269 (1983), 543-547.  Google Scholar

[39]

SIAM J. Sci. Comp., 33 (2011), 1643-1668. doi: 10.1137/100807697.  Google Scholar

[40]

Inverse Problems and Imaging, 5 (2011), 511-530. doi: 10.3934/ipi.2011.5.511.  Google Scholar

[41]

Physica D, 60 (1992), 259-268. Google Scholar

[42]

Int. J. Comput. Vis., 92 (2011), 265-280. doi: 10.1007/s11263-010-0357-3.  Google Scholar

[43]

SIAM J. Numer. Anal., 42 (2004), 686-713. doi: 10.1137/S0036142903422429.  Google Scholar

[44]

SSVM 2009, LNCS 5567, Springer, 42 (2009), 502-513. Google Scholar

[45]

Winston and Sons, Washington, DC, 1977.  Google Scholar

[46]

Inverse Problems and Imaging, 5 (2011), 237-261. doi: 10.3934/ipi.2011.5.237.  Google Scholar

[47]

Singapore: World Scientific, 2002. doi: 10.1142/9789812777096.  Google Scholar

[48]

Int. Conf. on Image Processing, (2008), 577-580. doi: 10.1109/ICIP.2008.4711820.  Google Scholar

[49]

Int. Conf. on Acoust. Speech and Signal Proc., (2006), 865-868. Google Scholar

[50]

IEEE trans. on Image Process., 19 (2010), 821-825. doi: 10.1109/TIP.2009.2034701.  Google Scholar

[51]

SIAM J. Imag. Sci., 3 (2010), 253-276. doi: 10.1137/090746379.  Google Scholar

[52]

UCLA CAM Report 08-34 (2008). Google Scholar

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