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Enhanced total generalized variation method based on moreau envelope

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Abstract

Image restoration with total generalized variation (TGV) regularization has been proved to be a creditable method and used broadly, but this method is imperfect in retaining image details. This paper proposes a non-convex and non-separable regularization term based on TGV enhancement, which can maintain the strict convexity of the total cost function and avoid the underestimation of the TGV method. The new regularization is obtained by subtracting the Moreau envelope of TGV from the TGV term, in which the former is defined by infimal convolution. To minimize the cost function, the proximal forward-backward splitting (FBS) algorithm is applied, and the alternating direction method of multipliers with adaptive parameter estimation (APE-ADMM), which avoids the drawbacks of FBS algorithm, is proposed by applying the variable splitting and the augment Lagrangian method (ALM). This proposed algorithm can update the regularization parameter adaptively. Experiments varify that the proposed method is more accurate in solution estimation than the other promoted ones, and the new ADMM algorithm with new regularization shows improved effects of image restoration.

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References

  1. Afonso M, Bioucas J, Figueiredo M (2011) An augmented lagrangian approach to the constrained optimization formulation of imaging inverse problems. IEEE Trans Image Process Publ IEEE Signal Process Soc 20:681–695. https://doi.org/10.1109/TIP.2010.2076294

    Article  MathSciNet  Google Scholar 

  2. Aujol JF (2009) Some first-order algorithms for total variation based image restoration. J Math Imaging Vis 34:307–327. https://doi.org/10.1007/s10851-009-0149-y

    Article  MathSciNet  Google Scholar 

  3. Bauschke H, Combettes P (2017) Convex analysis and monotone operator theory in hilbert spaces. Springer, New York

  4. Beck A, Teboulle M (2009) Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans Image Process 18:2419–2434. https://doi.org/10.1109/TIP.2009.2028250

    Article  MathSciNet  Google Scholar 

  5. Becker S, Combettes P (2013) An algorithm for splitting parallel sums of linearly composed monotone operators, with applications to signal recovery. arXiv:1305.5828

  6. Boyd S, Parikh N, Chu E, Peleato B, Eckstein J (2011) Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends Mach Learn 3:1–122. https://doi.org/10.1561/2200000016

    Article  Google Scholar 

  7. Bredies K, Kunisch K, Pock T (2010) Total generalized variation. SIAM J Imaging Sci 3:492–526. https://doi.org/10.1137/090769521

    Article  MathSciNet  Google Scholar 

  8. Chambolle A, Lions PL (1997) Image recovery via total variation minimization and related problems. Numer Math 76:167–188. https://doi.org/10.1007/s002110050258

    Article  MathSciNet  Google Scholar 

  9. Chan S, Khoshabeh R, Gibson K, Gill P, Nguyen T (2011) An augmented lagrangian method for total variation video restoration. IEEE Trans Image Process Publ IEEE Signal Process Soc 20:3097–3111. https://doi.org/10.1109/TIP.2011.2158229

    Article  MathSciNet  Google Scholar 

  10. Chan T, Esedoglu S, Park F (2010) A fourth order dual method for staircase reduction in texture extraction and image restoration problems. In: 2010 IEEE International Conference on Image Processing, pp 4137–4140

  11. Chan T, Marquina A, Mulet P (2000) High-order total variation-based image restoration. SIAM J Sci Comput 22:503–516. https://doi.org/10.1137/S1064827598344169

    Article  MathSciNet  Google Scholar 

  12. Combettes P L, Pesquet C (2009) Proximal splitting methods in signal processing. Fixed-Point Algorithm Inverse Probl Sci Eng 49:185–212. https://doi.org/10.1007/978-1-4419-9569-8_10

    Article  MathSciNet  Google Scholar 

  13. Dadashi V, Postolache M (2019) Forward-backward splitting algorithm for fixed point problems and zeros of the sum of monotone operators. Arabian J Math:1–11. https://doi.org/10.1007/s40065-018-0236-2

  14. Ding Y, Selesnick I (2015) Artifact-free wavelet denoising: Non-convex sparse regularization, convex optimization. IEEE Signal Process Lett 22:1364–1368. https://doi.org/10.1109/LSP.2015.2406314

    Article  Google Scholar 

  15. Easley G, Labate D, Colonna F (2009) Shearlet-based total variation diffusion for denoising. IEEE Trans Image Process Publ IEEE Signal Process Soc 18:260–268. https://doi.org/10.1109/TIP.2008.2008070

    Article  MathSciNet  Google Scholar 

  16. Gelb A, Hou X, Li Q (2019) Numerical analysis for conservation laws using l1minimization. J Sci Comput 81:1240–1265. https://doi.org/10.1007/s10915-019-00982-7

    Article  MathSciNet  Google Scholar 

  17. Getreuer P (2012) Total variation inpainting using split bregman. Image Process Line 2:147–157. https://doi.org/10.5201/ipol.2012.g-tvi

    Article  Google Scholar 

  18. Gholami A, Hosseini SM (2013) A balanced combination of tikhonov and total variation regularizations for reconstruction of piecewise-smooth signals. Signal Process 93:1945–1960. https://doi.org/10.1016/j.sigpro.2012.12.008

    Article  Google Scholar 

  19. Gu Y, Yang X, Gao Y (2019) A novel total generalized variation model for image dehazing. J Math Imaging Vis:1329–1341. https://doi.org/10.1007/s10851-019-00909-9

  20. He C, hu C, Yang X, He H, Zhang Q (2014) An adaptive total generalized variation model with augmented lagrangian method for image denoising. Math Probl Eng 2014:1–11. https://doi.org/10.1155/2014/157893

    Google Scholar 

  21. He C, hu C, Zhang W, Shi B (2014) A fast adaptive parameter estimation for total variation image restoration. IEEE Trans Image Process Publ IEEE Signal Process Soc 23:4954–4967. https://doi.org/10.1109/TIP.2014.2360133

    Article  MathSciNet  Google Scholar 

  22. Holt K (2014) Total nuclear variation and jacobian extensions of total variation for vector fields. IEEE Trans Image Process Publ IEEE Signal Process Soc 23:3975–3988. https://doi.org/10.1109/TIP.2014.2332397

    Article  MathSciNet  Google Scholar 

  23. Honglu Z, Tang L, Fang Z, Xiang C, Li C (2017) Nonconvex and nonsmooth total generalized variation model for image restoration. Signal Process 143:69–85. https://doi.org/10.1016/j.sigpro.2017.08.021

    Google Scholar 

  24. Huang Y, Dong Y (2014) New properties of forward-backward splitting and a practical proximal-descent algorithm. Appl Math Comput 237:60–68. https://doi.org/10.1016/j.amc.2014.03.062

    MathSciNet  MATH  Google Scholar 

  25. Knoll F, Bredies K, Pock T, Stollberger R (2011) Second order total generalized variation (tgv) for mri. Magn Reson Med Official J Soc Magn Reson Med Soc Magn Reson Med 65:480–491. https://doi.org/10.1002/mrm.22595

    Article  Google Scholar 

  26. Lanza A, Morigi S, Sgallari F (2016) Convex image denoising via non-convex regularization with parameter selection. J Math Imaging Vis 56:195–220. https://doi.org/10.1007/s10851-016-0655-7

    Article  MathSciNet  Google Scholar 

  27. Lanza A, Morigi S, Sgallari F (2015) Constrained tvpl2model for image restoration. J Sci Comput 68:64–91. https://doi.org/10.1007/s10915-015-0129-x

    Article  Google Scholar 

  28. Lee CO, Nam C, Park J (2018) Domain decomposition methods using dual conversion for the total variation minimization with l1fidelity term. J Sci Comput 78:951–970. https://doi.org/10.1007/s10915-018-0791-x

    Article  Google Scholar 

  29. M. Mohammadi M, Rojas C, Wahlberg B (2016) A class of nonconvex penalties preserving overall convexity in optimization-based mean filtering. IEEE Trans Signal Process 64:6650–6664. https://doi.org/10.1109/TSP.2016.2612179

    Article  MathSciNet  Google Scholar 

  30. Nasonov AV, Krylov A (2018) An improvement of bm3d image denoising and deblurring algorithm by generalized total variation. In: 2018 7th European workshop on visual information processing (EUVIP), pp 26–28

  31. Nikolova M, Ng M, Tam P (2010) Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction. IEEE Trans Image Process Publ IEEE Signal Process Soc 19:3073–3088. https://doi.org/10.1109/TIP.2010.2052275

    Article  MathSciNet  Google Scholar 

  32. Rudin L, Osher S, Fatemi E (1992) Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenom 60:259–268. https://doi.org/10.1016/0167-2789(92)90242-F

    Article  MathSciNet  Google Scholar 

  33. Selesnick I (2017) Sparse regularization via convex analysis. IEEE Trans Signal Process 65:4481–4494. https://doi.org/10.1109/TSP.2017.2711501

    Article  MathSciNet  Google Scholar 

  34. Selesnick I (2017) Total variation denoising via the moreau envelope. IEEE Signal Process Lett 24:216–220. https://doi.org/10.1109/LSP.2017.2647948

    Article  MathSciNet  Google Scholar 

  35. Selesnick I, Bayram I (2016) Enhanced sparsity by non-separable regularization. IEEE Trans Signal Process 64:2298–2313. https://doi.org/10.1109/TSP.2016.2518989

    Article  MathSciNet  Google Scholar 

  36. Selesnick I, Chen PY (2013) Total variation denoising with overlapping group sparsity. In: 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, pp 5696–5700

  37. Selesnick I, Parekh A, Bayram I (2015) Convex 1-d total variation denoising with non-convex regularization. IEEE Signal Process Lett 22:141–144. https://doi.org/10.1109/LSP.2014.2349356

    Article  Google Scholar 

  38. Setzer S, Steidl G, Teuber T (2011) Infimal convolution regularizations with discrete ii -type functionals. Commun Math Sci 9:797–827. https://doi.org/10.4310/CMS.2011.v9.n3.a7

    Article  MathSciNet  Google Scholar 

  39. Tikhonov A, Arsenin V (1977) Solution of ill-posed problem. Winston, New York

  40. Wali S, Liu Z, Wu C, Chang H (2018) A boosting procedure for variational-based image restoration. Numer Math Theory Methods Appl 11:49–73. https://doi.org/10.4208/nmtma.OA-2017-0046

    Article  MathSciNet  Google Scholar 

  41. Wang Y, Yang J, Yin W, Zhang Y (2008) A new alternating minimization algorithm for total variation image reconstruction. SIAM J Imaging Sci 1:248–272. https://doi.org/10.1137/080724265

    Article  MathSciNet  Google Scholar 

  42. Wang Z, Bovik A, Sheikh H, Simoncelli E (2004) Image quality assessment: From error visibility to structural similarity. IEEE Trans Image Process 13:600–612. https://doi.org/10.1109/TIP.2003.819861

    Article  Google Scholar 

  43. Wen Y-W, Chan R (2011) Parameter selection for total-variation-based image restoration using discrepancy principle. IEEE Trans Image Process Publ IEEE Signal Process Soc 21:1770–1781. https://doi.org/10.1109/TIP.2011.2181401

    Article  MathSciNet  Google Scholar 

  44. Wu C, Tai C (2010) Augmented lagrangian method, dual methods, and split bregman iteration for rof, vectorial tv, and high order models. SIAM J Imaging Sci 3:300–339. https://doi.org/10.1137/090767558

    Article  MathSciNet  Google Scholar 

  45. Yu Y, Peng J (2017) The moreau envelope based efficient first-order methods for sparse recovery. J Comput Appl Math 322:109–128. https://doi.org/10.1016/j.cam.2017.03.014

    Article  MathSciNet  Google Scholar 

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  1. School of Science, Beijing Jiaotong University, Beijing, China

    Mengmeng Zhou & Ping Zhao

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Zhou, M., Zhao, P. Enhanced total generalized variation method based on moreau envelope. Multimed Tools Appl 80, 19539–19566 (2021). https://doi.org/10.1007/s11042-021-10586-9

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