An algorithm for hybrid regularizers based image restoration with Poisson noise

Pham, Cong Thang ; Tran, Thi Thu Thao

Kybernetika, Tome 57 (2021) no. 3, pp. 446-473

Abstract (VO)

In this paper, a hybrid regularizers model for Poissonian image restoration is introduced. We study existence and uniqueness of minimizer for this model. To solve the resulting minimization problem, we employ the alternating minimization method with rigorous convergence guarantee. Numerical results demonstrate the efficiency and stability of the proposed method for suppressing Poisson noise.

Détail

DOI : 10.14736/kyb-2021-3-0446

Classification : 35A15, 94A08
Keywords: total variation; image denoising; image deblurring; alternating minimization method

Bibliographie

[1] Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: partial differential equations and the calculus of variations. (Second edition). Applied Mathematical Science 147, Springer-Verlag, New York 2006. | DOI

[2] Bardsley, J. M., Goldes, J.: Techniques for regularization parameter and hyper-parameter selection in PET and SPECT imaging. Inverse Probl. Sci. Engrg. 19 (2011), 2, 267-280. | DOI

[3] Boyd, S., Parikh, N., al., et: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3 (2010), 1-122. | DOI

[4] Bovik, A. C., Wang, Z.: Modern Image Quality Assessment, Synthesis Lectures on Image, Video, and Multimedia Processing. Morgan and Claypool Publishers, 2006. | DOI

[5] Chan, R. H., Chen, K.: Multilevel algorithm for a Poisson noise removal model with total-variation regularization. Int. J. Comput. Math. 84 (2007), 8, 1183-1198. | DOI

[6] Chen, D. Q., Cheng, L. Z.: Deconvolving Poissonian images by a novel hybrid variational model. J. Vis. Commun. Image R. 22 (2011), 7, 643-652. | DOI

[7] Chen, Y. M., Wunderli, T.: Adaptive total variation for image restoration in BV space. J. Math. Anal. Appl. 272 (2002), 1, 117-137. | DOI

[8] Dupe, F. X., Fadili, M. J., Starck, J L.: Deconvolution of confocal microscopy images using proximal iteration and sparse representations. In: I. S. Biomed. Imaging, Paris, France (2008), pp. 736-739.

[9] Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. Classics in Applied Mathematics, SIAM 1999.

[10] Eckstein, J., Bertsekas, D. P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Programming 55 (1992), 293-318. | DOI

[11] Frosioa, I., Borghese, N. A.: Compression and smart coding of offset and gain maps for intraoral digital x-ray sensors. Med. Phys. 36 (2009), 2, 464-479. | DOI

[12] He, C., Hu, C., Zhang, W., Shi, B.: A fast adaptive parameter estimation for total variation image restoration. IEEE Trans. Image Process. 23 (2014), 12, 4954-4967. | DOI

[13] Jiang, L., Huang, J., Lv, X. G., Liu, J.: Restoring Poissonian images by a combined first-order and second-order variation approach. J. Math. 2013 (2013), 274573. | DOI

[14] Jiang, L., Huang, J., Lv, X. G., Liu, J.: Alternating direction method for the high-order total variation-based Poisson noise removal problem. Numer. Algorithms 69 (2015), 3, 495-516. | DOI

[15] Le, T., Chartrand, R., Asaki, T.: A variational approach to constructing images corrupted by Poisson noise. J. Math. Imaging Vis. 27 (2007), 257-263. | DOI

[16] Li, F., Shen, C. M., Fan, J. S., Shen, C. L.: Image restoration combining a total variational filter and a fourth-order filter. J. Vis. Commun. Image Res. 18 (2007), 4, 322-330. | DOI

[17] Liu, Q., Yao, Z., Ke, Y.: Entropy solutions for a fourth-order nonlinear degenerate problem for noise removal. Nonlinear Anal. Theor. 67 (2007), 6, 1908-1918. | DOI

[18] Liu, X.: Augmented Lagrangian method for total generalized variation based Poissonian image restoration. Comput. Math. Appl. 71 (2016), 8, 1694-1705. | DOI

[19] Liu, X., Huang, L.: Total bounded variation-based Poissonian images recovery by split Bregman iteration. Math. Methods Appl. Sci. 35, (2012), 5, 520-529. | DOI

[20] Liu, X., Huang, L.: Poissonian image reconstruction using alternating direction algorithm. J. Electronic Imaging 22 (2013), 3, 033007. | DOI

[21] Lv, X. G., Jiang, L., Liu, J.: Deblurring Poisson noisy images by total variation with overlapping group sparsity. Appl. Math. Comput. 289 (2016), 20, 132-148. | DOI

[22] Ma, M., Zhang, J., al., et: Adaptive image restoration via a relaxed regularization of mean curvature. Math. Probl. Eng. (2020), 3416907. | DOI

[23] Micchelli, C. A., Shen, L., Xu, Y.: Proximity algorithms for image models: denoising. Inverse Probl. 27 (2011), 4, 045009. | DOI

[24] Osher, S., Scherzer, O.: G-norm properties of bounded variation regularization. Commun. Math. Sci. 2 (2004), 2, 237-254. | DOI

[25] Padcharoen, A., Kumama, P., Martinez-Moreno, J.: Augmented Lagrangian method for TV-$l$1-$l$2 based colour image restoration. J. Comput. Appl. Math.354 (2019), 507-519. | DOI

[26] Pham, C. T., Kopylov, A.: Multi-quadratic dynamic programming procedure of edge-preserving denoising for medical images. Int. Arch. Photogramm. Remote Sens. Spatial Inf. Sci. XL-5/W6 (2015), 101-106. | DOI

[27] Pham, C. T., Gamard, G., Kopylov, A., Tran, T. T. T.: An algorithm for image restoration with mixed noise using total variation regularization. Turk. J. Elec. Eng. Comp. Sci. 26 (2018), 2832-2846. | DOI

[28] Pham, C. T., Kopylov, A. V.: Tree-serial parametric dynamic programming with flexible prior model for image denoising. Comput. Opt. 42 (2018), 838-845. | DOI

[29] Pham, C. T., Tran, T. T. T., al., et.: An adaptive algorithm for restoring image corrupted by mixed noise. Cybern. Phys. 8 (2019), 73-82. | DOI

[30] Pham, C. T., Tran, T. T. T., Gamard, G.: An efficient total variation minimization method for image restoration. Informatica 31 (2020), 539-560. | DOI

[31] Sarder, P., Nehorai, A.: Deconvolution method for 3D fluorescence microscopy images. IEEE Signal Process. Magazine 23 (2006), 3, 32-45. | DOI

[32] Sawatzky, A., Brune, C., Kasters, T., Wabbeling, F., Burger, M.: EM-TV methods for inverse problems with Poisson noise. Lect. Notes Math. Springer 2090 (2013), 71-142. | DOI

[33] Setzer, S., Steidl, G., Teuber, T.: Deblurring Poissonian images by split Bregman techniques. J. Vis. Commun. Image R. 21 (2010), 3, 193-199. | DOI

[34] Yan, M.: EM-type algorithms for image reconstruction with background emission and Poisson noise. Lect. Notes Comput. Sci. Springer Berlin Heidelberg 6938 (2011), 33-42. | DOI

[35] Zhang, J., Ma, M., Wu, Z., Deng, C.: High-order total bounded variation model and its fast algorithm for poissonian image restoration. Math. Probl. Engrg. (2019), 2502731. | DOI

[36] Wang, Y., Yang, J., Yin, W., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci. 1 (2008), 3, 248-272. | DOI

[37] Wen, Y., Chan, R. H., Zeng, T.: Primal-dual algorithms for total variation based image restoration under Poisson noise. Sci. China Math. 59 (2016), 141-160. | DOI

[38] Woo, H., Yun, S.: Alternating minimization algorithm for speckle reduction with a shifting technique. IEEE T. Image Process. 21 (2012), 1701-1714. | DOI