Nonconvex fractional order total variation based image denoising model under mixed stripe and Gaussian noise

Myeongmin Kang; Miyoun Jung; Myeongmin Kang; Miyoun Jung

doi:10.3934/math.20241025

AIMS Mathematics

2024, Volume 9, Issue 8: 21094-21124. doi: 10.3934/math.20241025

Previous Article Next Article

Research article

Nonconvex fractional order total variation based image denoising model under mixed stripe and Gaussian noise

Myeongmin Kang ¹,
Miyoun Jung ^{2
,
,}

1.
Department of Mathematics, Chungnam National University, Daejeon 34134, Korea
2.
Department of Mathematics, Hankuk University of Foreign Studies, Yongin 17035, Korea

Received: 29 March 2024 Revised: 29 May 2024 Accepted: 07 June 2024 Published: 01 July 2024
MSC : 68U10, 65K10, 94A08, 49J40

In this paper, we propose a minimization-based image denoising model for the removal of mixed stripe and Gaussian noise. The objective function includes the prior information from both the stripe noise and image. Specifically, we adopted a unidirectional regularization term and a nonconvex group sparsity term for the stripe noise component, while we utilized a nonconvex fractional order total variation (FTV) regularization for the image component. The priors for stripes enable adequate extraction of periodic or non-periodic stripes from an image in the presence of high levels of Gaussian noise. Moreover, the nonconvex FTV facilitates image restoration with less staircase artifacts and well-preserved edges and textures. To solve the nonconvex problem, we employed an iteratively reweighted $ \ell_1 $ algorithm, and then the alternating direction method of multipliers was adopted for solving subproblems. This led to an efficient iterative algorithm, and its global convergence was proven. Numerical results show that the proposed model provides better denoising performance than existing models with respect to visual features and image quality evaluations.
- image denoising,
- stripe noise,
- Gaussian noise,
- fractional order total variation,
- nonconvex regularization
Citation: Myeongmin Kang, Miyoun Jung. Nonconvex fractional order total variation based image denoising model under mixed stripe and Gaussian noise[J]. AIMS Mathematics, 2024, 9(8): 21094-21124. doi: 10.3934/math.20241025

Related Papers:

Abstract

In this paper, we propose a minimization-based image denoising model for the removal of mixed stripe and Gaussian noise. The objective function includes the prior information from both the stripe noise and image. Specifically, we adopted a unidirectional regularization term and a nonconvex group sparsity term for the stripe noise component, while we utilized a nonconvex fractional order total variation (FTV) regularization for the image component. The priors for stripes enable adequate extraction of periodic or non-periodic stripes from an image in the presence of high levels of Gaussian noise. Moreover, the nonconvex FTV facilitates image restoration with less staircase artifacts and well-preserved edges and textures. To solve the nonconvex problem, we employed an iteratively reweighted $ \ell_1 $ algorithm, and then the alternating direction method of multipliers was adopted for solving subproblems. This led to an efficient iterative algorithm, and its global convergence was proven. Numerical results show that the proposed model provides better denoising performance than existing models with respect to visual features and image quality evaluations.

References

[1]	J. A. Richards, Remote sensing digital image analysis: an introduction, Springer Berlin, Heidelberg, 2013, 343–380. https://doi.org/10.1007/978-3-642-30062-2
[2]	I. Makki, R. Younes, C. Francis, T. Bianchi, M. Zucchetti, A survey of landmine detection using hyperspectral imaging, ISPRS J. Photogramm. Remote Sens., 124 (2017), 40–53. https://doi.org/10.1016/j.isprsjprs.2016.12.009 doi: 10.1016/j.isprsjprs.2016.12.009
[3]	H. Zhang, J. Li, Y. Huang, L. Zhang, A nonlocal weighted joint sparse representation classification method for hyperspectral imagery, IEEE J-STARS, 7 (2017), 2056–2065. https://doi.org/10.1109/JSTARS.2013.2264720 doi: 10.1109/JSTARS.2013.2264720
[4]	Y. Tarabalka, J. Chanussot, J. A. Benediktssons, Segmentation and classification of hyperspectral images using watershed transformation, Pattern Recogn., 43 (2010), 2367–2379. https://doi.org/10.1016/j.patcog.2010.01.016 doi: 10.1016/j.patcog.2010.01.016
[5]	D. W. J. Stein, S. G. Beaven, L. E. Hoff, E. M. Winter, A. P. Schaum, A. D. Stocker, Anomaly detection from hyperspectral imagery, IEEE Signal Proc. Mag., 19 (2002), 58–69. https://doi.org/10.1109/79.974730 doi: 10.1109/79.974730
[6]	M. D. Iordache, J. M. Bioucas-Dias, A. Plaza, Collaborative sparse regression for hyperspectral unmixing, IEEE Trans. Geosci. Remote Sens., 52 (2014), 341–354. https://doi.org/10.1109/TGRS.2013.2240001 doi: 10.1109/TGRS.2013.2240001
[7]	J. Chen, Y. Shao, H. Guo, W. Wang, B. Zhu, Destriping CMODIS data by power filtering, IEEE Trans. Geosci. Remote Sens., 41 (2003), 2119–2124. https://doi.org/10.1109/TGRS.2003.817206 doi: 10.1109/TGRS.2003.817206
[8]	J. Chen, H. Lin, Y. Shao, L. Yang, Oblique striping removal in remote sensing imagery based on wavelet transform, Int. J. Remote Sens., 27 (2006), 1717–1723. https://doi.org/10.1080/01431160500185516 doi: 10.1080/01431160500185516
[9]	R. Pande-Chhetri, A. Abd-Elrahman, De-striping hyperspectral imagery using wavelet transform and adaptive frequency domain filtering, ISPRS J. Photogramm. Remote Sens., 66 (2011), 620–636. https://doi.org/10.1016/j.isprsjprs.2011.04.003 doi: 10.1016/j.isprsjprs.2011.04.003
[10]	L. Sun, R. Neville, K. Staenz, H. P. White, Automatic destriping of Hyperion imagery based on spectral moment matching, J. Can. Remote Sens., 34 (2008), S68–S81. https://doi.org/10.5589/m07-067 doi: 10.5589/m07-067
[11]	M. Wegener, Destriping multiple sensor imagery by improved histogram matching, Int. J. Remote Sens., 11 (1990), 859–875. https://doi.org/10.1080/01431169008955060 doi: 10.1080/01431169008955060
[12]	H. Shen, L. Zhang, A MAP-based algorithm for destriping and inpainting of remotely sensed images, IEEE Trans. Geosci. Remote Sens., 47 (2009), 1492–1502. https://doi.org/10.1109/TGRS.2008.2005780 doi: 10.1109/TGRS.2008.2005780
[13]	M. Bouali, S. Ladjal, Toward optimal destriping of MODIS data using a unidirectional variational model, IEEE Trans. Geosci. Remote Sens., 49 (2011), 2924–2935. https://doi.org/10.1109/TGRS.2011.2119399 doi: 10.1109/TGRS.2011.2119399
[14]	Y. Chang, H. Fang, L. Yan, H. Liu, Robust destriping method with unidirectional total variation and framelet regularization, Opt. Express, 21 (2013), 23307–23323. https://doi.org/10.1364/OE.21.023307 doi: 10.1364/OE.21.023307
[15]	Y. Chang, L. Yan, H. Fang, H. Liu, Simultaneous destriping and denoising for remote sensing images with unidirectional total variation and sparse representation, IEEE Geosci. Remote Sens. Lett., 11 (2014), 1051–1055. https://doi.org/10.1109/LGRS.2013.2285124 doi: 10.1109/LGRS.2013.2285124
[16]	Y. Zhang, G. Zhou, L. Yan, T. Zhang, A destriping algorithm based on TV-Stokes and unidirectional total variation model, Optik, 127 (2016), 428–439. https://doi.org/10.1016/j.ijleo.2015.09.246 doi: 10.1016/j.ijleo.2015.09.246
[17]	M. Wang, X. Zheng, J. Pan, B. Wang, Unidirectional total variation destriping using difference curvature in MODIS emissive bands, Infrared Phys. Technol., 75 (2016), 1–11. https://doi.org/10.1016/j.infrared.2015.12.004 doi: 10.1016/j.infrared.2015.12.004
[18]	X. Liu, X. Lu, H. Shen, Q. Yuan, Y. Jiao, L. Zhang, Stripe noise separation and removal in remote sensing images by consideration of the global sparsity and local variational properties, IEEE Trans. Geosci. Remote Sens., 54 (2016), 3049-3060. https://doi.org/10.1109/TGRS.2015.2510418 doi: 10.1109/TGRS.2015.2510418
[19]	Y. Chang, L. Yan, T. Wu, S. Zhong, Remote sensing image stripe noise removal: from image decomposition perspective, IEEE Trans. Geosci. Remote Sens., 54 (2016), 7018–7031. https://doi.org/10.1109/TGRS.2016.2594080 doi: 10.1109/TGRS.2016.2594080
[20]	Y. Chen, T. Z. Huang, X. Zhao, L. J. Deng, J. Huang, Stripe noise removal of remote sensing images by total variation regularization and group sparsity constraint, Remote sens., 9 (2017), 559. https://doi.org/10.3390/rs9060559 doi: 10.3390/rs9060559
[21]	Y. Chen, T. Z. Huang, L. J. Deng, X. L. Zhao, M. Wang, Group sparsity based regularization model for remote sensing image stripe noise removal, Neurocomputing, 267 (2017), 95–106. https://doi.org/10.1016/j.neucom.2017.05.018 doi: 10.1016/j.neucom.2017.05.018
[22]	Y. Chen, T. Z. Huang, X. L. Zhao, Destriping of multispectral remote sensing image using low-rank tensor decomposition, IEEE J-STARS, 11 (2018), 4950–4967. https://doi.org/10.1109/JSTARS.2018.2877722 doi: 10.1109/JSTARS.2018.2877722
[23]	H. X. Dou, T. Z. Huang, L. J. Deng, X. L. Zhao, J. Huang, Directional $\ell_0$ sparse modeling for image stripe noise removal, Remote Sens., 10 (2018), 361. https://doi.org/10.3390/rs10030361 doi: 10.3390/rs10030361
[24]	S. Qiong, Y. Wang, X. Yan, H. Gu, Remote sensing images stripe noise removal by double sparse regulation and region separation, Remote Sens., 10 (2018), 998. https://doi.org/10.3390/rs10070998 doi: 10.3390/rs10070998
[25]	J. Wang, T. Z. Huang, T. H. Ma, X. L. Zhao, Y. Chen, A sheared low-rank model for oblique stripe removal, Appl. Math. Comput., 360 (2019), 167–180. https://doi.org/10.1016/j.amc.2019.03.066 doi: 10.1016/j.amc.2019.03.066
[26]	J. L. Wang, T. Z. Huang, X. L. Zhao, J. Huang, T. H. Ma, Y. B. Zheng, Reweighted block sparsity regularization for remote sensing images destriping, IEEE J-STARS, 12 (2019), 4951–4963. https://doi.org/10.1109/JSTARS.2019.2940065 doi: 10.1109/JSTARS.2019.2940065
[27]	J. H. Yang, X. L. Zhao, T. H. Ma, Y. Chen, T. Z. Huang, M. Ding, Remote sensing images destriping using unidirectional hybrid total variation and nonconvex low-rank regularization, J. Comput. Appl. Math., 363 (2020), 124–144. https://doi.org/10.1016/j.cam.2019.06.004 doi: 10.1016/j.cam.2019.06.004
[28]	X. Wu, H. Qu, L. Zheng, T. Gao, A remote sensing image destriping model based on low-rank and directional sparse constraint, Remote Sens., 13 (2021), 5126. https://doi.org/10.3390/rs13245126 doi: 10.3390/rs13245126
[29]	X. Liu, X. Lu, H. Shen, Q. Yuan, L. Zhang, Oblique stripe removal in remote sensing images via oriented variation, arXiv, 2018. https://doi.org/10.48550/arXiv.1809.02043
[30]	Q. Song, Z. Huang, H. Ni, K. Bai, Z. Li, Remote sensing images destriping with an enhanced low-rank prior and total variation regulation, Signal Image Video Process., 16 (2022), 1895–1903. https://doi.org/10.1007/s11760-022-02149-8 doi: 10.1007/s11760-022-02149-8
[31]	L. Song, H. Huang, Simultaneous destriping and image denoising using a nonparametric model with the EM algorithm, IEEE Trans. Image Process., 32 (2023), 1065–1077. https://doi.org/10.1109/TIP.2023.3239193 doi: 10.1109/TIP.2023.3239193
[32]	N. Kim, S. S. Han, C. S. Jeong, ADOM: ADMM-Based optimization model for stripe noise removal in remote sensing image, IEEE Access, 11 (2023), 106587–106606. https://doi.org/10.1109/ACCESS.2023.3319268 doi: 10.1109/ACCESS.2023.3319268
[33]	F. Yan, S. Wu, Q. Zhang, Y. Liu, H. Sun, Destriping of remote sensing images by an optimized variational model, Sensors, 23 (2023), 7529. https://doi.org/10.3390/s23177529 doi: 10.3390/s23177529
[34]	C. Wang, X. Zhao, Q. Wang, Z. Ma, P. Tang, An inexact proximal majorization-minimization algorithm for remote sensing image stripe noise removal, Numer. Algor., 2024. https://doi.org/10.1007/s11075-023-01743-2
[35]	L. I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithm, Phys. D, 60 (1992), 259–268. https://doi.org/10.1016/0167-2789(92)90242-F doi: 10.1016/0167-2789(92)90242-F
[36]	T. Chan, A. Marquina, P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503–516. https://doi.org/10.1137/S1064827598344169 doi: 10.1137/S1064827598344169
[37]	M. Lysaker, A. Lundervold, X. C. Tai, Noise removal using fourth-order partial differential equation with application to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12 (2003), 1579–1590. https://doi.org/10.1109/TIP.2003.819229 doi: 10.1109/TIP.2003.819229
[38]	K. Bredies, K. Kunisch, T. Pock, Total generalized variation, SIAM J. Imaging Sci., 3 (2010), 492–526. https://doi.org/10.1137/090769521 doi: 10.1137/090769521
[39]	F. Li, C. Shen, J. Fan, C. Shen, Image restoration combining a total variational filter and a fourth-order filter, J. Vis. Commun. Image Represent., 18 (2007), 322–330. https://doi.org/10.1016/j.jvcir.2007.04.005 doi: 10.1016/j.jvcir.2007.04.005
[40]	K. Papafitsoros, C. B. Sch$\ddot{o}$nlieb, A combined first and second order variational approach for image restoration, J. Math. Imaging Vis., 48 (2014), 308–338. https://doi.org/10.1007/s10851-013-0445-4 doi: 10.1007/s10851-013-0445-4
[41]	J. Bai, X. C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process., 16 (2007), 2492–2502. https://doi.org/10.1109/TIP.2007.904971 doi: 10.1109/TIP.2007.904971
[42]	J. Zhang, Z. Wei, L. Xiao, Adaptive fractional-order multi-scale method for image denoising, SIAM J. Imaging Sci., 43 (2012), 39–49. https://doi.org/10.1007/s10851-011-0285-z doi: 10.1007/s10851-011-0285-z
[43]	R. H. Chan, A. Lanza, S. Morigi, F. Sgallari, An adaptive strategy for the restoration of textured images using fractional order regularization, Numer. Math. Theor. Meth. Appl., 6 (2013), 276–296. https://doi.org/10.4208/nmtma.2013.mssvm15 doi: 10.4208/nmtma.2013.mssvm15
[44]	J. Zhang, Z. Hui, L. Xiao, A fast adaptive reweighted residual-feedback iterative algorithm for fractional order total variation regularized multiplicative noise removal of partly-textured images, Signal Process., 98 (2014), 381–395. https://doi.org/10.1016/j.sigpro.2013.12.009 doi: 10.1016/j.sigpro.2013.12.009
[45]	J. Zhang, K. Chen, A total fractional-order variation model for image restoration with nonhomogeneous boundary conditions and its numerical solution, SIAM J. Imaging Sci., 8 (2015), 2487–2518. https://doi.org/10.1137/14097121X doi: 10.1137/14097121X
[46]	A. Ullah, W. Chen, M. A. Khan, A new variational approach for restoring images with multiplicative noise, Comput. Math. Appl., 71 (2016), 2034–2050. https://doi.org/10.1016/j.camwa.2016.03.024 doi: 10.1016/j.camwa.2016.03.024
[47]	F. Dong, Y. Chen, A fractional-order derivative based variational framework for image denoising, Inverse Probl. Imag., 10 (2016), 27–50. https://doi.org/10.3934/ipi.2016.10.27 doi: 10.3934/ipi.2016.10.27
[48]	M. R. Chowdhury, J. Zhang, J. Qin, Y. Lou, Poisson image denoising based on fractional-order total variation, Inverse Probl. Imag., 14 (2020), 77–96. https://doi.org/10.3934/ipi.2019064 doi: 10.3934/ipi.2019064
[49]	Y. F. Pu, J. L. Zhou, X. Yuan, Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement, IEEE Trans. Image Process., 19 (2010), 491–511. https://doi.org/10.1109/TIP.2009.2035980 doi: 10.1109/TIP.2009.2035980
[50]	Z. Ren, C. He, Q. Zhang, Fractional order total variation regularization for image super-resolution, Signal Process., 93 (2013), 2408–2421. https://doi.org/10.1016/j.sigpro.2013.02.015 doi: 10.1016/j.sigpro.2013.02.015
[51]	D. Geman, G. Reynolds, Constrained restoration and recovery of discontinuities, IEEE Trans. Pattern Anal. Mach. Intell., 14 (1992), 367–383. https://doi.org/10.1109/34.120331 doi: 10.1109/34.120331
[52]	D. Geman, C. Yang, Nonlinear image recovery with half-quadratic regularization, IEEE Trans. Image Process., 4 (1995), 932–946. https://doi.org/10.1109/83.392335 doi: 10.1109/83.392335
[53]	M. Nikolova, M. K. Ng, C. P. Tam, Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction, IEEE Trans. Image Process., 19 (2010), 3073–3088. https://doi.org/10.1109/TIP.2010.2052275 doi: 10.1109/TIP.2010.2052275
[54]	S. Oh, H. Woo, S. Yun, M. Kang, Non-convex hybrid total variation for image denoising, J. Vis. Commun. Image Represent., 24 (2013), 332–344. https://doi.org/10.1016/j.jvcir.2013.01.010 doi: 10.1016/j.jvcir.2013.01.010
[55]	M. Kang, M. Kang, M. Jung, Nonconvex higher-order regularization based Rician noise removal with spatially adaptive parameters, J. Visual Commun. Image Represent., 32 (2015), 180–193. https://doi.org/10.1016/j.jvcir.2015.08.006 doi: 10.1016/j.jvcir.2015.08.006
[56]	T. Adam, R. Paramesran, Hybrid non-convex second-order total variation with applications to non-blind image deblurring, Signal Image Video Process., 14 (2020), 115–123. https://doi.org/10.1007/s11760-019-01531-3 doi: 10.1007/s11760-019-01531-3
[57]	Y. Sun, L. Lei, D. Guan, X. Li, G. Xiao, SAR image speckle reduction based on nonconvex hybrid total variation model, IEEE Trans. Geosci. Remote Sens., 59 (2020), 1231–1249. https://doi.org/10.1109/TGRS.2020.3002561 doi: 10.1109/TGRS.2020.3002561
[58]	P. Ochs, A. Dosovitskiy, T. Brox, T. Pock, On iteratively reweighted algorithms for nonsmooth nonconvex optimization in computer vision, SIAM J. Imaging Sci., 8 (2015), 331–372. https://doi.org/10.1137/140971518 doi: 10.1137/140971518
[59]	E. J. Candés, M. B. Wakin, S. P. Boyd, Enhancing sparsity by reweighted $\ell_1$ minimization, J. Fourier Anal. Appl., 14 (2008), 877–905. https://doi.org/10.1007/s00041-008-9045-x doi: 10.1007/s00041-008-9045-x
[60]	J. Eckstein, D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Program., 55 (1992), 293–318. https://doi.org/10.1007/BF01581204 doi: 10.1007/BF01581204
[61]	R. Glowinski, Numerical methods for nonlinear variational problems, Springer Berlin, Heidelberg, 1984. https://doi.org/10.1007/978-3-662-12613-4
[62]	S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2011), 1–122. https://doi.org/10.1561/2200000016 doi: 10.1561/2200000016
[63]	H. Carfantan, J. Idier, Statistical linear destriping of satellite-based pushbroom-type images, IEEE Trans. Geosci. Remote Sens., 48 (2010), 1860–1871. https://doi.org/10.1109/TGRS.2009.2033587 doi: 10.1109/TGRS.2009.2033587
[64]	K. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York, USA: John Wiley & Sons, 1993.
[65]	K. B. Oldham, J. Spanier, The fractional calculus: theory and applications of differentiation and integration to arbitrary order, New York, USA: Academic Press, 1974. https://doi.org/10.1016/s0076-5392(09)x6012-1
[66]	I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, London, UK: Academic Press, 1999. https://doi.org/10.1016/s0076-5392(99)x8001-5
[67]	L. Vese, T. F. Chan, Reduced non-convex functional approximations for image restoration & segmentation, UCLA CAM Report, 1997.
[68]	H. Attouch, J. Bolte, B. F. Svaiter, Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods, Math. Program., 137 (2013), 91–129. https://doi.org/10.1007/s10107-011-0484-9 doi: 10.1007/s10107-011-0484-9
[69]	L. P. D. Van den Dries, Tame topology and $o$-minimal structures, New York, NY, USA: Cambridge University Press, 1998. https://doi.org/10.1017/CBO9780511525919
[70]	A. Chambolle, T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vis., 40 (2011), 120–145. https://doi.org/10.1007/s10851-010-0251-1 doi: 10.1007/s10851-010-0251-1
[71]	T. Goldstein, S. Osher, The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci., 2 (2009), 323–343. https://doi.org/10.1137/080725891 doi: 10.1137/080725891
[72]	Z. Wang, A. C. Bovik, H. R. Sheikh, E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600–612. https://doi.org/10.1109/TIP.2003.819861 doi: 10.1109/TIP.2003.819861

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)