A nonconvex total variational model for the joint image segmentation and restoration of images corrupted by Rician noise

Myeongmin Kang; Myeongmin Kang

doi:10.3934/math.2026147

AIMS Mathematics

2026, Volume 11, Issue 2: 3594-3635. doi: 10.3934/math.2026147

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Research article

A nonconvex total variational model for the joint image segmentation and restoration of images corrupted by Rician noise

Myeongmin Kang ^,

Department of Mathematics, Chungnam National University, Daejeon 34134, Korea

Received: 28 September 2025 Revised: 06 January 2026 Accepted: 22 January 2026 Published: 06 February 2026
MSC : 65K10, 68U10, 94A08

In this paper, a novel variational model is proposed for image segmentation via joint restoration of images corrupted by blurring and Rician noise. The proposed model is built upon the piecewise constant Mumford–Shah framework and combines an appropriate data fidelity term with nonconvex total variation (NTV) regularization. The NTV regularization effectively denoises homogeneous regions while accurately preserving object boundaries to facilitate robust segmentation. To solve the resulting nonconvex optimization problem, a proximal alternating minimization algorithm is employed. In addition, an iteratively reweighted $ \ell_1 $ algorithm and the alternating direction method of multipliers are adopted to efficiently handle the corresponding subproblems. Numerical experiments demonstrate the effectiveness of the proposed model in achieving accurate and robust segmentation performance when compared with several state-of-the-art methods.
- image segmentation,
- image restoration,
- Rician noise,
- nonconvex total variation,
- proximal alternating minimization algorithm
Citation: Myeongmin Kang. A nonconvex total variational model for the joint image segmentation and restoration of images corrupted by Rician noise[J]. AIMS Mathematics, 2026, 11(2): 3594-3635. doi: 10.3934/math.2026147

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Abstract

In this paper, a novel variational model is proposed for image segmentation via joint restoration of images corrupted by blurring and Rician noise. The proposed model is built upon the piecewise constant Mumford–Shah framework and combines an appropriate data fidelity term with nonconvex total variation (NTV) regularization. The NTV regularization effectively denoises homogeneous regions while accurately preserving object boundaries to facilitate robust segmentation. To solve the resulting nonconvex optimization problem, a proximal alternating minimization algorithm is employed. In addition, an iteratively reweighted $ \ell_1 $ algorithm and the alternating direction method of multipliers are adopted to efficiently handle the corresponding subproblems. Numerical experiments demonstrate the effectiveness of the proposed model in achieving accurate and robust segmentation performance when compared with several state-of-the-art methods.

References

[1]	F. Li, M. K. Ng, T. Y. Zeng, C. Shen, A multiphase image segmentation method based on fuzzy region competition, SIAM J. Imaging Sci., 3 (2010), 277–299. https://doi.org/10.1137/080736752 doi: 10.1137/080736752
[2]	J. Zhang, J. Hu, Image segmentation based on 2D otsu method with histogram analysis, In: 2008 International conference on computer science and software engineering, Wuhan, China, 2008,105–108. https://doi.org/10.1109/CSSE.2008.206
[3]	S. Kim, M. Kang, Multiple-region segmentation without supervision by adaptive global maximum clustering, IEEE Trans. Image Process., 21 (2011), 1600–1612. https://doi.org/10.1109/TIP.2011.2179058 doi: 10.1109/TIP.2011.2179058
[4]	W. Cai, S. Chen, D. Zhang, Fast and robust fuzzy c-means clustering algorithms incorporating local information for image segmentation, Pattern Recogn., 40 (2007), 825–838. https://doi.org/10.1016/j.patcog.2006.07.011 doi: 10.1016/j.patcog.2006.07.011
[5]	M. Gong, Y. Liang, J. Shi, W. Ma, J. Ma, Fuzzy c-means clustering with local information and kernel metric for image segmentation, IEEE Trans. Image Process., 22 (2013), 573–584. https://doi.org/10.1109/TIP.2012.2219547 doi: 10.1109/TIP.2012.2219547
[6]	J. Carreira, C. Sminchisescu, CPMC: Automatic object segmentation using constrained parametric min-cuts, IEEE Trans. Pattern Anal., 34 (2012), 1312–1328. https://doi.org/10.1109/TPAMI.2011.231 doi: 10.1109/TPAMI.2011.231
[7]	J. Shi, J. Malik, Normalized cuts and image segmentation, IEEE Trans. Pattern Anal., 22 (2000), 888–905. https://doi.org/10.1109/34.868688 doi: 10.1109/34.868688
[8]	P. F. Felzenszwalb, D. P. Huttenlocher, Efficient graph-based image segmentation, Int. J. Comput. Vision, 59 (2004), 167–181. https://doi.org/10.1023/B:VISI.0000022288.19776.77 doi: 10.1023/B:VISI.0000022288.19776.77
[9]	D. Mumford, J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Commun. Pure Appl. Math., 42 (1989), 577–685. https://doi.org/10.1002/cpa.3160420503 doi: 10.1002/cpa.3160420503
[10]	T. Chan, L. Vese, An active contour model without edges, In: Scale-space theories in computer vision, Berlin, Heidelberg: Springer, 1999,141–151. https://doi.org/10.1007/3-540-48236-9_13
[11]	S.-H. Lee, J. K. Seo, Level set-based bimodal segmentation with stationary global minimum, IEEE Trans. Image Process., 15 (2006), 2843–2852. https://doi.org/10.1109/TIP.2006.877308 doi: 10.1109/TIP.2006.877308
[12]	X. Cai, R. Chan, T. Zeng, A two-stage image segmentation method using a convex variant of the Mumford–Shah model and thresholding, SIAM J. Imaging Sci., 6 (2013), 368–390. https://doi.org/10.1137/120867068 doi: 10.1137/120867068
[13]	L. Bar, N. Sochen, N. Kiryati, Variational pairing of image segmentation and blind restoration, In: Computer Vision - ECCV 2004, Berlin, Heidelberg: Springer, 2004,166–177. https://doi.org/10.1007/978-3-540-24671-8_13
[14]	X. Cai, Variational image segmentation model coupled with image restoration achievements, Pattern Recogn., 48 (2015), 2029–2042. https://doi.org/10.1016/j.patcog.2015.01.008 doi: 10.1016/j.patcog.2015.01.008
[15]	G. Gerig, O. Kubler, R. Kikinis, F. A. Jolesz, Nonlinear anisotropic filtering of MRI data, IEEE Trans. Med. Imaging, 11 (1992), 221–232. https://doi.org/10.1109/42.141646 doi: 10.1109/42.141646
[16]	K. Krissian, S. Aja-Fernández, Noise-driven anisotropic diffusion filtering of MRI, IEEE Trans. Image Process., 18 (2009), 2265–2274. https://doi.org/10.1109/TIP.2009.2025553 doi: 10.1109/TIP.2009.2025553
[17]	C. S. Anand, J. S. Sahambi, Wavelet domain non-linear filtering for MRI denoising, Magn. Reson. Imaging, 28 (2010), 842–861. https://doi.org/10.1016/j.mri.2010.03.013 doi: 10.1016/j.mri.2010.03.013
[18]	R. D. Nowak, Wavelet-based Rician noise removal for magnetic resonance imaging, IEEE Trans. Image Process., 8 (1999), 1408–1419. https://doi.org/10.1109/83.791966 doi: 10.1109/83.791966
[19]	D. W. Kim, C. Kim, D. H. Kim, D. H. Lim, Rician nonlocal means denoising for MRI images using nonparametric principal component analysis, EURASIP J. Image Vide., 2011 (2011), 15. https://doi.org/10.1186/1687-5281-2011-15 doi: 10.1186/1687-5281-2011-15
[20]	J. Yang, J. Fan, D. Ai, S. Zhou, S. Tang, Y. Wang, Brain MR image denoising for rician noise using pre-smooth non-local means filter, Biomed. Eng. Online, 14 (2015), 1–20. http://doi.org/10.1186/1475-925X-14-2 doi: 10.1186/1475-925X-14-2
[21]	S. Basu, T. Fletcher, R. Whitaker, Rician noise removal in diffusion tensor MRI, In: Medical image computing and computer-assisted intervention–MICCAI 2006, 2006, Berlin, Heidelberg: Springer, 117–125. https://doi.org/10.1007/11866565_15
[22]	P. Getreuer, M. Tong, L. A. Vese, A variational model for the restoration of MRI images corrupted by blur and Rician noise, In: Advances in visual computing. ISVC 2011, 2011, Berlin, Heidelberg: Springer, 686–698. https://doi.org/10.1007/978-3-642-24028-7_63
[23]	L. Chen, T. Zeng, A convex variational model for restoring blurred images with large Rician noise, J. Math. Imaging Vis., 53 (2015), 92–111. https://doi.org/10.1007/s10851-014-0551-y doi: 10.1007/s10851-014-0551-y
[24]	A. Martín, E. Schiavi, S. S. de León, On 1-Laplacian elliptic equations modeling magnetic resonance image Rician denoising, J. Math. Imaging Vis., 57 (2017), 202–224. https://doi.org/10.1007/s10851-016-0675-3 doi: 10.1007/s10851-016-0675-3
[25]	T. D. K. Phan, A spatially variant high-order variational model for Rician noise removal, PeerJ Comput. Sci., 9 (2023), e1579. https://doi.org/10.7717/peerj-cs.1579 doi: 10.7717/peerj-cs.1579
[26]	K. Brzostowski, R. Obuchowicz, Combining variational mode decomposition with regularisation techniques to denoise MRI data, Magn. Reson. Imaging, 106 (2024), 55–76. https://doi.org/10.1016/j.mri.2023.10.011 doi: 10.1016/j.mri.2023.10.011
[27]	D. Wei, S. Weng, F. Li, Nonconvex Rician noise removal via convergent plug-and-play framework, Appl. Math. Model., 123 (2023), 197–212. https://doi.org/10.1016/j.apm.2023.06.033 doi: 10.1016/j.apm.2023.06.033
[28]	X. Wu, S. Bricq, C. Collet, Brain MRI segmentation and lesion detection using generalized gaussian and Rician modeling, In: Proceedings volume 7962, medical imaging 2011: image processing, Lake Buena Vista (Orlando), Florida, United States, 2011, 796236. https://doi.org/10.1117/12.871384
[29]	L. Chen, Y. Li, T. Zeng, Variational image restoration and segmentation with Rician noise, J. Sci. Comput., 78 (2019), 1329–1352. https://doi.org/10.1007/s10915-018-0826-3 doi: 10.1007/s10915-018-0826-3
[30]	D. Jiang, W. Dou, L. Vosters, X. Xu, Y. Sun, T. Tan, Denoising of 3D MRI images with multi-channel residual learning of convolutional neural network, Jpn. J. Radiol., 36 (2018), 566–574. https://doi.org/10.1007/s11604-018-0758-8 doi: 10.1007/s11604-018-0758-8
[31]	S. Li, J. Zhou, D. Liang, Q. Liu, MRI denoising using progressively distribution-based neural network, Magn. Reson. Imaging, 71 (2020), 55–68. https://doi.org/10.1016/j.mri.2020.04.006 doi: 10.1016/j.mri.2020.04.006
[32]	X. You, N. Cao, H. Lu, M. Mao, W. Wang, Denoising of MRI images with Rician noise using a wider neural network and noise range division, Magn. Reson. Imaging, 64 (2019), 154–159. https://doi.org/10.1016/j.mri.2019.05.042 doi: 10.1016/j.mri.2019.05.042
[33]	C.-C. Wu, C.-H. Hsu, P. C. Wang, T.-W. Tu, Y.-Y. Hsu, Influence of Rician noise on cardiac MR image segmentation using deep learning, In: Intelligent systems design and applications. ISDA 2023, Cham: Springer, 2023,223–232. https://doi.org/10.1007/978-3-031-64813-7_24
[34]	L. I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259–268. https://doi.org/10.1016/0167-2789(92)90242-F doi: 10.1016/0167-2789(92)90242-F
[35]	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
[36]	M. C. Robini, A. Lachal, I. E. Magnin, A stochastic continuation approach to piecewise constant reconstruction, IEEE Trans. Image Process., 16 (2007), 2576–2589. https://doi.org/10.1109/TIP.2007.904955 doi: 10.1109/TIP.2007.904955
[37]	D. Krishnan, R. Fergus, Fast image deconvolution using hyper-Laplacian priors, In: Advances in neural information processing systems 22 (NIPS 2009), 2009, 1033–1041.
[38]	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
[39]	H. Attouch, J. Bolte, P. Redont, A. Soubeyran, Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the Kurdyka-Łojasiewicz inequality, Math. Oper. Res., 35 (2010), 438–457. https://doi.org/10.1287/moor.1100.0449 doi: 10.1287/moor.1100.0449
[40]	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
[41]	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
[42]	T. Sun, P. Yin, L. Cheng, H. Jiang, Alternating direction method of multipliers with difference of convex functions, Adv. Comput. Math., 44 (2018), 723–744. https://doi.org/10.1007/s10444-017-9559-3 doi: 10.1007/s10444-017-9559-3
[43]	T. Sun, H. Jiang, L. Cheng, W. Zhu, Iteratively linearized reweighted alternating direction method of multipliers for a class of nonconvex problems, IEEE Trans. Signal Process., 66 (2018), 5380–5391. https://doi.org/10.1109/TSP.2018.2868269 doi: 10.1109/TSP.2018.2868269
[44]	T. Sun, R. Barrio, M. Rodríguez, H. Jiang, Inertial nonconvex alternating minimizations for image deblurring, IEEE Trans. Image Process., 28 (2019), 6211–6224. https://doi.org/10.1109/TIP.2019.2917980 doi: 10.1109/TIP.2019.2917980
[45]	L. A. Vese, T. F. Chan, A multiphase level set framework for image segmentation using the Mumford and Shah model, Int. J. Comput. Vis., 50 (2002), 271–293. https://doi.org/10.1023/A:1020874308076 doi: 10.1023/A:1020874308076
[46]	Y. Yang, B. Wu, A new and fast multiphase image segmentation model for color images, Math. Probl. Eng., 2012 (2012), 494761, 494761. https://doi.org/10.1155/2012/494761 doi: 10.1155/2012/494761
[47]	Y. He, M. Y. Hussaini, J. Ma, B. Shafei, G. Steidl, A new fuzzy c-means method with total variation regularization for segmentation of images with noisy and incomplete data, Pattern Recogn., 45 (2012), 3463–3471. https://doi.org/10.1016/j.patcog.2012.03.009 doi: 10.1016/j.patcog.2012.03.009
[48]	R. Chan, H. Yang, T. Zeng, A two-stage image segmentation method for blurry images with poisson or multiplicative gamma noise, SIAM J. Imaging Sci., 7 (2014), 98–127. https://doi.org/10.1137/130920241 doi: 10.1137/130920241
[49]	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
[50]	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
[51]	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
[52]	Y. Chen, X. Ye, Projection onto a simplex, arXiv: 1101.6081.
[53]	B. N. Li, C. K. Chui, S. Chang, S. H. Ong, Integrating spatial fuzzy clustering with level set methods for automated medical image segmentation, Comput. Biol. Med., 41 (2011), 1–10. https://doi.org/10.1016/j.compbiomed.2010.10.007 doi: 10.1016/j.compbiomed.2010.10.007
[54]	S. Alpert, M. Galun, A. Brandt, R. Basri, Image segmentation by probabilistic bottom-up aggregation and cue integration, IEEE Trans. Pattern Anal., 34 (2012), 315–327. https://doi.org/10.1109/TPAMI.2011.130 doi: 10.1109/TPAMI.2011.130
[55]	P. Bideau, A. RoyChowdhury, R. R. Menon, E. Learned-Miller, The best of both worlds: combining CNNs and geometric constraints for hierarchical motion segmentation, In: 2018 IEEE/CVF Conference on computer vision and pattern recognition, Salt Lake City, UT, USA 2018,508–517. https://doi.org/10.1109/CVPR.2018.00060
[56]	K.-S. Chuang, H.-L. Tzeng, S. Chen, J. Wu, T.-J. Chen, Fuzzy C-means clustering with spatial information for image segmentation, Comput. Med. Imag. Grap., 30 (2006), 9–15. https://doi.org/10.1016/j.compmedimag.2005.10.001 doi: 10.1016/j.compmedimag.2005.10.001
[57]	P. Getreuer, Total variation deconvolution using split Bregman, Image Process. On Line, 2 (2012), 158–174. https://doi.org/10.5201/ipol.2012.g-tvdc doi: 10.5201/ipol.2012.g-tvdc

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