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Cartoon–Texture Image Decomposition Using Least Squares and Low-Rank Regularization

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Abstract

In this paper, we propose a novel model for the decomposition of cartoon–texture images, which integrates the edge-aware weighted least squares (WLS) with low-rank regularization. Unlike conventional methodologies that depend on total variation-based penalty functions, our model represents cartoon images using an edge-preserving WLS penalty. This approach effectively enhances edges and suppresses texture through iterative updates of an edge-preserving weight matrix. For the texture component, we introduce a low-rank penalty function to capture the structured regularity of texture patterns. By leveraging the repetitive nature of texture, our low-rank models can accurately represent these components. We employ a prediction–correction approach based on a three-block separable alternating direction multiplier method to solve the minimization problem, providing closed-form solutions for all subproblems. We also provide a convergence proof for the proposed algorithm. Numerical experiments validate the efficacy of our proposed method in successfully separating cartoon and texture components while preserving edges.

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Data Availability

No datasets were generated or analyzed during the current study.

Notes

  1. SNR is defined as \(\textrm{SNR} = 20\log _{10} \frac{\left\| f\right\| _2}{\left\| {\hat{f}}-f\right\| _2}\), where f and \({\hat{f}}\) are the true images and the reconstructed image, respectively.

References

  1. Bertsekas, D.: Convex Optimization Theory, vol. 1. Athena Scientific, Nashua (2009)

    MATH  Google Scholar 

  2. Biemond, J., Lagendijk, R.L.: Regularized iterative image restoration in a weighted Hilbert space. In: IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 11, pp. 1485–1488 (1986)

  3. Black, M.J., Sapiro, G., Marimont, D.H., Heeger, D.: Robust anisotropic diffusion. IEEE Trans. Image Process. 7(3), 421–432 (1998)

    Google Scholar 

  4. Cai, J.-F., Candès, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), 1956–1982 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Cai, X., Han, D., Yuan, X.: On the convergence of the direct extension of ADMM for three-block separable convex minimization models with one strongly convex function. Comput. Optim. Appl. 66(1), 39–73 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Catté, F., Lions, P.-L., Morel, J.-M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29(1), 182–193 (1992)

    MathSciNet  MATH  Google Scholar 

  7. Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1), 89–97 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Chambolle, A., Caselles, V., Cremers, D., Novaga, M., Pock, T.: An introduction to total variation for image analysis. Theor. Found. Numer. Methods Sparse Recovery 9(263–340), 227 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Chen, C., He, B., Ye, Y., Yuan, X.: The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent. Math. Program. 155(1), 57–79 (2016)

    MathSciNet  MATH  Google Scholar 

  10. Daubechies, I., DeVore, R., Fornasier, M., Güntürk, C.S.: Iteratively reweighted least squares minimization for sparse recovery. Commun. Pure Appl. Math. 63(1), 1–38 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Fan, Y.-R., Huang, T.-Z., Ma, T.-H., Zhao, X.-L.: Cartoon-texture image decomposition via non-convex low-rank texture regularization. J. Franklin Inst. 354(7), 3170–3187 (2017)

    MathSciNet  MATH  Google Scholar 

  12. Gazzola, S., Kilmer, M.E., Nagy, J.G., Semerci, O., Miller, E.L.: An inner-outer iterative method for edge preservation in image restoration and reconstruction. Inverse Probl. 36(12), 124004 (2020)

    MathSciNet  MATH  Google Scholar 

  13. Gilboa, G., Sochen, N., Zeevi, Y.Y.: Variational denoising of partly textured images by spatially varying constraints. IEEE Trans. Image Process. 15(8), 2281–2289 (2006)

    MATH  Google Scholar 

  14. Golub, G.H., Hansen, P.C., O’Leary, D.P.: Tikhonov regularization and total least squares. SIAM J. Matrix Anal. Appl. 21(1), 185–194 (1999)

    MathSciNet  MATH  Google Scholar 

  15. Golub, G.H., Van Loan, C.F.: Matrix Computations. JHU Press, Baltimore (2013)

    MATH  Google Scholar 

  16. Gu, S., Zhang, L., Zuo, W., Feng, X.: Weighted nuclear norm minimization with application to image denoising. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2862–2869 (2014)

  17. Gupta, B., Lamba, S.S.: An efficient anisotropic diffusion model for image denoising with edge preservation. Comput. Math. Appl. 93, 106–119 (2021)

    MathSciNet  MATH  Google Scholar 

  18. Han, D., Kong, W., Zhang, W.: A partial splitting augmented Lagrangian method for low patch-rank image decomposition. J. Math. Imaging Vis. 51(1), 145–160 (2015)

    MathSciNet  MATH  Google Scholar 

  19. Han, Y., Xu, C., Baciu, G., Li, M.: Lightness biased cartoon-and-texture decomposition for textile image segmentation. Neurocomputing 168, 575–587 (2015)

    MATH  Google Scholar 

  20. He, B., Tao, M., Yuan, X.: Alternating direction method with Gaussian back substitution for separable convex programming. SIAM J. Optim. 22(2), 313–340 (2012)

    MathSciNet  MATH  Google Scholar 

  21. He, B., Yuan, X.: A class of ADMM-based algorithms for three-block separable convex programming. Comput. Optim. Appl. 70(3), 791–826 (2018)

    MathSciNet  MATH  Google Scholar 

  22. Huang, G., Lanza, A., Morigi, S., Reichel, L., Sgallari, F.: Majorization-minimization generalized Krylov subspace methods for \(\ell _p\)–\(\ell _q\) optimization applied to image restoration. BIT Numer. Math. 57(2), 351–378 (2017)

    MATH  Google Scholar 

  23. Huang, Y., Ng, M.K., Wen, Y.-W.: A fast total variation minimization method for image restoration. Multiscale Model. Simul. 7(2), 774–795 (2008)

    MathSciNet  MATH  Google Scholar 

  24. Kamalaveni, V., Rajalakshmi, R.A., Narayanankutty, K.: Image denoising using variations of Perona–Malik model with different edge stopping functions. Procedia Comput. Sci. 58, 673–682 (2015)

    MATH  Google Scholar 

  25. Karras, D., Mertzios, G.: New PDE-based methods for image enhancement using SOM and Bayesian inference in various discretization schemes. Meas. Sci. Technol. 20(10), 104012 (2009)

    MATH  Google Scholar 

  26. Lanza, A., Morigi, S., Reichel, L., Sgallari, F.: A generalized Krylov subspace method for \(\ell _p-\ell _q\) minimization. SIAM J. Sci. Comput. 37(5), S30–S50 (2015)

    MATH  Google Scholar 

  27. Liang, X., Ren, X., Zhang, Z., Ma, Y.: Repairing sparse low-rank texture. In: 12th European Conference on Computer Vision, pp. 482–495. Springer (2012)

  28. Luo, Y., He, K., Xu, D., Shi, H., Yin, W.: Infrared and visible image fusion based on hybrid multi-scale decomposition and adaptive contrast enhancement. Signal Process. Image Commun. 130, 117228 (2024)

    Google Scholar 

  29. Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: the Fifteenth Dean Jacqueline B. Lewis Memorial Lectures, vol. 22. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  30. Ng, M.K., Chan, R.H., Tang, W.-C.: A fast algorithm for deblurring models with Neumann boundary conditions. SIAM J. Sci. Comput. 21(3), 851–866 (1999)

    MathSciNet  MATH  Google Scholar 

  31. Ono, S., Miyata, T., Yamada, I.: Cartoon-texture image decomposition using blockwise low-rank texture characterization. IEEE Trans. Image Process. 23(3), 1128–1142 (2014)

    MathSciNet  MATH  Google Scholar 

  32. Paige, C.C., Saunders, M.A.: LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. 8(1), 43–71 (1982)

    MathSciNet  MATH  Google Scholar 

  33. Pan, H., Wen, Y.-W., Huang, Y.: \(L_0\) gradient-regularization and scale space representation model for cartoon and texture decomposition. IEEE Trans. Image Process. (2024). https://doi.org/10.1109/TIP.2024.3403505

    Article  MATH  Google Scholar 

  34. Pang, Z.-F., Meng, G., Li, H., Chen, K.: Image restoration via the adaptive TVp regularization. Comput. Math. Appl. 80(5), 569–587 (2020)

    MathSciNet  MATH  Google Scholar 

  35. Pang, Z.-F., Zhang, H.-L., Luo, S., Zeng, T.: Image denoising based on the adaptive weighted TVp regularization. Signal Process. 167, 107325 (2020)

    MATH  Google Scholar 

  36. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)

    MATH  Google Scholar 

  37. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D Nonlinear Phenom. 60(1–4), 259–268 (1992)

    MathSciNet  MATH  Google Scholar 

  38. Schaeffer, H., Osher, S.: A low patch-rank interpretation of texture. SIAM J. Imaging Sci. 6(1), 226–262 (2013)

    MathSciNet  MATH  Google Scholar 

  39. Shi, B., Meng, G., Zhao, Z., Pang, Z.-F.: Image decomposition based on the adaptive direction total variation and G-norm regularization. Signal Image Video Process. 15(1), 155–163 (2021)

    MATH  Google Scholar 

  40. Tao, M., Yuan, X.: Recovering low-rank and sparse components of matrices from incomplete and noisy observations. SIAM J. Optim. 21(1), 57–81 (2011)

    MathSciNet  MATH  Google Scholar 

  41. Vese, L.A., Osher, S.J.: Image denoising and decomposition with total variation minimization and oscillatory functions. J. Math. Imaging Vis. 20(1), 7–18 (2004)

    MathSciNet  MATH  Google Scholar 

  42. Vogel, C., Oman, M.: Iterative method for total variation denoising. SIAM J. Sci. Comput. 17, 227–238 (1996)

    MathSciNet  MATH  Google Scholar 

  43. Wang, C., Xu, L., Liu, L.: Structure–texture image decomposition via non-convex total generalized variation and convolutional sparse coding. Vis. Comput. 39(3), 1121–1136 (2023)

    MATH  Google Scholar 

  44. Wang, W., Wang, J.: A variational model for cartoon–texture decomposition of a color image. J. Comput. Appl. Math. 449, 115932 (2024)

    MathSciNet  MATH  Google Scholar 

  45. Wen, Y.-W., Sun, H.-W., Ng, M.K.: A primal-dual method for the Meyer model of cartoon and texture decomposition. Numer. Linear Algebra Appl. 26(2), e2224 (2019)

    MathSciNet  MATH  Google Scholar 

  46. Wohlberg, B., Rodriguez, P.: An iteratively reweighted norm algorithm for minimization of total variation functionals. IEEE Signal Process. Lett. 14(12), 948–951 (2007)

    MATH  Google Scholar 

  47. Wolke, R., Schwetlick, H.: Iteratively reweighted least squares: algorithms, convergence analysis, and numerical comparisons. SIAM J. Sci. Stat. Comput. 9(5), 907–921 (1988)

    MathSciNet  MATH  Google Scholar 

  48. Yan, H.-Y., Zheng, Z.: Image cartoon–texture decomposition by a generalized non-convex low-rank minimization method. J. Franklin Inst. 361(2), 796–815 (2024)

    MathSciNet  MATH  Google Scholar 

  49. Zachevsky, I., Zeevi, Y.Y.J.: Statistics of natural stochastic textures and their application in image denoising. IEEE Trans. Image Process. 25(5), 2130–2145 (2016)

    MathSciNet  MATH  Google Scholar 

  50. Zhang, F., Ye, X., Liu, W.: Image decomposition and texture segmentation via sparse representation. IEEE Signal Process. Lett. 15, 641–644 (2008)

    MATH  Google Scholar 

  51. Zhang, H., He, W., Zhang, L., Shen, H., Yuan, Q.: Hyperspectral image restoration using low-rank matrix recovery. IEEE Trans. Geosci. Remote Sens. 52(8), 4729–4743 (2013)

    MATH  Google Scholar 

  52. Zhang, Y., Chen, L., Zhao, Z., Jia, J.: Multi-focus image fusion based on cartoon–texture image decomposition. Optik 127(3), 1291–1296 (2016)

    MATH  Google Scholar 

  53. Zhang, Z., Ganesh, A., Liang, X., Ma, Y.: Tilt: Transform invariant low-rank textures. Int. J. Comput. Vis. 99(1), 1–24 (2012)

    MathSciNet  MATH  Google Scholar 

  54. Zhang, Z., He, H.: A customized low-rank prior model for structured cartoon–texture image decomposition. Signal Process. Image Commun. 96, 116308 (2021)

    MATH  Google Scholar 

  55. Zhou, X., Yang, C., Zhao, H., Yu, W.: Low-rank modelling and its applications in image analysis. ACM Comput. Surv. 47(2), 1–33 (2014)

  56. Zhu, Z., Yin, H., Chai, Y., Li, Y., Qi, G.: A novel multi-modality image fusion method based on image decomposition and sparse representation. Inf. Sci. 432, 516–529 (2018)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was funded by the National Natural Science Foundation of China (Grant No. 12361089); the Scientific Research Fund Project of Yunnan Provincial Education Department (Grant No. 2024J0642); and Yunnan Fundamental Research Projects (Grant No. 202401AU070104).

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Authors and Affiliations

  1. School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan, People’s Republic of China

    Kexin Li

  2. School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan, 410081, People’s Republic of China

    You-wei Wen

  3. Department of Operations and Risk Management and School of Data Science, Lingnan University, Hong Kong, People’s Republic of China

    Raymond H. Chan

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  1. Kexin Li
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  3. Raymond H. Chan
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Kexin Li and You-wei Wen wrote the main manuscript text; Raymond. H Chan revised and improved the manuscript. All authors reviewed the manuscript.

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Correspondence to You-wei Wen.

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Li, K., Wen, Yw. & Chan, R.H. Cartoon–Texture Image Decomposition Using Least Squares and Low-Rank Regularization. J Math Imaging Vis 67, 5 (2025). https://doi.org/10.1007/s10851-024-01216-8

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