Digital Image Inpainting Analysis Based on the Cahn-Hilliard Model

Nada S. AL-Fartosi and Ahmed K. Al-Jaberi

References:

1. C. Schönlieb and A. Bertozzi, “Unconditionally stable schemes for higher order inpainting,” Commun. Math. Sci., vol. 9, no. 2, pp. 413-457, 2011.
2. A. Bertozzi, S. Esedoglu, and A. Gillette, “Analysis of a two-scale Cahn–Hilliard model for binary image inpainting,” Multiscale Model. Simul., vol. 6, no. 3, pp. 913-936, 2007.
3. A. L. Bertozzi, S. Esedoglu, and A. Gillette, “Inpainting of binary images using the Cahn–Hilliard equation,” IEEE Trans. Image Process., vol. 16, no. 1, pp. 285-291, 2006.
4. A. A. Al-Jaberi, S. A. Jassim, and N. Al-Jawad, “Inpainting large missing regions based on Seam Carving,” EAI Endorsed Trans. Ind. Networks Intell. Syst., vol. 5, no. 16, 2018.
5. L. Cherfils, H. Fakih, and A. Miranville, “A Cahn–Hilliard system with a fidelity term for color image inpainting,” J. Math. Imaging Vis., vol. 54, pp. 117-131, 2016.
6. N. Bullerjahn, “Error estimates for full discretization by an almost mass consEervation technique for Cahn–Hilliard systems with dynamic boundary conditions,” arXiv Prepr. arXiv2502.03847, 2025.
7. A. K. Al-Jaberi, A. Asaad, S. A. Jassim, and N. Al-Jawad, “Topological data analysis to improve exemplar-based inpainting,” in Mobile Multimedia/Image Processing, Security, and Applications 2018, SPIE, 2018, pp. 13-24.
8. E. Beretta, C. Cavaterra, M. Fornoni, and M. Grasselli, “Optimal control of the fidelity coefficient in a Cahn-Hilliard image inpai,” arXiv Prepr. arXiv2502.03025, 2025.
9. J. A. Carrillo, S. Kalliadasis, F. Liang, and S. P. Perez, “Enhancement of damaged-image prediction through Cahn–Hilliard image inpainting,” R. Soc. Open Sci., vol. 8, no. 5, p. 201294, 2021.
10. D. Jiang, M. Azaiez, A. Miranville, and C. Xu, “Nonlocal Cahn-Hilliard type model for image inpainting,” Comput. Math. with Appl., vol. 159, pp. 76-91, 2024.
11. A. Theljani, H. Houichet, and A. Mohamed, “An adaptive Cahn-Hilliard equation for enhanced edges in binary image inpainting,” J. Algorithm. Comput. Technol., vol. 14, p. 1748302620941430, 2020.
12. M. Burger, L. He, and C.-B. Schönlieb, “Cahn–Hilliard inpainting and a generalization for grayvalue images,” SIAM J. Imaging Sci., vol. 2, no. 4, pp. 1129-1167, 2009.
13. H. Qiumei, M. Jiaxuan, and X. Zhen, “Mass-preserving Spatio-temporal adaptive PINN for Cahn-Hilliard equations with strong nonlinearity and singularity,” arXiv Prepr. arXiv2404.18054, 2024.
14. J. Bosch and M. Stoll, “A fractional inpainting model based on the vector-valued Cahn–Hilliard equation,” SIAM J. Imaging Sci., vol. 8, no. 4, pp. 2352-2382, 2015.
15. A. L. Brkić, D. Mitrović, and A. Novak, “On the image inpainting problem from the viewpoint of a nonlocal Cahn-Hilliard type equation,” J. Adv. Res., vol. 25, pp. 67-76, 2020.
16. Q. Zou, “An image inpainting model based on the mixture of Perona–Malik equation and Cahn–Hilliard equation,” J. Appl. Math. Comput., vol. 66, no. 1, pp. 21-38, 2021.
17. A. K. Al-Jaberi, S. A. Jassim, and N. Al-Jawad, “Colourizing monochrome images,” in Mobile Multimedia/Image Processing, Security, and Applications 2018, SPIE, 2018, pp. 25-37.
18. A. Halim and B. V. R. Kumar, “An anisotropic PDE model for image inpainting,” Comput. Math. with Appl., vol. 79, no. 9, pp. 2701-2721, 2020.
19. J. W. Cahn, “On spinodal decomposition,” Acta Metall., vol. 9, no. 9, pp. 795-801, 1961.
20. J. W. Cahn, “On spinodal decomposition in cubic crystals,” Acta Metall., vol. 10, no. 3, pp. 179-183, 1962.
21. J. W. Cahn and J. E. Hilliard, “Free energy of a nonuniform system. I. Interfacial free energy,” J. Chem. Phys., vol. 28, no. 2, pp. 258-267, 1958.
22. C. Zhang, J. Tian, D. Li, X. Hou, and L. Wang, “An anisotropic PDE model for image inpainting photoplethysmography (IPPG) method,” Technol. Heal. Care, vol. 30, no. 1_suppl, pp. 391-402, 2022.
23. M. D. Fairchild, “Color Appearance Models.”
24. G. Sharma and R. Bala, Digital color imaging handbook, CRC Press, 2017.
25. K. Cheng, W. Feng, C. Wang, and S. M. Wise, “An energy stable fourth order finite difference scheme for the Cahn–Hilliard equation,” J. Comput. Appl. Math., vol. 362, pp. 574-595, 2019.
26. K. Gu, S. Wang, G. Zhai, W. Lin, X. Yang, and W. Zhang, “Analysis of distortion distribution for pooling in image quality prediction,” IEEE Trans. Broadcast., vol. 62, no. 2, pp. 446-456, 2016.
27. A. C. Bovik, Handbook of image and video processing, Academic Press, 2010.
28. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process., vol. 13, no. 4, pp. 600-612, 2004.
29. T. Barbu, Digital Image Processing, Analysis and Computer Vision Using Nonlinear Partial Differential Equations, vol. 1211, Springer Nature, 2025.
30. S. Prativadibhayankaram et al., “A study on the effect of color spaces in learned image compression,” in 2024 IEEE International Conference on Image Processing (ICIP), IEEE, 2024, pp. 3744-3750.
31. X. He, Y. Liu, P. Beckett, H. Uddin, A. Nirmalathas, and R. R. Unnithan, “A new CMY camera technology using Al-TiO2-Al nanorod filter mosaic integrated on a CMOS image sensor,” arXiv Prepr. arXiv2010.11680, 2020.
32. P. Hu, Y. Han, and J.-S. Pan, “An Advanced Bald Eagle Search Algorithm for Image Enhancement,” Comput. Mater. Contin., vol. 82, no. 3, 2025.
33. D. A. Kerr, “The CIE XYZ and xyY color spaces,” Colorimetry, vol. 1, no. 1, pp. 1-16, 2010.