Informatics and Applications
2025, Volume 19, Issue 4, pp 53-64COLOR IMAGE RESTORATION VIA THE LATTICE BOLTZMANN METHOD FOR ANISOTROPIC NONLINEAR DIFFUSION
- G. A. Chumarin
Abstract
The work proposes a method for restoring damaged regions of color three-channel images (the inpainting problem) based on the equation of nonlinear anisotropic diffusion. As the numerical solution algorithm, the lattice Boltzmann equation with five discrete velocities and multiple relaxation times is employed. The direction and intensity of the smoothing are determined using the structure matrix. A parallel implementation of the algorithm has been developed using MPI (Message Passing Interface) technology with image domain decomposition in a Cartesian topology. The application of the proposed method to images with defects of various shapes and sizes is examined. The results demonstrate the correctness of structural and color information restoration in the damaged regions. The accuracy of the method is evaluated on a test set of 10,000 images, and the execution times of sequential and parallel versions of the algorithm are compared.[+] References (42)
- McNamara, G., and G. Zanetti. 1988. Use of the Boltzmann equation to simulate lattice gas automata. Phys. Rev. Lett. 61(20):2332-2335. doi: 10.1103/ PhysRevLett.61.2332.
- Guo, Z., and C. Shu. 2013. Lattice Boltzmann method and its applications in engineering. Singapore: World Scientific Publs. 404 p. doi: 10.1142/8806.
- Kruger, T, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Silva, and E. Viggen. 2017. The lattice Boltzmann method: Principles and practice. Cham: Springer. 694 p. doi: 10.1007/978-3-319-44649-3.
- Succi, S. 2018. The lattice Boltzmann equation: For complex states of flowing matter. Oxford: Oxford University Press. 784 p. doi: 10.1093/oso/9780199592357.001.0001.
- Succi, S. 2022. Lattice fluid dynamics: Thirty-five years down the road. CR Mecanique 350:207-218. doi: 10.5802/crmeca. 161.
- Flekkoy, E. 1993. Lattice Bhatnagar-Gross-Krookmodels for miscible fluids. Phys. Rev. E 47(6):4247-4257. doi: 10.1103/PhysRevE.47.4247.
- Wolf-Gladrow, D. 1995. Lattice Boltzmann equation for diffusion. J. Stat. Phys. 79:1023-1032. doi: 10.1007/ BF02181215.
- Suga, S. 2006. Numerical schemes obtained from lattice Boltzmann equations for advection-diffusion equations. Int. J. Mod. Phys. C 17(11):1563-1577. doi: 10.1142/ S0129183106010030.
- Suga, S. 2009. Stability and accuracy of lattice Boltzmann schemes for anisotropic advection-diffusion equations. Int. J. Mod. Phys. C 20(04):633-650. doi: 10.1142/ S0129183109013856.
- Chai, Z., and T. S. Zhao. 2013. Lattice Boltzmann model for the convection-diffusion equation. Phys. Rev. E 87(6):063309. doi: 10.1103/PhysRevE.87.063309.
- Chen, Y., Z. Chai, and B. Shi. 2024. A general fourth- order mesoscopic multiple-relaxation-time lattice Boltzmann model and its macroscopic finite-difference scheme for two-dimensional diffusion equations. J. Comput. Phys. 509:113045. doi: 10.1016/j.jcp.2024.113045.
- Shi, B., and Z. Guo. 2009. Lattice Boltzmann model for nonlinear convection-diffusion equations. Phys. Rev. E 79(1):016701. doi: 10.1103/PhysRevE.79.016701.
- Yoshida, H., and M. Nagaoka. 2010. Multiple-relaxationtime lattice Boltzmann model for the convection and anisotropic diffusion equation. J. Comput. Phys. 229(20):7774-7795. doi: 10.1016/j.jcp.2010.06.038.
- Chai, Z., B. Shi, and Z. Guo. 2016. Amultiple-relaxation- time lattice Boltzmann model for general nonlinear anisotropic convection-diffusion equations. J. Sci. Com- put. 69:355-390. doi: 10.1007/s10915-016-0198-5.
- Perona, P., and J. Malik. 1990. Scale-space and edge detection using anisotropic diffusion. IEEE T. Pattern Anal. 12(7):629-639. doi: 10.1109/34.56205.
- Weickert, J. 1998. Anisotropic diffusion in image processing. Stuttgart: Teubner. 182 p.
- Jawerth, B., P. Lin, and E. Sinzinger. 1999. Lattice Boltzmann models for anisotropic diffusion of images. J. Math. Imaging Vis. 11:231-237. doi: 10.1023/A:1008304519705.
- Zhang, W, and B. Shi. 2012. Application of lattice Boltzmann method to image filtering. J. Math. Imaging Vis. 43:135-142. doi: 10.1007/s10851-011-0295-x.
- Rudin, L., S. Osher, and E. Fatemi. 1992. Nonlinear total variation based noise removal algorithms. Physica D 60(1-4):259-268. doi: 10.1016/0167-2789(92)90242-F
- Chang, Q., and T. Yang. 2009. Alattice Boltzmann method for image denoising. IEEE T. Image Process. 18(12):2797- 2802. doi: 10.1109/TIP.2009.2028369.
- Chen, J., Z. Chai, B. Shi, and W Zhang. 2014. Lattice Boltzmann method for filtering and contour detection of natural images. Comput. Math. Appl. 68(3):257-268. doi: 10.1016/j.camwa.2014.05.023.
- Michelet, J., M. M. Tekitek, and M. Berthier. 2022. Multiple relaxation time lattice Boltzmann schemes for advection-diffusion equations with application to radar image processing. J. Comput. Phys. 471:111612. doi: 10.1016/j.jcp.2022.111612.
- Chen, Yu, Z. Yan, and J. Shi. 2007. Application of lattice Boltzmann method to image segmentation. 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society. Lyon, France: IEEE. 6561- 6564. doi: 10.1109/IEMBS.2007.4353863.
- Chen, Yu, Z. Yan, and Y. Qian. 2008. An anisotropic diffusion model for medical image smoothing by using the lattice Boltzmann method. 7th Asian-Pacific Conference on Medical and Biological Engineering. Eds. Yi Peng, and W Xiaohong. IFMBE Proceedings ser. Berlin, Heidelberg: Springer. 19:255-259. doi: 10.1007/978-3-540- 79039-6_65.
- Chen, Yu, L. Navarro, Y. Wang, and G. Courbebaisse. 2014. Segmentation of the thrombus of giant intracranial aneurysms from CT angiography scans with lattice Boltzmann method. Med. Image Anal. 18(1):1-8. doi: 10.1016/j.media.2013.08.003.
- Nguyen, K. L., M. M.'Tekitek, P. Delachartre, and M. Berthier. 2019. Multiple relaxation time lattice Boltzmann models for multigrid phase-field segmentation of tumors in 3D ultrasound images. SIAM J. Imaging Sci. 12(3):1324-1346. doi: 10.1137/18M123462X.
- Ilyin, O. 2023. Lattice Boltzmann model for diffusion equation with reduced truncation errors: Applications to Gaussian filtering and image processing. Appl. Math. Comput. 456(1):128123. doi: 10.1016/j.amc.2023.128123.
- Ilyin, O. 2024. Lattice Boltzmann algorithm for vessel enhancement problems. Lobachevskii J. Mathematics 45:5823-5830. doi: 10.1134/S1995080224606714.
- Weickert, J. 1999. Coherence-enhancing diffusion filtering. Int. J. Comput. Vision 31:111-127. doi: 10.1023/ A:1008009714131.
- Chan, T. F, S. H. Kang, and J. Shen. 2003. Euler's elas- tica and curvature-based inpainting. SIAM J. Appl. Math. 63(2):564-592. doi: 10.1137/S0036139901390088.
- Chan, T. F, and J. Shen. 2005. Variational image inpainting. Communications Pure Applied Mathematics 58:579619. doi: 10.1002/cpa.20075.
- Bertozzi, A. L., S. Esedoglu, and A. Gillette. 2007. Inpainting of binary images using the Cahn-Hilliard equation. IEEET. Image Process. 16(1):285-291.doi: 10.1109/ TIP.2006.887728.
- Bertozzi, A. L., S. Esedoglu, and A. Gillette. 2007. Analysis of a two-scale Cahn-Hilliard model for image inpainting. Multiscale Model. Sim. 6(3):913-936. doi: 10.1137/060660631.
- Bertozzi, A. L., and C. B. Schonlieb. 2011. Unconditionally stable schemes for higher order inpainting. Commun. Math. Sci. 9(2):413-457. doi: 10.4310/CMS.2011. v9.n2.a4.
- Alvarez, L., and L. Mazorra. 1994. Signal and image restoration using shock filters and anisotropic diffusion. SIAMJ. Numer. Anal. 31(2):590-605. doi: 10.1137/ 0731032.
- Schaefer, K., and J. Weickert. 2024. Regularised diffusion-shock inpainting. J. Math. Imaging Vis. 66:447463. doi: 10.1007/s10851-024-01175-0.
- Novak, A., and N. Reinic. 2022. Shock filter as the classifier for image inpainting problem using the Cahn- Hilliard equation. Comput. Math. Appl. 123:105-14. doi: 10.1016/j.camwa.2022.07.021.
- Korolev, D. O., and O. G. Maleev. 2023. Issledovanie effektivnosti primeneniya binarnykh neyronnykh setey pri detektirovanii ob"ekta na izobrazhenii [Efficiency of binary neural networks for object detection on an image]. Informatika ieePrimeneniya - Inform. Appl. 17(3):88-92. doi: 10.14357/19922264230312.
- Kovalenko, A. S. 2024. Primenenie razlozheniya izobrazheniya s pomoshch'yu diskretnogo veyvlet- preobrazovaniya dlya postroeniya arkhitektury shumopodavlyayushchey neyronnoy seti [Image decomposition with discrete wavelet transform to design a denoising neural network]. Informatika i ee Primeneniya - Inform. Appl. 8(2):60-71. doi: 10.14357/19922264240209.
- Dostovalova, A. M. 2024. Neyrosetevoe kvadroderevo i ego primenenie dlya segmentirovaniya sputnikovykh izobrazheniy [Neural quadtree and its applications for sar imagery segmentation]. Informatika i ee Primenen iya-Inform. Appl. 18(4):77-85. doi: 10.14357/19922264240410.
- Zhou, B., A. Lapedriza, A. Khosla, A. Oliva, and A. Torralba. 2018. Places: A 10 million image database for scene recognition. IEEE T. Pattern Anal. 40(6):1452-1464. doi: 10.1109/TPAMI.2017.2723009.
- Zhou, W, A. Bovik, H. Sheikh, and E. Simoncelli. 2004. Image quality assessment: From error visibility to structural similarity. IEEE T. Image Process. 13(4):600-612. doi: 10.1109/TIP2003.819861.
[+] About this article
Title
COLOR IMAGE RESTORATION VIA THE LATTICE BOLTZMANN METHOD FOR ANISOTROPIC NONLINEAR DIFFUSIONJournal
Informatics and Applications2025, Volume 19, Issue 4, pp 53-64
Cover Date
2025-30-12DOI
10.14357/19922264250406Print ISSN
1992-2264Publisher
Institute of Informatics Problems, Russian Academy of SciencesAdditional Links
Key words
image restoration; inpainting; lattice Boltzmann equations; anisotropic diffusionAuthors
G. A. Chumarin
,
Author Affiliations
Federal Research Center "Computer Science and Control" of the Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation
M. V Lomonosov Moscow State University, 1-52 Leninskie Gory, Moscow 119991, GSP-1, Russian Federation