On the existence and uniqueness of weak solutions of a coupled diffusion system related to image restoration

Subit K. Jain; Sudeb Majee; Rajendra K. Ray

doi:10.3934/ipi.2023027

Article Contents

2024, Volume 18, Issue 1: 165-182. Doi: 10.3934/ipi.2023027

This issue Previous Article Linear inverse problems with nonnegativity constraints: Singularity of optimisers Next Article The Born approximation in the three-dimensional Calderón problem Ⅱ: Numerical reconstruction in the radial case

On the existence and uniqueness of weak solutions of a coupled diffusion system related to image restoration

1.
Department of Mathematics & Scientific Computing, National Institute of Technology Hamirpur, India
2.
School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, India

^*Corresponding author: Rajendra K. Ray
^*Corresponding author: Rajendra K. Ray

Received: September 2022

Revised: May 2023

Early access: May 31, 2023

Published online: May 31, 2023

Abstract / Introduction Full Text(HTML) Figure(8) / Table(2) Related Papers Cited by

Abstract

In this study, the existence and uniqueness of the weak solution of a coupled diffusion system is presented. Since the considered problem is coupled and nonlinear, first we consider a corresponding linearized problem and then use a weak convergence method with Schauder fixed-point theorem to prove the existence of a weak solution of the underlying problem in an appropriate Hilbert space. Moreover, the computational experiments show that the considered model could be applied to image restoration problems.

Keywords:

Mathematics Subject Classification: 35K55, 65M06, 68U10.

Citation:

Full Text(HTML)

Figure 1. Clean gray-scale images

Download: Full-size image PowerPoint slide

Figure 2. Restored images for different values of time step $ \tau $. Initially, images are degraded by Gaussian noise with mean $ \mu = 0 $ and standard deviation $ \sigma = 30 $

Download: Full-size image PowerPoint slide

Figure 3. A brick image corrupted by additive Gaussian noise with $ \mu = 0 $ and $ \sigma = 30 $ and restored by different models. (a) Noisy (b) PM: $ K = 10 $ (c) ROF (d) RD: $ \lambda = 0.02 $ (e) VBS: $ \lambda = 0.1, K = 6 $ (f) Proposed: $ k = 1.5 $

Download: Full-size image PowerPoint slide

Figure 4. Residual images of the clear brick image 1a and the images 3b–3f

Download: Full-size image PowerPoint slide

Figure 5. A starfish image corrupted by additive Gaussian noise with $ \mu = 0 $ and $ \sigma = 30 $ and restored by different models. (a) Noisy (b) PM: $ K = 10 $ (c) ROF (d) RD: $ \lambda = 0.05 $ (e) VBS: $ \lambda = 0.1, K = 2 $ (f) Proposed: $ k = 3 $

Download: Full-size image PowerPoint slide

Figure 6. Clean color images

Download: Full-size image PowerPoint slide

Figure 7. Peppers image corrupted by additive Gaussian noise with $ \mu = 0 $ and $ \sigma = 80 $ and restored by different models. (a) Noisy: PSNR = 11.74 (b) PM: PSNR = 22.29 (c) ROF: PSNR = 22.36 (d) RD: PSNR = 22.70 (e) VBS: PSNR = 22.55 (f) Proposed: PSNR = 22.81

Download: Full-size image PowerPoint slide

Figure 8. Caps image corrupted by additive Gaussian noise with $ \mu = 0 $ and $ \sigma = 50 $ and restored by different models. (a) Noisy: PSNR = 14.73 (b) PM: PSNR = 27.73 (c) ROF: PSNR = 27.91 (d) RD: PSNR = 27.95 (e) VBS: PSNR = 28.02 (f) Proposed: PSNR = 28.10

Download: Full-size image PowerPoint slide

Table 1. Calculated MSSIM and PSNR values for the restored images when the images are degraded by additive Gaussian noise with mean $ \mu = 0 $ and standard deviation $ \sigma = 30 $. Results are computed by the proposed model with different values of time step ($ \tau $)

Image	Time step ($ \tau $) $ \Rightarrow $	0.05	0.10	0.15	0.20	0.25	0.30
Brick	MSSIM	0.8625	0.8626	0.8626	0.8626	0.8626	0.8624
	PSNR	30.5638	30.5659	30.5636	30.5633	30.5623	30.5612
	CPU time (s)	71.61	31.30	20.83	15.64	12.41	10.18
Starfish	MSSIM	0.8138	0.8137	0.8137	0.8135	0.8135	0.8134
	PSNR	26.9196	26.9276	26.9245	26.9241	26.9217	26.9199
	CPU time (s)	19.59	10.58	8.73	6.61	5.64	5.27

| Show Table

DownLoad: CSV

Table 2. Comparison of MSSIM and PSNR values for various approaches and proposed model. Clean image is degraded by additive Gaussian noise with mean $ \mu = 0 $ and different values of standard deviation $ \sigma $

Image	$ \sigma $	PM model[26]		ROF model[30]		RD model[16]		VBS model[28]		Proposed Model
Image	$ \sigma $	MSSIM	PSNR	MSSIM	PSNR	MSSIM	PSNR	MSSIM	PSNR	MSSIM	PSNR
Brick	10	0.9422	34.18	0.9450	34.27	0.9447	34.22	0.9456	34.38	0.9465	34.55
	30	0.8313	30.03	0.8446	30.17	0.8442	30.33	0.8489	30.24	0.8626	30.56
	50	0.7359	27.91	0.7522	28.14	0.7446	28.03	0.7515	28.05	0.7689	28.25
Starfish	10	0.9232	32.24	0.9213	32.11	0.9232	32.11	0.9244	32.02	0.9286	32.62
	30	0.8006	26.46	0.7967	26.52	0.7904	26.56	0.8081	26.87	0.8135	26.92
	50	0.7064	23.93	0.7128	23.94	0.7094	24.05	0.7225	24.11	0.7257	24.19

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	R. Adam, Sobolev spaces, in: Pure and Applied Mathematics Series of Monographs and Textbooks, Vol. 65, Academic Press, Inc., New York, San Francisco, London,, 1975.
[2]	L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel, Axioms and fundamental equations of image processing, Archive for Rational Mechanics and Analysis, 123 (1993), 199-257. doi: 10.1007/BF00375127.
[3]	A. Araújo, S. Barbeiro and P. Serranho, Stability of finite difference schemes for complex diffusion processes, SIAM Journal on Numerical Analysis, 50 (2012), 1284-1296. doi: 10.1137/110825789.
[4]	A. Araújo, S. Barbeiro and P. Serranho, Stability of finite difference schemes for nonlinear complex reaction–diffusion processes, IMA Journal of Numerical Analysis, 35 (2015), 1381-1401. doi: 10.1093/imanum/dru037.
[5]	G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Applied Mathematical Sciences, 147. Springer, New York, 2006.
[6]	G. Baravdish, O. Svensson, M. Gulliksson and Y. Zhang, Damped second order flow applied to image denoising, IMA Journal of Applied Mathematics, 84 (2019), 1082-1111. doi: 10.1093/imamat/hxz027.
[7]	A. Belahmidi and A. Chambolle, Time-delay regularization of anisotropic diffusion and image processing, ESAIM: Mathematical Modelling and Numerical Analysis, 39 (2005), 231-251. doi: 10.1051/m2an:2005010.
[8]	Y. Cao, J. Yin, Q. Liu and M. Li, A class of nonlinear parabolic-hyperbolic equations applied to image restoration, Nonlinear Analysis: Real World Applications, 11 (2010), 253-261. doi: 10.1016/j.nonrwa.2008.11.004.
[9]	F. Catté, P.-L. Lions, J.-M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM Journal on Numerical Analysis, 29 (1992), 182-193. doi: 10.1137/0729012.
[10]	A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numerische Mathematik, 76 (1997), 167-188. doi: 10.1007/s002110050258.
[11]	C. Elliott and S. Smitheman, Numerical analysis of the TV regularization and $ {H}^{-1} $ fidelity model for decomposing an image into cartoon plus texture, IMA Journal of Numerical Analysis, 29 (2009), 651-689. doi: 10.1093/imanum/drn025.
[12]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 2015.
[13]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. doi: 10.1090/gsm/019.
[14]	R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2002.
[15]	P. Guidotti, Anisotropic diffusions of image processing from perona-malik on, Advanced Studies in Pure Mathematics, 99.
[16]	Z. Guo, J. Yin and Q. Liu, On a reaction–diffusion system applied to image decomposition and restoration, Mathematical and Computer Modelling, 53 (2011), 1336-1350. doi: 10.1016/j.mcm.2010.12.031.
[17]	S. K. Jain and R. K. Ray, Edge detectors based telegraph total variational model for image filtering, Information Systems Design and Intelligent Applications, (2016), 119-126. doi: 10.1007/978-81-322-2755-7_13.
[18]	S. K. Jain, R. K. Ray and A. Bhavsar, A comparative study of iterative solvers for image de-noising, Proceedings of the 3rd International Conference on Frontiers of Intelligent Computing: Theory and Applications (FICTA) 2014, (2015), 307-314.
[19]	S. K. Jain, R. K. Ray and A. Bhavsar, Iterative solvers for image denoising with diffusion models: A comparative study, Comput. Math. Appl., 70 (2015), 191-211. doi: 10.1016/j.camwa.2015.04.009.
[20]	J. L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris, 1968.
[21]	X. Liu, L. Huang and Z. Guo, Adaptive fourth-order partial differential equation filter for image denoising, Applied Mathematics Letters, 24 (2011), 1282-1288. doi: 10.1016/j.aml.2011.01.028.
[22]	H. Luo, L. Zhu and H. Ding, Coupled anisotropic diffusion for image selective smoothing, Signal Processing, 86 (2006), 1728-1736. doi: 10.1016/j.sigpro.2005.09.019.
[23]	K. Mikula, Image processing with partial differential equations, Modern Methods in Scientific Computing and Applications, 75 (2002), 283-321.
[24]	M. Nitzberg and T. Shiota, Nonlinear image filtering with edge and corner enhancement, IEEE Transactions on Pattern Analysis and Machine Intelligence, 14 (1992), 826-833. doi: 10.1109/34.149593.
[25]	J. Nolen, Partial Differential Equations and Diffusion Processes, Technical report, Technical report, Stanford University, Department of Mathematics, 2009.
[26]	P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639. doi: 10.1109/34.56205.
[27]	V. S. Prasath and A. Singh, A hybrid convex variational model for image restoration, Applied Mathematics and Computation, 215 (2010), 3655-3664. doi: 10.1016/j.amc.2009.11.003.
[28]	V. S. Prasath and D. Vorotnikov, On a system of adaptive coupled pdes for image restoration, Journal of Mathematical Imaging and Vision, 48 (2014), 35-52. doi: 10.1007/s10851-012-0386-3.
[29]	V. Ratner and Y. Y. Zeevi, Image enhancement using elastic manifolds, Image Analysis and Processing, 2007. ICIAP 2007. 14th International Conference on, (2007), 769-774. doi: 10.1109/ICIAP.2007.4362869.
[30]	L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.
[31]	A. Siddig, Z. Guo, Z. Zhou and B. Wu, An image denoising model based on a fourth-order nonlinear partial differential equation, Comput. Math. Appl., 76 (2018), 1056-1074. doi: 10.1016/j.camwa.2018.05.040.
[32]	Y.-H. R. Tsai and S. Osher, Total variation and level set methods in image science, Acta Numerica, 14 (2005), 509-573. doi: 10.1017/S0962492904000273.
[33]	Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861.
[34]	J. Weickert, A review of nonlinear diffusion filtering, Scale-Space Theory in Computer Vision, (1997), 1-28.
[35]	J. Weickert, Anisotropic Diffusion in Image Processing, European Consortium for Mathematics in Industry. B. G. Teubner, Stuttgart, 1998.
[36]	J. Weickert, Applications of nonlinear diffusion in image processing and computer vision, Acta Math. Univ. Comenianae, 70 (2001), 33-50.
[37]	A. P. Witkin, Scale-space filtering: A new approach to multi-scale description, Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP'84., 9 (1984), 150-153. doi: 10.1109/ICASSP.1984.1172729.
[38]	Y.-L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Transactions on Image Processing, 9 (2000), 1723-1730. doi: 10.1109/83.869184.
[39]	R. Zanella, F. Porta, V. Ruggiero and M. Zanetti, Serial and parallel approaches for image segmentation by numerical minimization of a second-order functional, Applied Mathematics and Computation, 318 (2018), 153-175. doi: 10.1016/j.amc.2017.07.021.
[40]	W. Zhang, J. Li and Y. Yang, A class of nonlocal tensor telegraph-diffusion equations applied to coherence enhancement, Comput. Math. Appl., 67 (2014), 1461-1473. doi: 10.1016/j.camwa.2014.02.013.