Abstract
In this paper, we consider integral equation model for image restoration. Out-of-focus models are formulated to Fredholm integral equations of the first kind. This inverse problem is ill-posed, and regularization methods must be applied for approximating the reconstructed image. In general, extrapolation for increasing the accuracy of regularization methods is much less considered. In this paper, we apply linear extrapolation for Lavrentiev regularization for image restoration. This linear extrapolation is presented in Hamarik et al. (J Inv Ill-Posed Probl 15:277–294, 2007) in order to increase accuracy of solution of Fredholm integral equation of the first kind. Also, we apply Graves and Prenter method as another regularization method because kernel of integral equation is self-adjoint and symmetric. In addition, since kernel of integral equation in out-of-focus models is a Gaussian function with small standard deviation, approximating these integrals requires careful numerical treatment, and to meet this challenge, we apply numerical quadrature based on graded mesh that is applied in Ma and Xu (J Sci Comput, 2018). Finally, experimental results of extrapolated Lavrentiev method and Graves and Prenter method in terms of visual quality of reconstructed images and relative \({L^2}\) error are presented.
Similar content being viewed by others
References
Hamarik, U., Palm, R., Raus, T.: Use of extrapolation in regularization methods. J. Inv. Ill-Posed Probl. 15, 277–294 (2007)
Ma, Y., Xu, Y.: Computing integrals Involved the Gaussian function with a small standard deviation. J. Sci. Comput. (2018)
Reeves, Stanley J.: Image restoration: fundamentals of image restoration. Acad. Press Libr. Signal Process. 4, 165–192 (2014)
Kabanikhin, S.I., Romanov, V.G., Vasin, V.V.: Pioneering papers by M. M. Lavrentiev. J. Inv. Ill-Posed Probl. 15, 441–450 (2007)
Plato, R.: On the discrepancy principle for iterative and parametric methods to solve linear ill-posed equations. Numer. Math. 75, 99–120 (1996)
Lu, Y., Shen, L., Xu, Y.: Integral equation models for image restoration: high accuracy methods and fast algorithms. Inverse Probl. 26(4), 045006 (2010)
Graves, J., Prenter, P.M.: Numerical iterative filters applied to first kind Fredholm integral equations. Numer. Math. 30, 281–299 (1978)
Hansen, P.C.: Deconvolution and regularization with Toeplitz matrices. Numer. Algorithm 29, 323–378 (2002)
Bertero, M., Brianzi, P., Pike, E.: Super-resolution in confocal scanning microscopy. Inverse Probl. 3, 195–212 (1987)
Gonzalez, R., Woods, R.: Digital Image Processing. Addison-Wesley, Boston (1993)
Chaudhuri, S., Rajagopalan, A.: Depth from Defocus: A real aperture imaging approach. Springer, New York (1999)
Stein, E., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princton (1971)
Mesgarani, H., Aghazadeh, N., Parmour, P.: Aitken extrapolation and epsilon algorithm for an accelerated solution of weakly singular nonlinear Volterra integral equations. Phys. Scr. 81, 025006 (2010)
Landweber, L.: An iteration formula for Fredholm integral equations of the first kind. Am. J. Math. 73, 615–624 (1951)
Krasnosel’skii, M.A., Emelin, I.V., Veretennikov, AYu.: On regularization of ill-posed problems by stop-rules of iterative procedures with random errors. Numer. Funct. Anal. Optim. 5(2), 199–215 (1982)
Strand, Otto Neali: Theory and methods related to the singular-function expansion and Landweber’s iteration for integral equations of the first kind. SIAM J. Numer. Anal. 11, 789–825 (1974)
Nemirovskiy, A.S., Polyak, B.T.: Iterative methods for solving linear ill-posed problems under precise information I. Eng. Cybern. 20(3), 1–11 (1984)
Jack, G., Prenter, P.M.: Numerical iterative filters applied to first kind Fredholm integral equations. Numer. Math. 30, 281–299 (1978)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Mesgarani, H., Parmour, P. Application of numerical solution of linear Fredholm integral equation of the first kind for image restoration. Math Sci 17, 371–378 (2023). https://doi.org/10.1007/s40096-022-00456-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40096-022-00456-2
Keywords
- Fredholm integral equation
- Inverse problems
- Ill-posed problems
- Regularization method
- Numerical integration