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A collocation method solving integral equation models for image restoration. (English) Zbl 1347.65197

Summary: We propose a collocation method for solving integral equations which model image restoration from out-of-focus images. Restoration of images from out-of-focus images can be formulated as an integral equation of the first kind, which is an ill-posed problem. We employ the Tikhonov regularization to treat the ill-posedness and obtain results of a well-posed second kind integral equation whose integral operator is the square of the original operator. The present of the square of the integral operator requires high computational cost to solve the equation. To overcome this difficulty, we convert the resulting second kind integral equation into an equivalent system of integral equations which do not involve the square of the integral operator. A multiscale collocation method is then applied to solve the system. A truncation strategy for the matrices appearing in the resulting discrete linear system is proposed to design a fast numerical solver for the system of integral equations. A quadrature method is used to compute the entries of the resulting matrices. We estimate the computational cost of the numerical method and its approximate accuracy. Numerical experiments are presented to demonstrate the performance of the proposed method for image restoration.

MSC:

65R20 Numerical methods for integral equations
65R32 Numerical methods for inverse problems for integral equations
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
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References:

[1] P.M. Anselone, Compact operator approximation theory and applications to integral equations , Prentice-Hill, Englewood Cliffs, 1995.
[2] M. Bertero, P. Brianzi and E. Pike, Super-resolution in confocal scanning microcopy , Inv. Prob. 3 (1987), 195-212. · Zbl 0674.65101
[3] Z. Chen, S. Ding, Y. Xu and H. Yang, Multiscale collocation methods for ill-posed integral equations via a coupled system , Inv. Prob. 28 (2012), 025006. · Zbl 1244.65237
[4] Z. Chen, C.A. Micchelli and Y. Xu, A construction of interpolating wavelets on invariant sets , Math. Comp. 68 (1999), 1569-1587. · Zbl 1051.65009
[5] —-, The Petrov-Galerkin methods for second kind integral equations II: Multiwavelet scheme , Adv. Comp. Math. 7 (1997), 199-233. · Zbl 0915.65134
[6] —-, Fast collocation method for second kind integral equation , SIAM J. Num. Anal. 40 (2002), 49-55. · Zbl 1016.65107
[7] Z. Chen, C.A. Micchelli and Y. Xu, Multiscale methods for Fredholm integral equations , Cambrige University Press, Cambrige, 2015. · Zbl 1332.65188
[8] Z. Chen, B. Wu and Y. Xu, Error control strategies for numerical integration in fast collocation methods , Northeast Math. J. 21 (2005), 233-252. · Zbl 1079.65132
[9] Z. Chen, Y. Xu and H. Yang, Fast collocation method for solving ill-posed integral equation of the first kind , Inv. Prob. 24 (2008), 065007. · Zbl 1167.65073
[10] W. Fang and M. Lu, A fast collocation method for an inverse boundary value problem , Inter. J. Numer. Meth. Engin. 59 (2004), 1563-1585. · Zbl 1059.65100
[11] R. Gonzalez and R. Woods, Digital image processing , Addison-Wesley, Boston, 1993.
[12] W.C. Groetsch, Uniform convergence of regularization methods for Fredholm equations of the first kind , J. Austr. Math. Soc. 39 (1985), 282-286. · Zbl 0591.65090
[13] —-, The theory of Tikhonov regularization for Fredholm equation of the first kind , Res. Notes Math. 105 , Pitman, Boston, 1984.
[14] C.P. Hansen, Deconvolution and regularization with Topelitz matrices , Numer. Alg. 29 (2002), 323-378. · Zbl 1002.65145
[15] H. Kaneko and Y. Xu, Gauss-type quadratures for weakly singular integral and their application to Fredholm integral equations of the second kind , Math. Comp. 62 (1994), 739-753. · Zbl 0799.65023
[16] Y. Lu, L. Shen and Y. Xu, Integral equation models for image restoration: High accuracy methods and fast algorithms , Inv. Prob. 26 (2010), 045006. · Zbl 1256.94012
[17] S.V. Maass, P. Pereverzev, R. Ramlau and S.G. Soodky, An adaptive discretization for Tikhonov-Phillips regularizaton with a posteriori parameter selection , Numer. Math. 87 (2001), 485-502.
[18] C.A. Micchelli and Y. Xu, Using the matrix refinement equation for the construction of wavelets on invariant sets , Appl. Comp. Harmonic Anal. 1 (1994), 344-375. · Zbl 0815.42019
[19] —-, Reconstruction and decomposition algorithms for biorthogonal multiwalveles , Multidim. Syst. Signal Proc. 8 (1997), 31-69. · Zbl 0872.42010
[20] C.A. Micchelli, Y. Xu and Y. Zhao, Galerkin methods for second-kind integral equations , J. Comp. Appl. Math. 86 (1997), 251-270. · Zbl 0913.65129
[21] S. Pereverzev and E. Shock, On the adaptive selection of the parameter in regularization of ill-posed problems , SIAM J. Numer. Anal. 43 (2005), 2060-2076. · Zbl 1103.65058
[22] R. Plato, Iterative and parametric methods for linear ill-posed equation Habilitationsschrift , Fachb. Math., Berlin, 1995.
[23] —-, The Galerkin scheme for Lavrentiev’s \(m\)-times iterated method to solve the linear accretive Volterra integral equations of the first kind , BIT 37 (1997), 404-423. · Zbl 0882.65135
[24] R. Plato and G. Vainikko, On the regularization of projection methods for solving ill-posed problems , Numer. Math. 57 (1990), 63-79. · Zbl 0675.65053
[25] P.M. Rajan, Convergence anaylysis of a regularized approximation for solving Fredholm integral equations of the first kind , J. Math. Anal. Appl. 279 (2003), 522-530. · Zbl 1021.65070
[26] U. Tautenhahn, On the method of Lavrentiev regularization for nonlinear ill-posed problem , Inv. Prob. 18 (2002), 191-207. · Zbl 1005.65058
[27] A. Tikhonov and V. Arsenin, Solution to ill-posed problems , Wiley, New York, 1977. · Zbl 0354.65028
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