### Abstract

We propose a collocation method for solv- ing 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 equa- tion into an equivalent system of integral equations which do not involve the square of the integral operator. A mul- tiscale collocation method is then applied to solve the sys- tem. 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 re- sulting 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.

Original language | English (US) |
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Pages (from-to) | 263-307 |

Number of pages | 45 |

Journal | Journal of Integral Equations and Applications |

Volume | 28 |

Issue number | 2 |

DOIs | |

State | Published - 2016 |

### Keywords

- Image restoration
- Inverse problems

### ASJC Scopus subject areas

- Numerical Analysis
- Applied Mathematics

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## Cite this

*Journal of Integral Equations and Applications*,

*28*(2), 263-307. https://doi.org/10.1216/JIE-2016-28-2-263