### Abstract

Image restoration problems are often solved by finding the minimizer of a suitable objective function. Usually this function consists of a data-fitting term and a regularization term. For the least squares solution, both the data-fitting and the regularization terms are in the ℓ1 norm. In this paper, we consider the least absolute deviation (LAD) solution and the least mixed norm (LMN) solution. For the LAD solution, both the data-fitting and the regularization terms are in the ℓ1 norm. For the LMN solution, the regularization term is in the ℓ1 norm but the data-fitting term is in the ℓ2 norm. Since images often have nonnegative intensity values, the proposed algorithms provide the option of taking into account the nonnegativity constraint. The LMN and LAD solutions are formulated as the solution to a linear or quadratic programming problem which is solved by interior point methods. At each iteration of the interior point method, a structured linear system must be solved. The preconditioned conjugate gradient method with factorized sparse inverse preconditioners is employed to solve such structured inner systems. Experimental results are used to demonstrate the effectiveness of our approach. We also show the quality of the restored images, using the minimization of mixed ℓ1-ℓ2 and ℓ1-ℓ2 norms, is better than that using only the ℓ2 norm,

Original language | English (US) |
---|---|

Pages (from-to) | 1881-1902 |

Number of pages | 22 |

Journal | SIAM Journal on Scientific Computing |

Volume | 27 |

Issue number | 6 |

DOIs | |

State | Published - Nov 27 2006 |

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### All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Applied Mathematics

### Cite this

*SIAM Journal on Scientific Computing*,

*27*(6), 1881-1902. https://doi.org/10.1137/040615079

}

*SIAM Journal on Scientific Computing*, vol. 27, no. 6, pp. 1881-1902. https://doi.org/10.1137/040615079

**Efficient minimization methods of mixed ℓ2-ℓ1 and ℓ2-ℓ1 norms for image restoration.** / Fu, Haoying; Ng, Michael K.; Nikolova, Mila; Barlow, Jesse L.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Efficient minimization methods of mixed ℓ2-ℓ1 and ℓ2-ℓ1 norms for image restoration

AU - Fu, Haoying

AU - Ng, Michael K.

AU - Nikolova, Mila

AU - Barlow, Jesse L.

PY - 2006/11/27

Y1 - 2006/11/27

N2 - Image restoration problems are often solved by finding the minimizer of a suitable objective function. Usually this function consists of a data-fitting term and a regularization term. For the least squares solution, both the data-fitting and the regularization terms are in the ℓ1 norm. In this paper, we consider the least absolute deviation (LAD) solution and the least mixed norm (LMN) solution. For the LAD solution, both the data-fitting and the regularization terms are in the ℓ1 norm. For the LMN solution, the regularization term is in the ℓ1 norm but the data-fitting term is in the ℓ2 norm. Since images often have nonnegative intensity values, the proposed algorithms provide the option of taking into account the nonnegativity constraint. The LMN and LAD solutions are formulated as the solution to a linear or quadratic programming problem which is solved by interior point methods. At each iteration of the interior point method, a structured linear system must be solved. The preconditioned conjugate gradient method with factorized sparse inverse preconditioners is employed to solve such structured inner systems. Experimental results are used to demonstrate the effectiveness of our approach. We also show the quality of the restored images, using the minimization of mixed ℓ1-ℓ2 and ℓ1-ℓ2 norms, is better than that using only the ℓ2 norm,

AB - Image restoration problems are often solved by finding the minimizer of a suitable objective function. Usually this function consists of a data-fitting term and a regularization term. For the least squares solution, both the data-fitting and the regularization terms are in the ℓ1 norm. In this paper, we consider the least absolute deviation (LAD) solution and the least mixed norm (LMN) solution. For the LAD solution, both the data-fitting and the regularization terms are in the ℓ1 norm. For the LMN solution, the regularization term is in the ℓ1 norm but the data-fitting term is in the ℓ2 norm. Since images often have nonnegative intensity values, the proposed algorithms provide the option of taking into account the nonnegativity constraint. The LMN and LAD solutions are formulated as the solution to a linear or quadratic programming problem which is solved by interior point methods. At each iteration of the interior point method, a structured linear system must be solved. The preconditioned conjugate gradient method with factorized sparse inverse preconditioners is employed to solve such structured inner systems. Experimental results are used to demonstrate the effectiveness of our approach. We also show the quality of the restored images, using the minimization of mixed ℓ1-ℓ2 and ℓ1-ℓ2 norms, is better than that using only the ℓ2 norm,

UR - http://www.scopus.com/inward/record.url?scp=33751208047&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33751208047&partnerID=8YFLogxK

U2 - 10.1137/040615079

DO - 10.1137/040615079

M3 - Article

AN - SCOPUS:33751208047

VL - 27

SP - 1881

EP - 1902

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

SN - 1064-8275

IS - 6

ER -