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

Haoying Fu, Michael K. Ng, Mila Nikolova, Jesse L. Barlow

Research output: Contribution to journalArticle

126 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)1881-1902
Number of pages22
JournalSIAM Journal on Scientific Computing
Volume27
Issue number6
DOIs
StatePublished - Nov 27 2006

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Image Restoration
Image reconstruction
Norm
Data Fitting
Least Absolute Deviation
Regularization
Term
Interior Point Method
Conjugate gradient method
Quadratic programming
Preconditioned Conjugate Gradient Method
Least-squares Solution
Linear programming
Linear systems
Nonnegativity
Quadratic Programming
Minimizer
Preconditioner
Objective function
Linear Systems

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

Cite this

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

In: SIAM Journal on Scientific Computing, Vol. 27, No. 6, 27.11.2006, p. 1881-1902.

Research output: Contribution to journalArticle

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