An ℓp-ℓq minimization method with cross-validation for the restoration of impulse noise contaminated images

Abstract Discrete ill-posed problems arise in many areas of science and engineering. Their solutions, if they exist, are very sensitive to perturbations in the data. Regularization aims to reduce this sensitivity. Many regularization methods replace the original problem by a minimization problem with a fidelity term and a regularization term. Recently, the use of a p -norm to measure the fidelity term and a q -norm to measure the regularization term has received considerable attention. The relative importance of these terms is determined by a regularization parameter. When the perturbation in the available data is made up of impulse noise and a sparse solution is desired, it often is beneficial to let 0 p , q 1 . Then the p - and q -norms are not norms. The choice of a suitable regularization parameter is crucial for the quality of the computed solution. It therefore is important to develop methods for determining this parameter automatically, without user-interaction. However, the latter has so far not received much attention when the data is contaminated by impulse noise. This paper discusses two approaches based on cross validation for determining the regularization parameter in this situation. Computed examples that illustrate the performance of these approaches when applied to the restoration of impulse noise contaminated images are presented.

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