A Generalized Krylov Subspace Method for $\ell_p$-$\ell_q$ Minimization

Abstract

This paper presents a new efficient approach for the solution of the ℓp-ℓq minimization problem based on the application of successive orthogonal projections onto generalized Krylov subspaces of increasing dimension. The subspaces are generated according to the iteratively reweighted least-squares strategy for the approximation of ℓp/ℓq-norms by weighted ℓ 2-norms. Computed image restoration examples illustrate that it suffices to carry out only a few iterations to achieve highquality restorations. The combination of a low iteration count and a modest storage requirement makes the proposed method attractive.

Fractional Tikhonov regularization with a nonlinear penalty term

“…An automatic strategy for the choice of the parameter q in for image restoration problems is provided in [18]. Computed examples in [6,17,18] show ℓ q -quasi-norm (q < 1) regularization to yield more accurate image restorations than ℓ 1 -norm regularization. This is due to the fact that the ℓ q -quasinorm gives sparser restorations and better preserves edges than the ℓ 1 -norm.…”

“…The minimization problem (1.14) or the minimization problem with the TVnorm penalty term replaced by (1.15) can be solved by iteratively reweighted norm methods; see, e.g., [14,17,24]. These methods compute the desired solution by solving a sequence of weighted least-squares problems.…”

“…In the computed examples described in Section 4, we use the method described by Wen and Chan [25]. When, instead, the penalty term in (1.6) is replaced by the term (1.15) for some 0 < q < 2, we apply the method described in [17] to solve the equivalent ℓ 2 -ℓ q problem…”

“…The method in [17] also can be applied to solve (3.2). This method determines a sequence of approximate solutions in nested subspaces using the iteratively reweighted norm method described in [24], and replaces the penalty terms in (3.3) and (3.2) by weighted Euclidean norms, in which a diagonal weighting matrix is updated until a solution of (3.3) or (3.2) has been determined.…”

“…This method determines a sequence of approximate solutions in nested subspaces using the iteratively reweighted norm method described in [24], and replaces the penalty terms in (3.3) and (3.2) by weighted Euclidean norms, in which a diagonal weighting matrix is updated until a solution of (3.3) or (3.2) has been determined. The subspaces are generalized Krylov subspaces made up of the span of the residual vectors of the normal equations for each weighting matrix; see [17] for details.…”