Half-Quadratic Algorithm for ℓp - ℓq Problems with Applications to TV-ℓ1 Image Restoration and Compressive Sensing

In this paper, we consider the \(\ell _p\)-\(\ell _q\) minimization problem with \(0<p,q\le 2\). The problem has been studied extensively in image restoration and compressive sensing. In the paper, we first extend the half-quadratic algorithm from \(\ell _1\)-norm to \(\ell _p\)-norm with \(0<p<2\). Based on this, we develop a half-quadratic algorithm to solve the \(\ell _p\)-\(\ell _q\) problem. We prove that our algorithm is indeed a majorize-minimize approach. From that we derive some convergence results of our algorithm, e.g. the objective function value is monotonically decreasing and convergent. We apply the proposed approach to TV-\(\ell _1\) image restoration and compressive sensing in magnetic resonance (MR) imaging applications. The numerical results show that our algorithm is fast and efficient in restoring blurred images that are corrupted by impulse noise, and also in reconstructing MR images from very few \(k\)-space data.

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