## A new alternating minimization algorithm for total variation image reconstruction.(English)Zbl 1187.68665

Summary: We propose, analyze, and test an alternating minimization algorithm for recovering images from blurry and noisy observations with total variation (TV) regularization. This algorithm arises from a new half-quadratic model applicable to not only the anisotropic but also the isotropic forms of TV discretizations. The per-iteration computational complexity of the algorithm is three fast Fourier transforms. We establish strong convergence properties for the algorithm including finite convergence for some variables and relatively fast exponential (or $$q$$-linear in optimization terminology) convergence for the others. Furthermore, we propose a continuation scheme to accelerate the practical convergence of the algorithm. Extensive numerical results show that our algorithm performs favorably in comparison to several state-of-the-art algorithms. In particular, it runs orders of magnitude faster than the lagged diffusivity algorithm for TV-based deblurring. Some extensions of our algorithm are also discussed.

### MSC:

 68U10 Computing methodologies for image processing 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 65J22 Numerical solution to inverse problems in abstract spaces 65K10 Numerical optimization and variational techniques 65T50 Numerical methods for discrete and fast Fourier transforms 90C25 Convex programming

### Software:

FTVd; RecPF; ForWaRD
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