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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:
68U10Image processing (computing aspects)
65J22Inverse problems (numerical methods in abstract spaces)
65K10Optimization techniques (numerical methods)
65T50Discrete and fast Fourier transforms (numerical methods)
90C25Convex programming