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Alternating direction method for image inpainting in wavelet domains. (English) Zbl 1234.68448
Summary: Image inpainting in wavelet domains refers to the recovery of an image from incomplete and/or inaccurate wavelet coefficients. To reconstruct the image, total variation (TV) models have been widely used in the literature, and they produce high-quality reconstructed images. In this paper, we consider an unconstrained, TV-regularized, $\ell_2$-data-fitting model to recover the image. The model is solved by the alternating direction method (ADM). At each iteration, the ADM needs to solve three subproblems, all of which have closed-form solutions. The per-iteration computational cost of the ADM is dominated by two Fourier transforms and two wavelet transforms, all of which admit fast computation. Convergence of the ADM iterative scheme is readily obtained. We also discuss extensions of this ADM scheme to solving two closely related constrained models. We present numerical results to show the efficiency and stability of the ADM for solving wavelet domain image inpainting problems. Numerical results comparing the ADM with some recent algorithms are also reported.

##### MSC:
 68U10 Image processing (computing aspects) 65J22 Inverse problems (numerical methods in abstract spaces) 65K10 Optimization techniques (numerical methods) 65T50 Discrete and fast Fourier transforms (numerical methods) 90C25 Convex programming
FTVd; RecPF
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