In this paper we are particularly interested in the image inpainting problemusing directional complex tight wavelet frames. Under the assumption that framecoefficients of images are sparse, several iterative thresholding algorithmsfor the image inpainting problem have been proposed in the literature. Theoutputs of such iterative algorithms are closely linked to solutions of severalconvex minimization models using the balanced approach which simultaneouslycombines the $l_1$-regularization for sparsity of frame coefficients and the$l_2$-regularization for smoothness of the solution. Due to the redundancy of atight frame, elements of a tight frame could be highly correlated andtherefore, their corresponding frame coefficients of an image are expected toclose to each other. This is called the grouping effect in statistics. In thispaper, we establish the grouping effect property for frame-based convexminimization models using the balanced approach. This result on grouping effectpartially explains the effectiveness of models using the balanced approach forseveral image restoration problems. Inspired by recent development ondirectional tensor product complex tight framelets (TP-CTFs) and theirimpressive performance for the image denoising problem, in this paper wepropose an iterative thresholding algorithm using a single tight frame derivedfrom TP-CTFs for the image inpainting problem. Experimental results show thatour proposed algorithm can handle well both cartoons and texturessimultaneously and performs comparably and often better than several well-knownframe-based iterative thresholding algorithms for the image inpainting problemwithout noise. For the image inpainting problem with additive zero-mean i.i.d.Gaussian noise, our proposed algorithm using TP-CTFs performs superior thanother known state-of-the-art frame-based image inpainting algorithms.
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