Image Inpainting Using Directional Tensor Product Complex Tight Framelets

Different from an orthonormal basis, a tight frame is an overcomplete and energy-preserving system with flexibility and redundancy. Many image restoration methods employing tight frames have been developed and studied in the literature. Tight wavelet frames have been proven to be useful in many applications. In this paper we are particularly interested in the image inpainting problem using directional complex tight wavelet frames. Under the assumption that frame coefficients of images are sparse, several iterative thresholding algorithms for the image inpainting problem have been proposed in the literature. The outputs of such iterative algorithms are closely linked to solutions of several convex minimization models using the balanced approach which simultaneously combines the l1-regularization for sparsity of frame coefficients and the l2-regularization for smoothness of the solution. Due to the redundancy of a tight frame, elements of a tight frame could be highly correlated and therefore, their corresponding frame coefficients of an image are expected to close to each other. This is called the grouping effect in statistics. In this paper, we establish the grouping effect property for frame-based convex minimization models using the balanced approach. This result on grouping effect partially explains the effectiveness of models using the balanced approach for several image restoration problems. Since real-world natural images usually have two layers consisting of cartoons and textures, methods using simultaneous cartoon and texture inpainting are popular in the literature by using two combined tight frames: one tight frame (often built from wavelets, curvelets or shearlets) provides sparse representations for cartoons, and the other tight frame (often built from discrete cosine transform) offers sparse approximation for textures. Inspired by recent development on directional tensor product complex tight framelets (TP-CTFs) and their impressive performance for the image denoising problem, in this paper we propose an iterative thresholding algorithm using a single tight frame derived from TP-CTFs for the image inpainting problem. Experimental results show that our proposed algorithm can handle well both cartoons and textures simultaneously and performs comparably and often better than several well-known frame-based iterative thresholding algorithms for the image inpainting problem without noise. For the image inpainting problem with additive zero-mean i.i.d. Gaussian noise, our proposed algorithm using TP-CTFs performs superior than other known state-of-the-art frame-based image inpainting algorithms.