Image Denoising Using Tensor Product Complex Tight Framelets with Increasing Directionality

Tensor product real-valued wavelets have been employed in many applications such as image processing with impressive performance. Though edge singularities are ubiquitous and play a fundamental role in image processing and many other two-dimensional problems, tensor product real-valued wavelets are known to be only sub-optimal since they can only capture edges well along the coordinate axis directions (that is, the horizontal and vertical directions in dimension two). Among several approaches in the literature to enhance the performance of tensor product real-valued wavelets, the dual tree complex wavelet transform (DT-CWT), proposed by Kingsbury [16] and further developed by Selesnick et al. [24], is one of the most popular and successful enhancements of the classical tensor product real-valued wavelets by employing a correlated pair of orthogonal wavelet filter banks. The two-dimensional DT-CWT is obtained via tensor product and offers improved directionality with 6 directions. In this paper we shall further enhance the performance of DT-CWT for the problem of image denoising. Using framelet-based approach and the notion of discrete affine systems, we shall propose a family of tensor product complex tight framelets TP-CTFn for all integers n > 3 with increasing directionality, where n refers to the number of filters in the underlying one-dimensional complex tight framelet filter bank. For dimension two, such tensor product complex tight framelet TP-CTFn offers 1 2 (n − 1)(n − 3) + 4 directions when n is odd, and 1 2 (n− 4)(n+ 2) + 6 directions when n is even. In particular, we shall show that TP-CTF4, which is different to DT-CWT in both nature and design, provides an alternative to DT-CWT. Indeed, we shall see that TP-CTF4 behaves quite similar to DT-CWT by offering 6 directions in dimension two, employing the tensor product structure, and enjoying slightly less redundancy than DT-CWT. When TP-CTF4 is applied to image denoising, its performance is comparable to DT-CWT. Moreover, better results on image denoising can be obtained by using other TP-CTFn, for example, n = 6, which has 14 directions in dimension two. Moreover, TP-CTFn allows us to further improve DT-CWT by using TP-CTFn as the first stage filter bank in DT-CWT. Experiments on image denoising using TP-CTFn and detailed comparison with DT-CWT will be provided in this paper.