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. 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 suboptimal 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 [Phil. Trans. R. Soc. Lond. A, 357 (1999), pp. 2543–2560] and further developed by Selesnick, Baraniuk, and Kingsbury [IEEE Signal Process. Mag., 22 (2005), pp. 123–151], 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 essentially via tensor product and offers improved directionality with six directions. In this paper we shall further enhance the performance of the DT-CWT for the problem of image denoising. Using a 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 a 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 from the DT-CWT in both nature and design, provides an alternative to the DT-CWT. Indeed, we shall see that TP-CTF4 behaves quite similar to the DT-CWT by offering six directions in dimension two, employing the tensor product structure, and enjoying slightly less redundancy than the DT-CWT. Then we shall apply TP-CTFn to the problem of image denoising. We shall see that the performance of TP-CTF4 for image denoising is comparable to that of the DT-CWT. 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 the DT-CWT by using TP-CTFn as the first stage filter bank in the DT-CWT. We shall also provide discussion and comparison of TP-CTFn with several generalizations of the DT-CWT, shearlets, and directional nonseparable tight framelets.