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Tensor product complex tight framelets with increasing directionality. (English) Zbl 1295.42023

Summary: 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 (\(DT-\mathbb{C}WT\)), proposed by N. Kingsbury [Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 357, No. 1760, 2543–2560 (1999; Zbl 0976.68527)] and further developed by I. W. Selesnick, R. G. Baraniuk, and N. G. Kingsbury [IEEE Signal Process. Mag. 22, 123–151 (2005)], 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 \(DT-\mathbb{C}WT\) is obtained essentially via tensor product and offers improved directionality with six directions. In this paper we shall further enhance the performance of the \(DT-\mathbb{C}WT\) 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 \(TP-\mathbb{C}TF_n\) for all integers \(n\geq 3\) 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 \(TP-\mathbb{C}TF_n\) offers \(\frac{1}{2}(n-1)(n-3)+4\) directions when \(n\) is odd and \(\frac{1}{2}(n-4)(n+2)+6\) directions when \(n\) is even. In particular, we shall show that \(TP-\mathbb{C}TF_4\), which is different from the \(DT-\mathbb{C}WT\) in both nature and design, provides an alternative to the \(DT-\mathbb{C}WT\). Indeed, we shall see that \(TP-\mathbb{C}TF_4\) behaves quite similar to the \(DT-\mathbb{C}WT\) by offering six directions in dimension two, employing the tensor product structure, and enjoying slightly less redundancy than the \(DT-\mathbb{C}WT\). Then we shall apply \(TP-\mathbb{C}TF_n\) to the problem of image denoising. We shall see that the performance of \(TP-\mathbb{C}TF_4\) for image denoising is comparable to that of the \(DT-\mathbb{C}WT\). Better results on image denoising can be obtained by using other \(TP-\mathbb{C}TF_n\), for example, \(n=6\), which has 14 directions in dimension two. Moreover, \(TP-\mathbb{C}TF_n\) allows us to further improve the \(DT-\mathbb{C}WT\) by using \(TP-\mathbb{C}TF_n\) as the first stage filter bank in the \(DT-\mathbb{C}WT\). We shall also provide discussion and comparison of \(TP-\mathbb{C}TF_n\) with several generalizations of the \(DT-\mathbb{C}WT\), shearlets, and directional nonseparable tight framelets.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15 General harmonic expansions, frames
65T60 Numerical methods for wavelets
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory

Citations:

Zbl 0976.68527
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