Directional tensor product complex tight framelets with low redundancy

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• 作者：Bin Han^a ; ^{bhan@ualberta.ca" class="auth_mail" title="E-mail the corresponding author} ; Zhenpeng Zhao^a ; ^{zzhao7@ualberta.ca" class="auth_mail" title="E-mail the corresponding author} ; Xiaosheng Zhuang^b ; ^{xzhuang7@cityu.edu.hk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：42C40 ; 42C15 ; 94A08 ; 68U10
• 刊名：Applied and Computational Harmonic Analysis
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：41
• 期：2
• 页码：603-637
• 全文大小：1287 K

文摘

Having the advantages of redundancy and flexibility, various types of tight frames have already shown impressive performance in applications such as image and video processing. For example, the undecimated wavelet transform, which is a particular case of tight frames, is known to have good performance for the denoising problem. Empirically, it is widely known that higher redundancy rate of a tight frame often leads to better performance in applications. The wavelet/framelet transform is often implemented in an undecimated fashion for the purpose of better performance in practice. Though high redundancy rate of a tight frame can improve performance in applications, as the dimension increases, it also makes the computational cost skyrocket and the storage of frame coefficients increase exponentially. This seriously restricts the usefulness of such tight frames for problems in moderately high dimensions such as video processing in dimension three. Inspired by the directional tensor product complex tight framelets TP-CTF_m${\text{TP-}\mathbb{C}\text{TF}}_m$ with m≥3$m\ge 3$ in  and  and their impressive performance for image processing in  and , in this paper we introduce directional tensor product complex tight framelets ${\text{TP-}\mathbb{C}\text{TF}}_m^{\downarrow }$ (called reduced TP-CTF_m${\text{TP-}\mathbb{C}\text{TF}}_m$) with low redundancy. Such ${\text{TP-}\mathbb{C}\text{TF}}_m^{\downarrow }$ are particular examples of tight framelet filter banks with mixed sampling factors. In particular, we shall develop a directional tensor product complex tight framelet ${\text{TP-}\mathbb{C}\text{TF}}_6^{\downarrow }$ such that it performs nearly as well as the original TP-CTF₆${\text{TP-}\mathbb{C}\text{TF}}_6$ in [20] for image/video denoising/inpainting but it has significantly lower redundancy rates than TP-CTF₆${\text{TP-}\mathbb{C}\text{TF}}_6$ in every dimension. The ${\text{TP-}\mathbb{C}\text{TF}}_6^{\downarrow }$ in d   dimensions not only offers good directionality as the original TP-CTF₆${\text{TP-}\mathbb{C}\text{TF}}_6$ does but also has the low redundancy rate $\frac{3^d-1}{2^d-1}$ (e.g., the redundancy rates are 253">$2,2\frac{2}{3},3\frac{5}{7},5\frac{1}{3}$ and 74790f3a6fbc2a91017d12b642d">$7\frac{\mathrm{<font\; color="red">25</font>}}{31}$ for dimension 7474727b" title="Click to view the MathML source">d=1,…,5$d=1,\dots ,5$, respectively), in comparison with the redundancy rate $2^d\times \frac{3^d-1}{2^d-1}$ of TP-CTF₆${\text{TP-}\mathbb{C}\text{TF}}_6$ in dimension d  . Moreover, our numerical experiments on image/video denoising and inpainting show that the performance using our proposed ${\text{TP-}\mathbb{C}\text{TF}}_6^{\downarrow }$ is often comparable with or sometimes better than several state-of-the-art frame-based methods which have much higher redundancy rates than that of ${\text{TP-}\mathbb{C}\text{TF}}_6^{\downarrow }$.