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Directional tensor product complex tight framelets with low redundancy

机译:方向性张量积复杂的紧框架,具有低冗余

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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-CTFm with m >= 3 in [15,20] and their impressive performance for image processing in [20,33], in this paper we introduce directional tensor product complex tight framelets TP-CTFm down arrow (called reduced TP-CTFm) with low redundancy. Such TP-CTFm down arrow are particular examples of tight framelet filter banks with mixed sampling factors. In particular, we shall develop a directional tensor product complex tight framelet TP-CTF6 down arrow such that it performs nearly as well as the original TP-CTF6 in [20] for image/video denoising/inpainting but it has significantly lower redundancy rates than TP-CTF6 in every dimension. The TP-CTF6 down arrow in d dimensions not only offers good directionality as the original TP-CTF6 does but also has the low redundancy rate 3(d)-1/2(d)-1 (e.g., the redundancy rates are 2, 2 2/3, 3 5/7 and 7 25/31 for dimension d = 1,..., 5, respectively), in comparison with the redundancy rate 2(d) x 3(d)-1/2(d)-1 of TP-CTF6 in dimension d. Moreover, our numerical experiments on image/video denoising and inpainting show that the performance using our proposed TP-CTF6 down arrow 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 TP-CTF6 down arrow. (C) 2015 Elsevier Inc. All rights reserved.
机译:具有冗余和灵活性的优点,各种类型的紧帧在图像和视频处理等应用中已显示出令人印象深刻的性能。例如,众所周知,未压缩的小波变换是紧帧的一种特殊情况,对于去噪问题具有良好的性能。根据经验,众所周知,紧密帧的更高冗余率通常会导致应用程序中更好的性能。小波/小帧变换通常以未抽取的方式实现,目的是在实践中获得更好的性能。尽管紧密帧的高冗余率可以提高应用程序的性能,但是随着尺寸的增加,它也使计算成本飞速增长,并且帧系数的存储量呈指数增长。这就严重限制了这种紧密帧在中等尺寸的问题(例如三维尺寸的视频处理)中的有用性。受到[15,20]中m> = 3的定向张量积复杂紧框架TP-CTFm及其在[20,33]中对图像处理的出色表现的启发,在本文中,我们介绍了定向张量积复杂紧框架TP-CTFm CTFm向下箭头(称为精简TP-CTFm),具有低冗余。此类TP-CTFm向下箭头是具有混合采样因子的紧密小框架滤波器组的特定示例。特别是,我们将开发方向张量积复杂的紧框架TP-CTF6向下箭头,使其性能与[20]中的原始TP-CTF6在图像/视频降噪/修复方面几乎一样,但其冗余率比TP-CTF6的各个方面。 d尺寸的TP-CTF6向下箭头不仅提供了与原始TP-CTF6相同的良好方向性,而且还具有较低的冗余率3(d)-1/2(d)-1(例如,冗余率为2,与冗余率2(d)x 3(d)-1/2(d)相比,尺寸d = 1,...,5分别为2 2 / 3、3 5/7和7 25/31)尺寸d中TP-CTF6的-1)。此外,我们在图像/视频降噪和修复方面的数值实验表明,使用我们建议的TP-CTF6向下箭头所获得的性能通常可与甚至可以优于几种基于冗余的帧率更高的基于帧的方法。比TP-CTF6向下箭头大。 (C)2015 Elsevier Inc.保留所有权利。

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