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New overcomplete wavelet transforms and wavelet based deconvolution.

机译：新的超完备小波变换和基于小波的反卷积。

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摘要

The dyadic DWT provides an octave-band frequency decomposition and is very effective for processing piecewise smooth signals. But in fine (high-frequency) scales, the frequency resolution is quite low -- an undesired feature for signals which are not regarded as 'smooth' in general (e.g. audio, texture). The discrete wavelet packet transform (DWPT) avoids this problem by iterating on the highpass filters as well. However, due to critical sampling, both DWT and DWPT are highly shift-varying. In addition, they do not provide directional analysis/synthesis functions when they are extended to 2D (or higher dimensions) via tensor products. We extend the the dual-tree complex wavelet transform (DT- C WT), introduced by Kingsbury, to a dual-tree complex wavelet packet transform (DT- C WPT). The DT- C WPT we propose retains the desired properties of the DT- C WT (like near shift-invariance, directional analysis/synthesis for multidimensional signals) while offering an improved frequency resolution, which can be tailored to the particular signal family at hand.;Another transform studied in this thesis involves wavelet frames with a rational dilation factor. Given an admissable wavelet psi(t), the dyadic wavelet basis consists of dyadic dilates and translates of psi( t), given by 2j/2y 2jt-n j,n∈Z . However the basis functions at different scales are quite dissimilar (compare psi(t) and 2 psi(2t) for example). This abrupt change between neighboring scales might be a problem for certain signals. There are two issues. First, the dilation factor (which is 2 for the dyadic DWT) should be decreased, so as to make the change between each scale more gradual. Second, the frequency resolution, which is influenced by the dilation factor, due to the completeness requirement of the basis (or redundancy considerations of the frame) should be improved. In order to address these, we propose the rational DWT, which consists of iterated FBs with rational sampling factors. The filter design problems encountered in such a setting are inherently different from those for integer dilation DWTs, and the techniques for integer dilation DWT design are not applicable. Several schemes to overcome these problems are proposed in this thesis.;A third topic considered is on iterated non-perfect reconstruction (non-PR) FBs. In a nutshell, we investigate the frame bounds of iterated non-perfect reconstruction filter banks and provide frame bounds valid for iterated FBs with an arbitrary number of stages using the frame bounds of the underlying frame on the real line. Conversely, given the frame bounds of the iterated FB, we derive bounds for the underlying wavelet frame.;Lastly, we consider a modification to the 'thresholded Landweber' algorithm, which has drawn attention recently for the solution of wavelet regularized inverse problems. The modification we discuss accelerates the convergence of the algorithm, by taking advantage of the different behavior of the blurring operator in different subbands.
机译：双向DWT提供了倍频程频率分解，对于处理分段平滑信号非常有效。但是在精细的（高频）标度上，频率分辨率非常低-对于通常不被视为``平滑''的信号（例如音频，纹理），这是不希望有的功能。离散小波包变换（DWPT）也通过迭代高通滤波器来避免此问题。但是，由于关键采样，DWT和DWPT都具有很大的移位变化。此外，当它们通过张量积扩展到2D（或更高尺寸）时，它们不提供方向分析/综合功能。我们将金斯伯里（Kingsbury）引入的双树复数小波变换（DT-C WT）扩展到双树复数小波包变换（DT- C WPT）。我们建议的DT- C WPT保留了DT- C WT的所需属性（例如，多维信号的近移不变性，方向分析/合成），同时提供了改进的频率分辨率，可以针对手头的特定信号系列进行定制本文研究的另一种变换涉及具有合理扩张因子的小波帧。给定一个允许的小波psi（t），二进小波基由二进位扩张和psi（t）的平移组成，由2j / 2y 2jt-n j，n∈Z给出。但是，不同尺度下的基函数完全不同（例如，比较psi（t）和2 psi（2t））。对于某些信号，相邻刻度之间的这种突然变化可能是一个问题。有两个问题。首先，应减小膨胀因子（对于二进位DWT为2），以使每个标度之间的变化更加渐进。其次，由于基础的完整性要求（或帧的冗余考虑），应提高受扩频因子影响的频率分辨率。为了解决这些问题，我们提出了合理的DWT，它由具有合理采样因子的迭代FB组成。在这种情况下遇到的滤波器设计问题与整数膨胀DWT的固有问题是不同的，并且整数膨胀DWT设计的技术不适用。本文提出了几种克服这些问题的方案。第三个要考虑的问题是迭代非完美重构（non-PR）FB。简而言之，我们研究了迭代的非完美重建滤波器组的帧边界，并使用实线上的基础帧的帧边界提供了具有任意级数的迭代FB的有效帧边界。相反，给定迭代FB的帧边界，我们可以得出底层小波框架的边界。我们讨论的修改通过利用模糊算子在不同子带中的不同行为来加速算法的收敛。

著录项

  • 作者

    Bayram, Ilker.;

  • 作者单位

    Polytechnic Institute of New York University.;

  • 授予单位 Polytechnic Institute of New York University.;
  • 学科 Engineering Electronics and Electrical.
  • 学位 Ph.D.
  • 年度 2009
  • 页码 208 p.
  • 总页数 208
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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