关于双树复小波变换的可移性
On the Shiftability of Dual-Tree Complex Wavelet Transforms
摘要：
双树复小波变换（DT-CWT）具有比传统离散小波变换更好的平移不变性。我们提出了一个幅相表示的DT-CWT，其中，除其他事项外，提供了一个直接的解释的改进的移位不变性。该表示法是基于分数希尔伯特变换（fHT）算子组的移位动作，其将任意相移的概念从正弦曲线扩展到有限能量信号（特别是小波）。特别是，我们的DT-CWT的移位特性的fHT的移位。表示的核心是fHT群的某些基本不变性，即平移、伸缩和范数，它们在建立变换的关键属性中起着决定性的作用。事实证明，这些基本不变性是这个群体所独有的。接下来，通过引入fHT算子的Bedrosian定理的推广，我们推导出对通过低通函数的调制获得的特定小波族的fHT的移位作用的明确理解（例如，Shannon和Gabor小波）。这实际上将相应的双树变换与窗口傅立叶分析的框架联系起来。最后，我们将这些想法扩展到多维设置，通过引入一个方向性的扩展的fHT，分数方向希尔伯特变换。特别是，我们推导出一个信号表示涉及方向选择性小波与适当的相移，这有助于解释沿某些优先方向的变换的改进的移位不变性的叠加。
英文摘要：
The dual-tree complex wavelet transform (DT-CWT) is known to exhibit better shift-invariance than the conventional discrete wavelet transform. We propose an amplitude-phase representation of the DT-CWT which, among other things, offers a direct explanation for the improvement in the shift-invariance. The representation is based on the shifting action of the group of fractional Hilbert transform (fHT) operators, which extends the notion of arbitrary phase-shifts from sinusoids to finite-energy signals (wavelets in particular). In particular, we characterize the shiftability of the DT-CWT in terms of the shifting property of the fHTs. At the heart of the representation are certain fundamental invariances of the fHT group, namely that of translation, dilation, and norm, which play a decisive role in establishing the key properties of the transform. It turns out that these fundamental invariances are exclusive to this group. Next, by introducing a generalization of the Bedrosian theorem for the fHT operator, we derive an explicitly understanding of the shifting action of the fHT for the particular family of wavelets obtained through the modulation of lowpass functions (e.g., the Shannon and Gabor wavelet). This, in effect, links the corresponding dual-tree transform with the framework of windowed-Fourier analysis. Finally, we extend these ideas to the multi-dimensional setting by introducing a directional extension of the fHT, the fractional directional Hilbert transform. In particular, we derive a signal representation involving the superposition of direction-selective wavelets with appropriate phase-shifts, which helps explain the improved shift-invariance of the transform along certain preferential directions.
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