测不准原理与最优稀疏小波变换
Uncertainty principles and optimally sparse wavelet transforms
摘要：
本文介绍了一种新的小波变换局部化框架，如一维小波变换和Shearlet变换。我们的目标是设计非自适应窗函数，在某种意义上促进稀疏性。为此，我们引入了一个框架，用于分析窗口函数的本地化方面。我们的局部化理论在两个方面与传统理论不同。首先，我们区分群生成元和测量局部化的算子（称为可观测量）。其次，我们定义了一个信号变换的不确定性的基础上作为一个整体的窗口，而不是定义一个单独的窗口的不确定性。我们表明，窗口函数的不确定性，在信号空间中，是密切相关的本地化的再生核的小波变换，在相空间。因此，我们表明，使用不确定性最小化窗口函数，结果在某种意义上是最佳稀疏的表示。
英文摘要：
In this paper we introduce a new localization framework for wavelet transforms, such as the 1D wavelet transform and the Shearlet transform. Our goal is to design nonadaptive window functions that promote sparsity in some sense. For that, we introduce a framework for analyzing localization aspects of window functions. Our localization theory diverges from the conventional theory in two ways. First, we distinguish between the group generators, and the operators that measure localization (called observables). Second, we define the uncertainty of a signal transform based on a window as a whole, instead of defining the uncertainty of an individual window. We show that the uncertainty of a window function, in the signal space, is closely related to the localization of the reproducing kernel of the wavelet transform, in phase space. As a result, we show that using uncertainty minimizing window functions, results in representations which are optimally sparse in some sense.
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