The Uncertainty Principle Associated with the Continuous Shearlet Transform
The Uncertainty Principle Associated with the Continuous Shearlet Transform
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DOI：
10.1142/s021969130800229x
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发表时间：
2008-03
期刊：
影响因子：
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通讯作者：
S. Dahlke;Gitta Kutyniok;P. Maass;C. Sagiv;H. Stark;G. Teschke
中科院分区：
文献类型：
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作者：
S. Dahlke;Gitta Kutyniok;P. Maass;C. Sagiv;H. Stark;G. Teschke
寻找高维信号的最优表示，特别是方向表示，是目前研究的热点。由于经典的小波变换不能提供精确的方向信息，因此过去提出了几种新的表示系统，包括脊波、曲波和最近的Shearlets。本文研究并可视化了连续Shearlet变换。此外，我们的目标是推导母Shearlet函数，以确保相关变换参数的最佳精度。为此，我们首先证明了该变换与来自所谓的Shearlet群的酉群表示相关联，并计算了相关的容许条件。这使我们能够利用一般不确定性原理来推导母Shearlet函数，这些函数可以最小化为Shearlet群的无穷小生成器（缩放、剪切和平移）导出的不确定性关系。我们进一步讨论了在加权l2空间下保证所导出的最小值的平方可积性的方法。此外，我们还研究了最小值是否满足可容许性条件，从而提出了一种平衡最小值和可容许性的方法。
Finding optimal representations of signals in higher dimensions, in particular directional representations, is currently the subject of intensive research. Since the classical wavelet transform does not provide precise directional information in the sense of resolving the wavefront set, several new representation systems were proposed in the past, including ridgelets, curvelets and, more recently, Shearlets. In this paper we study and visualize the continuous Shearlet transform. Moreover, we aim at deriving mother Shearlet functions which ensure optimal accuracy of the parameters of the associated transform. For this, we first show that this transform is associated with a unitary group representation coming from the so-called Shearlet group and compute the associated admissibility condition. This enables us to employ the general uncertainty principle in order to derive mother Shearlet functions that minimize the uncertainty relations derived for the infinitesimal generators of the Shearlet group: scaling, shear and translations. We further discuss methods to ensure square-integrability of the derived minimizers by considering weighted L2-spaces. Moreover, we study whether the minimizers satisfy the admissibility condition, thereby proposing a method to balance between the minimizing and the admissibility property.