Clifford（几何）代数小波变换
Clifford (Geometric) Algebra Wavelet Transform
摘要：
由于Clifford（几何）代数傅里叶变换（CFT）是全局的，本文引入了局部Clifford（几何）代数（GA）小波的概念。我们展示了如何为$n=2,3 (\mod 4)$连续$Cl_n$值的容许小波可以构造使用的相似群$SIM(n)$。我们严格地以真实几何解释为目标，并因此用与$-1$成直角的GA叶片替换虚单元$i \in \C$。由于非交换性而产生的后果。我们表示的容许条件的$Cl_{n}$ CFT，然后推导出一组重要的属性，如伸缩，平移和旋转协方差，再生核，并显示如何反转Clifford小波变换。作为一个明确的例子，我们介绍Clifford Gabor小波。我们进一步发明了广义Clifford小波不确定性原理。CFTs和Clifford小波到$Cl_{0,n'}, n' = 1,2 (\mod 4)$的扩展是直接的。关键词：克利福德几何代数，克利福德小波变换，多维小波，连续小波，相似群。
英文摘要：
While the Clifford (geometric) algebra Fourier Transform (CFT) is global, we introduce here the local Clifford (geometric) algebra (GA) wavelet concept. We show how for $n=2,3 (\mod 4)$ continuous $Cl_n$-valued admissible wavelets can be constructed using the similitude group $SIM(n)$. We strictly aim for real geometric interpretation, and replace the imaginary unit $i \in \C$ therefore with a GA blade squaring to $-1$. Consequences due to non-commutativity arise. We express the admissibility condition in terms of a $Cl_{n}$ CFT and then derive a set of important properties such as dilation, translation and rotation covariance, a reproducing kernel, and show how to invert the Clifford wavelet transform. As an explicit example, we introduce Clifford Gabor wavelets. We further invent a generalized Clifford wavelet uncertainty principle. Extensions of CFTs and Clifford wavelets to $Cl_{0,n'}, n' = 1,2 (\mod 4)$ appear straight forward. Keywords: Clifford geometric algebra, Clifford wavelet transform, multidimensional wavelets, continuous wavelets, similitude group.
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