Wavelets on the n-sphere and related manifolds
Wavelets on the n-sphere and related manifolds
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DOI：
10.1063/1.532481
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发表时间：
1998-08-01
影响因子：
1.3
通讯作者：
Vandergheynst, P
中科院分区：
文献类型：
--
作者：
Antoine, JP;Vandergheynst, P
我们提出了一个纯粹的群论推导的连续小波变换（CWT）的（n-1）-球面Sn-1。基于与平方可积群表象相关联的一般相干态的构造。通过Iwasawa分解，将CWT的参数空间X（类似于SO（n）xR*（+））嵌入到广义Lorentz群SO 0（n，1）中，使得X类似于或等于SO 0（n，1）IN，其中N类似于或等于Rn-1.然后，由作用在Sn-1上有限能量信号空间L-2（Sn-1，d μ）中的SO_0（n，1）的一个合适的酉表示导出Sn-1上的连续小波变换，证明了它在X上是平方可积的.我们发现一个必要条件的小波的容许性，在一个零均值条件的形式，这需要所有通常的过滤性能的CWT。接下来，通过在半径为R的球面上重做构造并对R->无穷大进行群压缩，得到Sn-1上的这种连续小波变换的欧几里得极限，由此恢复平坦欧几里得空间上的通常连续小波变换。最后，我们讨论了这一构造在双叶双曲面Hn-1 SO_0（n-1，1）/SO（n-1）及其它黎曼对称空间中的推广. (C)1998年美国物理学会。[S0022-2488（98）00308-9]。
We present a purely group-theoretical derivation of the continuous wavelet transform (CWT) on the (n-1)-sphere Sn-1. based on the construction of general coherent states associated to square integrable group representations. The parameter space of the CWT, X similar to SO(n)xR*(+), is embedded into the generalized Lorentz group SO0(n,1) via the Iwasawa decomposition, so that X similar or equal to SO0(n,1)IN, where N similar or equal to Rn-1. Then the CWT on Sn-1 is derived from a suitable unitary representation of SO0(n,1) acting in the space L-2(Sn-1,d mu) of finite energy signals on Sn-1, which turns out to be square integrable over X. We find a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition, which entails all the usual filtering properties of the CWT. Next the Euclidean limit of this CWT on Sn-1 is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R-->infinity, from which one recovers the usual CWT on flat Euclidean space. Finally, we discuss the extension of this construction to the two-sheeted hyperboloid Hn-1SO0(n-1,1)/SO(n-1) and some other Riemannian symmetric spaces. (C) 1998 American Institute of Physics. [S0022-2488(98)00308-9].