Ieee Transactions on Signal Processing 1 Exact Wavelets on the Ball
Ieee Transactions on Signal Processing 1 Exact Wavelets on the Ball
复制标题
Ieee 信号处理汇刊 1 球上的精确小波
DOI：
--
复制
发表时间：
--
期刊：
影响因子：
--
通讯作者：
J. McEwen
中科院分区：
文献类型：
--
作者：
B. Leistedt;J. McEwen
我们在三维球（即实心球）上发展了精确的小波变换，我们称之为小波变换。为此，我们首先利用阻尼拉盖尔多项式构造径向线上的精确谐波变换，并推导出相应的正交规则。结合球谐变换，这种方法引出了球上的采样定理和一种新的三维分解，我们称之为傅里叶-拉盖尔变换。我们将这个新的变换与著名的傅里叶-贝塞尔分解联系起来，并证明傅里叶-拉盖尔基的带限性是精确计算傅里叶-贝塞尔分解的充分条件。然后，我们通过谐波平铺在球上构造小旗子变换，这是精确的，这要归功于傅里叶-拉盖尔变换的准确性（小旗子这个名字就是由此而来的）。相应的小波核在实调和空间中具有紧致的局部化性质，其角孔径在径向平移下保持不变。我们引入了一种多分辨率算法来快速执行小波变换，同时在球上的最小样本数量中捕获每个小波尺度上的所有信息。我们对这些新工具的实现实现了浮点精度，并且是公开的。我们通过数值实验证明了这些库的速度和准确性，并通过一个简单的去噪例子说明了它们的功能。
—We develop an exact wavelet transform on the three-dimensional ball (i.e. on the solid sphere), which we name the flaglet transform. For this purpose we first construct an exact harmonic transform on the radial line using damped Laguerre polynomials and develop a corresponding quadrature rule. Combined with the spherical harmonic transform, this approach leads to a sampling theorem on the ball and a novel three-dimensional decomposition which we call the Fourier-Laguerre transform. We relate this new transform to the well-known Fourier-Bessel decomposition and show that band-limitness in the Fourier-Laguerre basis is a sufficient condition to compute the Fourier-Bessel decomposition exactly. We then construct the flaglet transform on the ball through a harmonic tiling, which is exact thanks to the exactness of the Fourier-Laguerre transform (from which the name flaglets is coined). The corresponding wavelet kernels have compact localisation properties in real and harmonic space and their angular aperture is invariant under radial translation. We introduce a multiresolution algorithm to perform the flaglet transform rapidly, while capturing all information at each wavelet scale in the minimal number of samples on the ball. Our implementation of these new tools achieves floating point precision and is made publicly available. We perform numerical experiments demonstrating the speed and accuracy of these libraries and illustrate their capabilities on a simple denoising example.