We present a joint $\mathbb {SO}(3)$-spectral domain filtering framework using the directional spatially localized spherical harmonic transform (DSLSHT), for the estimation and enhancement of random anisotropic signals on the sphere contaminated by random anisotropic noise. We design an optimal filter for filtering the DSLSHT representation of the noise-contaminated signal in the joint $\mathbb {SO}(3)$-spectral domain. The filter is optimal in the sense that the filtered representation in the joint domain is the minimum mean square error estimate of the DSLSHT representation of the underlying (noise-free) source signal. We also derive a least square solution for the estimate of the source signal from the filtered representation in the joint domain. We demonstrate the capability of the proposed filtering framework using the Earth topography map in the presence of anisotropic, zero-mean, uncorrelated Gaussian noise, and compare its performance with the joint spatial-spectral domain filtering framework.

中文翻译：

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存在各向异性噪声时球面信号的联合 SO(3)-谱域滤波

我们提出联合 $\mathbb {SO}(3)$- 使用定向空间局部球谐变换 (DSLSHT) 的谱域滤波框架，用于估计和增强被随机各向异性噪声污染的球体上的随机各向异性信号。我们设计了一个最佳滤波器来过滤关节中噪声污染信号的 DSLSHT 表示$\mathbb {SO}(3)$-谱域。在联合域中的滤波表示是底层（无噪声）源信号的 DSLSHT 表示的最小均方误差估计的意义上，滤波器是最佳的。我们还从联合域中的滤波表示导出源信号估计的最小二乘解。我们在存在各向异性、零均值、不相关高斯噪声的情况下使用地球地形图证明了所提出的过滤框架的能力，并将其性能与联合空间-谱域过滤框架进行了比较。