球面上的最优逼近抽样：改进与变化
Optimal-Dimensionality Sampling on the Sphere: Improvements and Variations
摘要：
为了精确地表示和重建带限信号的球，最近已经提出了一种最佳维数采样方案，该方案要求最佳样本数等于信号在谱（谐波）域中的自由度数。与最优维采样相关联的球谐变换（SHT）的计算需要以迭代方式对一系列线性系统进行反演。反演的稳定性取决于沿同纬的样品等纬环的位置。在这项工作中，我们已经开发出一种方法来放置这些等纬度环的样本，其目的是提高良好的空调的线性系统中所涉及的计算的SHT。我们还提出了一种多通SHT算法，以迭代地提高带限信号的SHT的准确性。此外，我们审查的变化，计算复杂性和提高精度的SHT的嵌入所提出的方法。通过数值实验，我们说明了所提出的变化和改进的SHT算法对应的最优维数采样方案显着提高了SHT的精度。
英文摘要：
For the accurate representation and reconstruction of band-limited signals on the sphere, an optimal-dimensionality sampling scheme has been recently proposed which requires the optimal number of samples equal to the number of degrees of freedom of the signal in the spectral (harmonic) domain. The computation of the spherical harmonic transform (SHT) associated with the optimal-dimensionality sampling requires the inversion of a series of linear systems in an iterative manner. The stability of the inversion depends on the placement of iso-latitude rings of samples along co-latitude. In this work, we have developed a method to place these iso-latitude rings of samples with the objective of improving the well-conditioning of the linear systems involved in the computation of the SHT. We also propose a multi-pass SHT algorithm to iteratively improve the accuracy of the SHT of band-limited signals. Furthermore, we review the changes in the computational complexity and improvement in accuracy of the SHT with the embedding of the proposed methods. Through numerical experiments, we illustrate that the proposed variations and improvements in the SHT algorithm corresponding to the optimal-dimensionality sampling scheme significantly enhance the accuracy of the SHT.
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