论文标题
在常规网格上的感应矩阵的相互连贯性上的紧密界限
Tight bounds on the mutual coherence of sensing matrices for Wigner D-functions on regular grids
论文作者
论文摘要
旋转组功能近似的许多实际采样模式在参数轴上使用常规样品。在本文中，我们将与一类规则模式相对应的传感矩阵与量子力学中的角动量分析相关的相互一致性分析，并为其提供简单的下限。量子力学中的角动量分析出现了Wigner D功能的产物。我们首先将产物表示为单个Wigner D功能和角动量系数的线性组合，也称为Wigner 3J符号。使用组合身份，我们表明在某些条件下，样品的带宽和数量，在零阶的传感矩阵的内部产物（在零阶的内部产物）等于两个Legendre多项式的内部产物，主导了相互的相干性项，并为其固定了较低的结合。换句话说，对于一类常规采样模式，我们为可以通过分析计算的传感矩阵列的内部产物提供了一个下限。我们从数值上验证了我们的理论结果，并表明相互连贯性的下限大于welch结合。此外，我们提供可以实现球形谐波的下限的算法。
Many practical sampling patterns for function approximation on the rotation group utilizes regular samples on the parameter axes. In this paper, we relate the mutual coherence analysis for sensing matrices that correspond to a class of regular patterns to angular momentum analysis in quantum mechanics and provide simple lower bounds for it. The products of Wigner d-functions, which appear in coherence analysis, arise in angular momentum analysis in quantum mechanics. We first represent the product as a linear combination of a single Wigner d-function and angular momentum coefficients, otherwise known as the Wigner 3j symbols. Using combinatorial identities, we show that under certain conditions on the bandwidth and number of samples, the inner product of the columns of the sensing matrix at zero orders, which is equal to the inner product of two Legendre polynomials, dominates the mutual coherence term and fixes a lower bound for it. In other words, for a class of regular sampling patterns, we provide a lower bound for the inner product of the columns of the sensing matrix that can be analytically computed. We verify numerically our theoretical results and show that the lower bound for the mutual coherence is larger than Welch bound. Besides, we provide algorithms that can achieve the lower bound for spherical harmonics.
