Coherence Bounds for Sensing Matrices in Spherical Harmonics Expansion

Arya Bangun, Arash Behboodi, Rudolf Mathar

Abstract: The mutual coherence provides a basis for deriving recovery guarantees in compressed sensing. In this paper, the mutual coherence of spherical harmonics sensing matrices is examined for a class of sensing patterns common in practice and is used as a figure of merit for designing sensing matrices. We will show that for each sampling pattern, the coherence is lower bounded by the inner product of two Legendre polynomials with different degrees. In some practical situation, it is desirable to have sampling points on a sphere follow a regular pattern, hence, facilitating the measurement process. It will be shown that for a class of sampling patterns, the mutual coherence would be at its maximum, yielding the worst performance. Finally, the sampling strategy is proposed to achieve the derived lower bound.