Sparse recovery of spherical harmonic expansions from uniform distribution on sphere

We analyse the characteristics of spherical harmonics to derive a tighter bound on the minimum number of required measurements to accurately recover a sparse signal in spherical harmonic domain. We numerically show the coherence of spherical harmonic matrix can be reduced from a polynomial order of N^1/4 or N^1/6 (both achieved by preconditioning) to a logarithmic order, i.e., log²(L) with respect to the degree of spherical harmonics L. Hence, one can, with high probability, recover s-sparse spherical harmonic expansions from M ≥ s log³ N measurements randomly sampled from the uniform sin θ dθ d_φ measure on sphere.