Sparse recovery in Wigner-D basis expansion

Arya Bangun, Arash Behboodi, Rudolf Mathar

Abstract: We are concerned with the recovery of s-sparse Wigner-D expansions in terms of N Wigner-D functions. Considered as a generalization of spherical harmonics, Wigner-D functions are eigen-functions of Laplace-Beltrami operator and form an orthonormal system. However, since they are not uniformly bounded, the existing results on Bounded Orthonormal System (BOS) do not apply. Using previously introduced preconditioning technique, a new orthonormal and bounded system is obtained for which Restricted Isometry Property (RIP) property can be established. We show that the number of sufficient samples for sparse recovery scales with N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/6</sup> s log <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> (s) log(N). The phase transition diagram for this problem is also presented. We will also discuss the application of our results in the spherical near-field antenna measurement.