In this paper, the goal is to design deterministic sampling patterns on the sphere and the rotation group and, thereby, construct sensing matrices for sparse recovery of band-limited functions. It is first shown that random sensing matrices, which consists of random samples of Wigner D-functions, satisfy the Restricted Isometry Property (RIP) with proper preconditioning and can be used for sparse recovery on the rotation group. The mutual coherence, however, is used to assess the performance of deterministic and regular sensing matrices. We show that many of widely used regular sampling patterns yield sensing matrices with the worst possible mutual coherence, and therefore are undesirable for sparse recovery. Using tools from angular momentum analysis in quantum mechanics, we provide a new expression for the mutual coherence, which encourages the use of regular elevation samples. We construct low coherence deterministic matrices by fixing the regular samples on the elevation and minimizing the mutual coherence over the azimuth-polarization choice. It is shown that once the elevation sampling is fixed, the mutual coherence has a lower bound that depends only on the elevation samples. This lower bound, however, can be achieved for spherical harmonics, which leads to new sensing matrices with better coherence than other representative regular sampling patterns. This is reflected as well in our numerical experiments where our proposed sampling patterns perfectly match the phase transition of random sampling patterns.that are obtained from sampling functions in finite-dimensional function spaces. The sensing matrix entries in these applications are samples of orthonormal basis functions of the ambient space. Fourier matrices [5], matrices from trigonometric polynomials [6], orthogonal polynomials [7,8] and spherical harmonics [9,10] are some examples of these matrices. Fortunately, when the orthonormal functions are uniformly bounded, also called Bounded Orthonormal Systems (BOSs), a similar recovery guarantee can be obtained. If the samples are taken randomly from a certain probability measure, BOS matrices are proven to satisfy the RIP property [4, Chapter 12]. If the orthonormal functions are uniformly bounded by K, the required number of measurements scales with K 2 .This randomness in the measurement process, however, is inadmissible in many applications, for instance when the measurement process involves movements of mechanical devices. Random measurements require arbitrary movements that are possibly harmful to the measurement device. In these applications, the measurement process should be designed by considering the physical characteristics of the measurement device. An example, which is the main motivation of the current work, is the antenna measurement application. The samples in antenna measurements are taken using a robotic arm or which samples of a smooth trajectory are preferred over random samples. Therefore, regular sampling patterns like equiangular patterns are widely used for the measurement proces...

Abstract

Many practical sampling patterns for function approximation on the rotation group utilizes regular samples on the parameter axes. In this paper, we analyze the mutual coherence 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.

No abstract