Sensing Matrix Design and Sparse Recovery on the Sphere and the Rotation Group

Abstract

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...

Tight bounds on the mutual coherence of sensing matrices for Wigner D-functions on regular grids

“…Consequently, the resulting deterministic sampling points can be implemented into a real-system to carry out measurements on the spherical surface, as discussed in [10,11]. In this article, we confirm mathematically the numerical findings of [8]. Our proof relies on using results for angular momentum analysis in quantum mechanics and properties of Wigner 3j symbols.…”

“…In many applications, the goal is to recover a function defined on a group, say on the sphere S 2 and the rotation group SO(3), from only a few samples [7][8][9][10]27]. This problem can be seen as a linear inverse problem with structured sensing matrices that contain samples of spherical harmonics and Wigner D-functions.…”

“…In the area of compressed sensing and sparse signal recovery, there are recovery guarantee results for random sampling patterns [9,27] based on proving Restricted Isometry Property (RIP) of the corresponding sensing matrix. Regular deterministic sampling patterns are, however, more prevalent in practice due to their easier deployment [7,8]. RIP based results cannot be used for analyzing deterministic sensing matrices, because it has been shown in [6,33] that verifying RIP is computationally hard.…”

“…In those articles, the authors attempt to derive a compact formulation of the product and represent it as a linear combination of a single orthogonal polynomial with some coefficients. This representation reveals interesting properties that can also be applied in quantum mechanics [14,30], geophysics [32], machine learning [24], and low-coherence sensing matrices [7,8,10]. In quantum mechanics, these coefficients are used to calculate the addition of angular momenta, i.e., the interaction between two charged particles, and such coefficients are called the Clebsch-Gordan coefficients or Wigner 3j symbols.…”

“…In this work, we are interested in coherence analysis and sparse recovery tasks. It has been shown in [8] that a wide class of modular symmetric regular sampling patterns, like equiangular sampling, yield highly coherent sensing matrices and thereby perform poorly for signal recovery tasks. Besides, the coherence was shown to be affected by the choice of elevation sampling pattern independent of azimuth and polarization sampling patterns.…”