Name
Title | ||
|---|---|---|
Spherical Designs and Nonconvex Minimization for Recovery of Sparse Signals on the Sphere. |
Abstract | ||
|---|---|---|
This paper considers the use of spherical designs and nonconvex minimization for recovery of sparse signals on the unit sphere S-2. The available information consists of low order, potentially noisy, Fourier coefficients for S-2. As Fourier coefficients are integrals of the product of a function and spherical harmonics, a good cubature rule is essential for the recovery. A spherical t-design is a set of points on S-2, which are nodes of an equal weight cubature rule integrating exactly all spherical polynomials of degree <= t. We will show that a spherical t-design provides a sharp error bound for the approximation signals. Moreover, the resulting coefficient matrix has orthonormal rows. In general the l(1) minimization model for recovery of sparse signals on S-2 using spherical harmonics has infinitely many minimizers, which means that most existing sufficient conditions for sparse recovery do not hold. To induce the sparsity, we replace the l(1)-norm by the l(q)-norm (0 < q < 1) in the basis pursuit denoise model. Recovery properties and optimality conditions are discussed. Moreover, we show that the penalty method with a starting point obtained from the reweighted l(1) method is promising to solve the l(q) basis pursuit denoise model. Numerical performance on nodes using spherical t-designs and t(epsilon)-designs (extremal fundamental systems) are compared with tensor product nodes. We also compare the basis pursuit denoise problem with q = 1 and 0 < q < 1. |
Year | DOI | Venue |
|---|---|---|
2018 | 10.1137/17M1147378 | SIAM JOURNAL ON IMAGING SCIENCES |
Keywords | Field | DocType |
sparse recovery,quasi-norm,spherical design,nonconvex minimization,spherical cubature,reweighted l(1) | Coefficient matrix,Polynomial,Mathematical analysis,Spherical harmonics,Orthonormal basis,Minification,Fourier series,Spherical design,Mathematics,Unit sphere | Journal |
Volume | Issue | ISSN |
11 | 2 | 1936-4954 |
Citations | PageRank | References |
0 | 0.34 | 0 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Xiaojun Chen | 1 | 1298 | 107.51 |
| Robert S. Womersley | 2 | 258 | 74.51 |