On the Absence of Uniform Recovery in Many Real-World Applications of Compressed Sensing and the Restricted Isometry Property and Nullspace Property in Levels
The purpose of this paper is twofold. The first is to point out that the property of uniform recovery, meaning that all sparse vectors are recovered, does not hold in many applications where compressed sensing is successfully used. This includes fields like magnetic resonance imaging (MRI), nuclear magnetic resonance computerized tomography, electron tomography, radio interferometry, helium atom scattering, and fluorescence microscopy. We demonstrate that for natural compressed sensing matrices involving a level based reconstruction basis (e.g., wavelets), the number of measurements required to recover all $s$-sparse signals for reasonable $s$ is excessive. In particular, uniform recovery of all $s$-sparse signals is quite unrealistic. This realization explains why the restricted isometry property (RIP) is insufficient for explaining the success of compressed sensing in various practical applications. The second purpose of the paper is to introduce a new framework based on a generalized RIP and a generalized nullspace property that fit the applications where compressed sensing is used. We demonstrate that the shortcomings previously used to prove that uniform recovery is unreasonable no longer apply if we instead ask for structured recovery that is uniform only within each of the levels. To examine this phenomenon, a new tool, termed the “restricted isometry property in levels” (RIP$_L$) is described and analyzed. Furthermore, we show that with certain conditions on the RIP$_L$, a form of uniform recovery within each level is possible. Fortunately, recent theoretical advances made by Li and Adcock demonstrate the existence of large classes of matrices that satisfy the RIP$_L$. Moreover, such matrices are used extensively in applications such as MRI. Finally, we conclude the paper by providing examples that demonstrate the optimality of the results obtained.