On Compressive orthonormal Sensing

The Compressive Sensing (CS) approach for recovering sparse signal with orthonormal measurements has been studied under various notions of coherence. However, existing notions of coherence either do not exploit the structure of the underlying signal, or are too complicated to provide an explicit sampling scheme for all orthonormal basis sets. Consequently, there is lack of understanding of key factors that guide the sampling of CS with orthonormal measurements and achieve as low sample complexity as possible. In this paper, we introduce a new notion of π-coherence that exploits both the sparsity structure of the signal and the local coherence. Based on π-coherence, we propose a sampling scheme that is adapted to the underlying true signal and is applicable for CS under all orthonormal basis. Our scheme outperforms (up to a constant factor) existing sampling schemes for orthonormal measurements, and achieves a near-optimal sample complexity (up to certain logarithm factors) for several popular choices of orthonormal basis. Furthermore, we characterize the necessary conditions on the sampling schemes for CS with orthonormal measurements. We then propose a practical multi-phase implementation of our sampling scheme, and verify its advantage over existing sampling schemes via application to magnetic resonance imaging (MRI) in medical science.