Restricted Entropy and Spectrum Properties for the Compressively Sensed Domain in Hyperspectral Imaging

Abstract

The restricted isometry property (RIP) and restricted conformal property (RCP) are two fundamental properties widely used in compressive sensing (CS) for sampling sparse signals. These two properties are sufficient to preserve the magnitude of a signal vector and the angle between two signal vectors in the compressively sensed band domain (CSBD), respectively. This article derives two new CS properties for hyperspectral signal vectors (HSVs), to be called the restricted entropy property (REP) and restricted spectrum property (RSP), which can be shown to preserve the entropy of an HSV and the spectral similarity between two HSVs in CSBD in correspondence to RIP and RCP, respectively. These two properties enable hyperspectral analysis algorithms to be directly applied to CSBD without loss of data integrity, while the dimensionality of CSBD can be considerably reduced. Most importantly, REP and RSP preserve exploitation algorithm performance without the need for decompression so as to avoid specifying a sparse basis.