Restricted entropy and spectrum properties for hyperspectral imaging

The restricted isometry property (RIP) and restricted conformal property (RCP) are two fundamental properties of compressive sensing (CS) for sampling sparse signals. These properties guarantee that the magnitude of a signal vector and the angle between two signal vectors in the original data space (ODS) be preserved in the compressively sensed band domain (CSBD), respectively. This chapter derives two new CS properties for hyperspectral imaging, to be called restricted entropy property (REP) and restricted spectrum property (RSP) which can be shown to preserve the entropy of a hyperspectral signature vector (HSV) and the spectral similarity between two HSVs in both ODS and CSBD in correspondence to RIP and RCP, respectively. As a result of REP and RSP, many hyperspectral analysis algorithms can be performed directly in CSBD without loss of data integrity, while the dimensionality of ODS can be significantly reduced to that of CSBD. Most importantly, REP and RSP preserve hyperspectral exploitation algorithm performance without the need for decompression as to avoid the need of specifying a sparse basis.