N-FINDER for Finding Endmembers in Compressively Sensed Band Domain

Abstract

N-finder algorithm (N-FINDR) has been widely used for finding endmembers in hyperspectral imagery. Since N-FINDR must find all endmembers simultaneously, it requires exhausting all possible $p$ -endmember combinations among the entire data samples with $p$ being the number of endmembers required to be found. Accordingly, directly implementing N-FINDR is practically infeasible. To mitigate this dilemma, two recently developed algorithms called sequential N-FINDR (SQ N-FINDR) and successive N-FINDR (SC N-FINDR) were developed. However, even such an exhaustive search issue can be resolved numerically, another challenging issue for N-FINDR, which remains unsolved, is spectral dimensionality reduction. Because a $p$ -vertex simplex is embedded in a ( $p-1$ )-dimensional spectral data space, N-FINDR does not require full spectral dimensionality to calculate simplex volume (SV). This article presents a compressive sensing (CS) approach to N-FINDR that can find a $p$ -vertex simplex with the maximal SV by SQ/SC N-FINDR in a compressively sensed band domain (CSBD). In particular, to make this idea work, a new CS-based property called restricted SV property (RSVP) can be shown to be preserved in CSBD via a sensing matrix. It is this property that allows what N-FINDR and SQ/SC N-FINDR can achieve in the original data space (ODS) to be also achieved in CSBD. To further show the utility of SQ/SC N-FINDR in both ODS and CSBD as well as SV preserved by RSVP, a series of experiments are conducted for performance analysis.