Enhancing Pure-Pixel Identification Performance via Preconditioning

Abstract

In this paper, we analyze different preconditionings designed to enhance robustness of pure-pixel search algorithms, which are used for blind hyperspectral unmixing and which are equivalent to near-separable nonnegative matrix factorization algorithms. Our analysis focuses on the successive projection algorithm (SPA), a simple, efficient, and provably robust algorithm. Recently, a provably robust preconditioning was proposed by Gillis and Vavasis [SIAM J. Optim., 25 (2015), pp. 677--698] which requires the resolution of a semidefinite program (SDP). Since solving the SDP in high precisions can be time consuming, we generalize the robustness analysis to approximate solutions of the SDP showing that a high accuracy solution is not crucial for robustness, paving the way for faster preconditionings. This first contribution also allows us to provide a robustness analysis for two other preconditionings. The first one is prewhitening, which can be interpreted as an optimal solution of the same SDP with additional constraints. We analyze the robustness of prewhitening, which allows us to characterize situations in which it performs competitively with the SDP-based preconditioning. The second one is based on SPA itself and can be interpreted as an optimal solution of a relaxation of the SDP. It is extremely fast when competing with the SDP-based preconditioning on several synthetic data sets.