Spectral unmixing using nonnegative matrix factorization with smoothed L0 norm constraint

Abstract

Spectral unmixing (SU) is a hot topic in remote sensing image interpretation, where the linear mixing model (LMM) is discussed widely for its validity and simplicity [1]. SU often includes two facts as follows: 1) endmembers extraction; 2) abundances estimation. Mathematically, in the SU model, the collections, the endmember signatures, and the abundances are nonnegative [1]. Therefore, nonnegative matrix factorization (NMF) has a great potential to solve SU, especially for LMM [2]. In fact, NMF (or NMF like) algorithms have been widely discussed in SU, such as NMF based on minimum volume constraint (NMF-MVC) [1], NMF based on minimum distance constraint (NMF-MDC) [3], and so on. These methods have advantages and disadvantages, respectively. In light of that the abundances are often sparse and sparse NMF tends to result more determinate factors, NMF with sparseness constraint has attracted more and more attentions [4-6].To solve SU using sparse NMF practically, one problem should be addressed firstly, that is how to select the functions to measure the sparseness feature. Since the abundance suffers from sum-to-one constraint physically, the widely used measure based on L1 norm constraint may be degenerate [7, 8]. As the smoothed L0 norm of the signals can reflect the sparseness intuitively and it is easy to be optimized, we focus on NMF with smoothed L0 norm constraint (NMF-SL0) in this work [9]. The rest of this paper is organized as follows. In Section II, typical SU and NMF models are presented. Section III describes the L0-based sparse NMF for solving SU, together with the gradient based optimization algorithm NMF-SL0. Simulations using synthetic mixtures and real hyperspectral images are presented in Section IV. Finally, conclusions are summarized in Section V.