Dimensionality reduction, classification, and spectral mixture analysis using non-negative underapproximation
Optical Engineering: The Journal of SPIE
Nonnegative Matrix Factorization (NMF) and its variants have recently been successfully used as dimensionality reduction techniques for identification of the materials present in hyperspectral images. In this paper, we present a new variant of NMF called Nonnegative Matrix Underapproximation (NMU): it is based on the introduction of underapproximation constraints which enables one to extract features in a recursive way, like PCA, but preserving nonnegativity. Moreover, we explain why these
... ional constraints make NMU particularly wellsuited to achieve a parts-based and sparse representation of the data, enabling it to recover the constitutive elements in hyperspectral data. We experimentally show the efficiency of this new strategy on hyperspectral images associated with space object material identification, and on HYDICE and related remote sensing images. (NMF) is closely related to an older approach based on the geometric interpretation of the distribution of spectral signatures: they are located inside a low-dimensional simplex which vertices are the pure pixel signatures (i.e., the signatures of each individual material). 3, 4 † Any invertible matrix D such that V D ≥ 0 and D −1 W ≥ 0 generates an equivalent solution. ‡ Unless P = NP, drawback 1. can not be 'resolved' since the underlying problem of spectral unmixing is of combinatorial nature 15 and can be shown to be equivalent to a NP-hard problem. 7 * * Note that Λ does not correspond to the Lagrangian dual variables of min x≥0,y≥0,xy T ≤M ||M − xy T ||1. However, this formulation is closely related to the Lagrangian relaxation and allows to use the same derivations as Algorithm 1.