Band Selection for Hyperspectral Images Using Non-Negativity Constraints

Abstract

This paper presents a new factorization technique for hyperspectral signal processing based on a constrained singular value decomposition (SVD) approach. Hyperpectral images typically have a large number of contiguous bands that are highly correlated. Likewise the field of view typically contains a limited number of materials and the spectra are also correlated. Only a selected number of bands, the extreme bands that include the dominant materials spectral signatures, are needed to express the data. Factorization can provide a means for interpretation and compression of the spectral data. Hyperspectral images are represented as non-negative matrices by graphic concatenation, with the pixels arranged into columns and each row corresponding to a spectral band. SVD and principal component analysis enjoy a broad range of applications, including, rank estimation, noise reduction, classification and compression, with the resulting singular vectors forming orthogonal basis sets for subspace projection techniques. A key property of non-negative matrices is that their columns/rows form non-negative cones, with any non-negative linear combination of the columns/rows belonging to the cone. Data sets of spectral images and time series reside in non-negative orthants and while subspaces spanned by SVD include all orthants, SVD projections can be constrained to the non-negative orthants. In this paper we utilize constraint sets that confine projections of SVD singular vectors to lie within the cones formed by the spectral data. The extreme vectors of the cone are found and these vectors form a basis for the factorization of the data. The approach is illustrated in an application to hyperspectral data of a mining area collected by an airborne sensor.