Hyperspectral Unmixing with Rare Endmembers via Minimax Nonnegative Matrix Factorization

Abstract

Hyperspectral images are used for ground-cover classification because many materials can be identified by their spectral signature, even in images with low spatial resolution. Pixels in such an image are often modeled as a convex combination of vectors, called endmembers, that correspond to the reflectance of a material to different wavelengths of light. This is the so-called linear mixing model. Since reflectance is inherently nonnegative, the task of unmixing hyperspectral pixels can be posed as a low-rank nonnegative matrix factorization (NMF) problem, where the data matrix is decomposed into the product of the estimated endmembers and their abundances in the scene. The standard NMF problem then minimizes the residual of the decomposition. Thus, using NMF works well when materials are present in similar amounts, but if some materials are under-represented, they may be missed with this formulation. Alternatively, we propose a novel hyperspectral unmixing model using a collection of NMF subproblems solved for patches of the original image. The endmembers are estimated jointly, such that the the maximum residual across all patches is minimized. In this paper we estimate the solution to the patch-based minimax NMF model, and show that it can estimate rare endmembers with superior accuracy.