A Stepwise Analytical Gradient Descent Search for Hyperspectral Unmixing and Its Parallel Code Vectorization

Spectral mixture analysis (SMA) is a very important task for hyper-spectral image analysis, in general, and subpixel data extraction, in particular. In this paper we present a new methodology for spectral unmixing, where a vector of fractions, corresponding to a set of endmembers (EMs), is estimated for each pixel in the image. The process first provides an initial estimate of the fraction vector, followed by an iterative procedure that converges to an optimal solution. Specifically, gradient descent (GD) optimization is applied to the spectral angle mapper (SAM) objective function, so as to reduce significantly the estimation error due to amplitude (i.e., magnitude) variations in EM spectra, caused by the illumination change effect. Analytical derivations of the objective function’s gradient and the optimal step size, in each iteration, are presented. To reduce the running time, we have implemented our unmixing module via code vectorization, i.e., the entire process is "folded" into a single loop, and the fractions for all of the pixels are solved for simultaneously. We call this new scheme vectorized code gradient descent unmixing (VCGDU). Its performance was compared to the commonly used fully constrained least squares unmixing (FCLSU) and the fast state-of-art method, sparse unmixing by variable splitting and augmented Lagrangian (SUnSAL), based on the alternating direction method of multipliers (ADMM). The comparison was carried out on a real Airborne Visible/Infrared Imaging Spectrometer image and several synthetic pixel sets composed of subsets of EMs (from the real image) whose fractions at each pixel were set at random. Considering all of the potential EMs at each pixel (without knowing specifically which EMs or how many of them are actually mixed in the pixel), we show that the accuracy due to VCGDU is higher than that obtained by FCLSU and SUnSAL for a relatively large A Stepwise Analytical Gradient Descent Search for Hyperspectral Unmixing and Its Parallel Code Vectorization Fadi Kizel, Maxim Shoshany, and Nathan S. Netanyahu