Alternating Direction Algorithms for Constrained Sparse Regression: Application to Hyperspectral Unmixing

TL;DR

This paper introduces two new algorithms based on the alternating direction method of multipliers to efficiently solve convex optimization problems in hyperspectral unmixing, improving speed and accuracy over existing methods.

Contribution

The paper presents SUnSAL and C-SUnSAL algorithms, novel applications of ADMM for constrained sparse regression and unmixing problems in hyperspectral imaging.

Findings

C-SUnSAL effectively solves CBP and CBPDN problems.

SUnSAL efficiently handles CLS and FCLS problems.

Algorithms outperform existing methods in speed and accuracy.

Abstract

Convex optimization problems are common in hyperspectral unmixing. Examples include: the constrained least squares (CLS) and the fully constrained least squares (FCLS) problems, which are used to compute the fractional abundances in linear mixtures of known spectra; the constrained basis pursuit (CBP) problem, which is used to find sparse (i.e., with a small number of non-zero terms) linear mixtures of spectra from large libraries; the constrained basis pursuit denoising (CBPDN) problem, which is a generalization of BP that admits modeling errors. In this paper, we introduce two new algorithms to efficiently solve these optimization problems, based on the alternating direction method of multipliers, a method from the augmented Lagrangian family. The algorithms are termed SUnSAL (sparse unmixing by variable splitting and augmented Lagrangian) and C-SUnSAL (constrained SUnSAL). C-SUnSAL solves the CBP and CBPDN problems, while SUnSAL solves CLS and FCLS, as well as a more general version thereof, called constrained sparse regression (CSR). C-SUnSAL and SUnSAL are shown to outperform off-the-shelf methods in terms of speed and accuracy.