Hyperspectral BSS Using GMCA With Spatio-Spectral Sparsity Constraints

Generalized morphological component analysis (GMCA) is a recent algorithm for multichannel data analysis which was used successfully in a variety of applications including multichannel sparse decomposition, blind source separation (BSS), color image restoration and inpainting. Building on GMCA, the purpose of this contribution is to describe a new algorithm for BSS applications in hyperspectral data processing. It assumes the collected data is a mixture of components exhibiting sparse spectral signatures as well as sparse spatial morphologies, each in specified dictionaries of spectral and spatial waveforms. We report on numerical experiments with synthetic data and application to real observations which demonstrate the validity of the proposed method.

I. Introduction

Over the last few years, the use of multichannel sensors has spread widely in a variety of research fields ranging from astronomy to geophysics. This has raised interest in methods for the coherent processing of multivariate data, as well as more specific approaches for hyperspectral data. In this context, the data matrix is composed of images of size observed in different wavelength bands. A widely used approach to model such data consists in assuming that each row of is the linear combination of so-called sources: where is known as a source and models for the contribution of the th source in the th channel. The term stands for noise or source imperfections. By defining the so-called mixing matrix the entries of which are and the source matrix the rows of which are the sources , the data are more concisely modeled as follows:

$${\bf X} = {\bf AS} + {\bf N}$$ where models some additive noise contribution. Blind source separation methods then aim at estimating both and from the data . Several statistical approaches have been applied to solve this problem. In a nutshell, designing an effective blind source separation method reduces to finding a measure of diversity between the sources. In the last two decades, the mainstream approach has been independent component analysis (ICA—see [1], [2] and references therein). These statistical approaches aim at designing blind source separation methods that enforce the statistical indepedence of the sought after sources.