Nonlinear unmixing by using non-euclidean metrics in a linear unmixing chain

In the linear mixing model, many techniques for endmember extraction are based on the assumption that pure pixels exist in the data, and form the extremes of a simplex embedded in the data cloud. These endmembers can then be obtained by geometrical approaches, such as looking for the largest simplex, or by maximal orthogonal subspace projections. Also obtaining the abundances of each pixel with respect to these endmembers can be completely written in geometrical terms. While these geometrical algorithms assume Euclidean geometry, it has been shown that using different metrics can offer certain benefits, such as dealing with nonlinear mixing effects by using geodesic or kernel distances, or dealing with correlations and colored noise by using Mahalanobis metrics. In this paper, we demonstrate how a linear unmixing chain based on maximal orthogonal subspace projections and simplex projection can be written in terms of distance geometry, so that other metrics can be easily employed. This yields a very flexible processing chain: by using other metrics, the same unmixing methodology can be used to deal with a wide range of unmixing models and scenarios. As an example, metrics are provided for dealing with intimate mixtures, nonlinear dimensionality reduction, and colored noise.