Nonlinear Unmixing by Using Different Metrics in a Linear Unmixing Chain

Several popular endmember extraction and unmixing algorithms are based on the geometrical interpretation of the linear mixing model, and assume the presence of pure pixels in the data. These endmembers can be identified by maximizing a simplex volume, or finding maximal distances in subsequent subspace projections, while unmixing can be considered a simplex projection problem. Since many of these algorithms can be written in terms of distance geometry, where mutual distances are the properties of interest instead of Euclidean coordinates, one can design an unmixing chain where other distance metrics are used. Many preprocessing steps such as (nonlinear) dimensionality reduction or data whitening, and several nonlinear unmixing models such as the Hapke and bilinear models, can be considered as transformations to a different data space, with a corresponding metric. In this paper, we show how one can use different metrics in geometry-based endmember extraction and unmixing algorithms, and demonstrate the results for some well-known metrics, such as the Mahalanobis distance, the Hapke model for intimate mixing, the polynomial post-nonlinear model, and graph-geodesic distances. This offers a flexible processing chain, where many models and preprocessing steps can be transparently incorporated through the use of the proper distance function.