1 Non-linear Spectral Unmixing by Geodesic Simplex Volume Maximization Rob Heylen, Member, IEEE, Dˇ zevdet Burazerovi´ c, Member, IEEE and Paul Scheunders, Member, IEEE Abstract—Spectral mixtures observed in hyperspectral im- agery often display non-linear mixing effects. Since most tra- ditional unmixing techniques are based upon the linear mixing model, they perform poorly in finding the correct endmembers and their abundances in the case of non-linear spectral mixing. In this paper, we present an unmixing algorithm that is capable of extracting endmembers and determining their abundances in hyperspectral imagery under non-linear mixing assumptions. The algorithm is based upon simplex volume maximization, and uses shortest-path distances in a nearest-neighbor graph in spectral space, hereby respecting the non-trivial geometry of the data manifold in the case of non-linearly mixed pixels. We demonstrate the algorithm on an artificial data set, the AVIRIS Cuprite data set, and a hyperspectral image of a heathland area in Belgium. Index Terms—Hyperspectral imaging, Spectral analysis, Man- ifolds I. I NTRODUCTION One of the inescapable implications of hyperspectral remote sensing is that a single pixel can often record only a mixed signature of distinct surface materials. This creates the need for unmixing [1], or decomposition of the observed pixel spectrum into its constituent spectra, ideally corresponding to individual materials. Besides identifying these pure spectra, or endmembers, a second important aspect is the estimation of their respective abundances in each observed pixel. Spectral unmixing is generally done under the assumption of linear mixing: the spectrum of each pixel consists of a linear combination of endmembers, with abundances that are positive and sum to one. In practice, such a model fits the situation where the endmember materials appear in the pixel as spatially segregated regions. This interpretation also yields a physical explanation for the constraints on the abundances: Any given material cannot have a negative contribution to a pixel, and the sum of all contributions has to equal one. In the literature, the linear mixture model has been prevalently used, and continues to be a popular starting point in formulating general frameworks [2] and dealing with specific problems [3], [4]. However, one often encounters situations where the linear mixing model is no longer adequate for describing the spectral mixing effects. Examples are secondary and higher-order reflections, shallow water environments (an example is shown in Fig. 1), intricate mineral mixtures [1], ... Unmixing in these cases has often been handled by extensively modeling the source of the non-linear effects (e.g. multiple reflectance and scattering [5], [6]), or by employing more model-independent R. Heylen, D. Burazerovi´ c and P. Scheunders are with the IBBT-Visielab, University of Antwerp, Universiteitsplein 1, 2610 Wilrijk, Belgium Fig. 1. Scatter plot of band 10 (710 nm) and band 16 (884 nm) of partly submerged grassland. The data manifold has a highly nontrivial shape, and does not resemble a linear simplex, indicating complex non-linear mixing interactions are present. methods for dealing with non-linearity (e.g. kernel-based pro- cessing [7] and artificial neural networks [8], [9], [10], [11]). In [12] one proposed a combined approach, as it used the linear mixture model to find the endmembers, and a neural network to refine their abundances. An alternative strategy for dealing with non-linearities in hy- perspectral data sets is performing a non-linear dimensionality reduction, yielding a linear space of reduced dimensionality, followed by traditional algorithms based on the linear mixing assumption. Most non-linear dimensionality reduction algo- rithms are data-driven and unsupervised, and use a geometri- cally oriented approach based on manifold learning [13]. Most of these approaches consist of a mapping that preserves some global or local relationship from a high-dimensional manifold (constituted by the source data) while projecting it to a lower- dimensional linear space. The subsequent linear operations may relate to any of the conventional unmixing, classification or compression techniques often performed on hyperspectral data, yielding a two-step process for coping with non-linear data sets. For instance, several non-linear projections (Local Linear Embedding [14], Laplacian Eigenmaps [15] and Local Tangent Space Alignment [16]) were recently compared as a precursor to a K-nearest neighbor classifier [17]. ISOMAP has been used as a preprocessing step in classification problems [18], and in combination with linear unmixing for target detection [19]. An important disadvantage of most non-linear dimension- ality reduction techniques is their high computational cost and memory requirements, making then rather impractical for use with sizable hyperspectral scenes. This problem was