A spectral unmixing algorithm for distributed endmembers with applications to biomedical imaging

Spectral unmixing algorithms tend to make the simplifying assumptions that each type of material (endmember) in a spectral library may be represented by a single reference spectrum and that the mixing process is linear. While these assumptions are convenient in that they allow techniques of linear algebra to be used, they lack realism as each material type in a spectral image will in general emit a distribution of spectra while the mixing itself need not be linear. We describe a 'common sense' spectral unmixing algorithm for the general case where endmembers are described by arbitrary D-dimensional probability distributions and the mixing can be non-linear. As an application we outline an unsupervised procedure for deriving the fractional material content of every pixel in an image and identifying anomalies given no a priori knowledge. Accurate endmember distributions are obtained by first masking out impure pixels using locally normalised Sobel and Laplacian filters and then Performing single-link hierarchical clustering on the pure pixels which remain. The most probable endmember decomposition for a given target spectrum is found by selecting an appropriate set of endmembers based on the target's immediate neighbourhood, and performing a constrained maximum likelihood search over the space of fractional abundances. We also explain how the procedure may be applied to subpixel and anomaly detection. To illustrate our ideas the techniques described are applied to biomedical images throughout.