Identifiable Solutions to Foreground Signature Extraction from Hyperspectral Images in an Intimate Mixing Scenario

Hyperspectral imaging considers the measurement of spectral signatures in near and far field settings. In the far field setting, the interactions of material spectral signatures are typically modeled using linear mixing. In the near field setting, material signatures frequently interact in a nonlinear manner (e.g., intimate mixing). An important task in hyperspectral imaging is to estimate the distribution and spectral signatures of materials present in hyperspectral data, i.e., unmixing. Motivated by forensics, this work considers a specific unmixing task, namely, the problem of foreground material signature extraction in an intimate mixing setting where thin layers of foreground material are deposited on other (background) materials. The unmixing task presents a fundamental challenge of unique (identifiable) recovery of material signatures in this and other settings. We propose a novel model for this intimate mixing setting and explore a framework for the task of foreground material signature extraction with identifiability guarantees under this model. We identify solution criteria and data conditions under which a foreground material signature can be extracted up to scaling and elementwise-inverse variations with theoretical guarantees in a noiseless setting. We present algorithms based on two solution criteria (volume minimization and endpoint member identification) to recover foreground material signatures under these conditions. Numerical experiments on real and synthetic data illustrate the efficacy of the proposed algorithms.