Multi-Image Unsupervised Spectral Analysis

Large data sets delivered by imaging spectrometers are interesting in many ways in the Planetary Sciences. Due to the size of the data, which often prohibits conventional exploratory data analysis, unsupervised analysis methods could be a way of gathering interesting information contained in the data. In this work, we investigate some of the opportunities and limitations of unsupervised analysis based on non-negative matrix approximation [2] in planetary settings. Since typically there often is no ground truth to compare to, unsupervised rather than supervised methods allow to extract new information from data sets. Often, the practicability of these methods suffered from low performance, which made large-scale analyses almost prohibitively expensive. New research and implementation strategies [1] for non-negative matrix factorisation make it possible to extract sources and relative abundances for typical planetary data sets with reasonable resources. In this work, we try to give an impression of some of the trade-offs and opportunities involved. Nonnegative matrix factorisation is a technique which has enjoyed considerable research and been used in many application areas, from document clustering to spectral analysis. By considering P pixels of an hyperspectral image acquired at L frequency bands, the observed spectra are gathered in a P × L data matrix X. Each row of this matrix contains a measured spectrum at a pixel with spatial index p = 1, . . . , P . According to the linear mixing model, the pth spectrum, 1 ≤ p ≤ P , can be expressed as a linear combination of ri, 1 ≤ ri ≤ R, pure spectra of the surface components. Using matrix notations, this linear spectral mixing model can be written as