Automatic endmember selection and nonlinear spectral unmixing of Lunar analog minerals

Abstract

While the interpretation of spectral reflectance data has been widely applied to detect the presence of minerals, determining and quantifying the abundances of minerals contained by planetary surfaces is still an open problem. With this paper we address one of the two main questions arising from the spectral unmixing problem. While the mathematical mixture model has been extensively researched, considerably less work has been committed to the selection of endmembers from a possibly huge database or catalog of potential endmembers. To solve the endmember selection problem we define a new spectral similarity measure that is not purely based on the reconstruction error, i.e. the squared difference between the modeled and the measured reflectance spectrum. To select reasonable endmembers, we extend the similarity measure by adding information extracted from the spectral absorption bands. This will allow for a better separation of spectrally similar minerals. Evaluating all possible subsets of a possibly very large catalog that contain at least one endmember leads to an exponential increase in computational complexity, rendering catalogs of 20–30 endmembers impractical. To overcome this computational limitation, we propose the usage of a genetic algorithm that, while initially starting with random subsets, forms new subsets by combining the best subsets and, to some extent, does a local search around the best subsets by randomly adding a few endmembers. A Monte-Carlo simulation based on synthetic mixtures and a catalog size varying from three to eight endmembers demonstrates that the genetic algorithm is expected to require less combinations to be evaluated than an exhaustive search if the catalog comprises 10 or more endmembers. Since the genetic algorithm evaluates some combinations multiple times, we propose a simple modification and store previously evaluated endmember combinations. The resulting algorithm is shown to never require more function evaluations than a full exhaustive search and the number of required function evaluations appears to grow less than exponentially. It thus requires considerably less time than an exhaustive search because the number of function evaluations is a hardware independent measure of the computational complexity. To evaluate the spectral similarity measure, we created a spectral reflectance catalog of selected lunar analog minerals. Based on precisely prepared mixtures of two to three components, we show that the proposed spectral similarity measure selects less false endmembers from the catalog than a similarity measure that is purely based on the reconstruction error.