Sub-pixel target spectra estimation and detection using functions of multiple instances

The Functions of Multiple Instances (FUMI) method for learning target pattern and non-target patterns is introduced and extended. The FUMI method differs significantly from traditional supervised learning algorithms because only functions of target patterns are available. Moreover, these functions are likely to involve other non-target patterns. In this paper, data points which are convex combinations of a target prototype and several non-target prototypes are considered. The Convex-FUMI (C-FUMI) method learns the target and non-target patterns, the number of non-target patterns, and the weights (or proportions) of all the prototypes for each data point. For hyperspectral image analysis, the target and non-target prototypes estimated using C-FUMI are the end-members for the target material and non-target (background) materials. For this method, training data need only binary labels indicating whether a data point contains or does not contain some proportion of the target endmember; the specific target proportions for the training data are not needed. In this paper, the C-FUMI algorithm is extended to incorporate weights for training data such that target and non-target training data sets are balanced (resulting in the Weighted C-FUMI algorithm). After learning the target prototype using the binary-labeled training data, target detection is performed on test data. Results showing sub-pixel explosives detection and sub-pixel target detection on simulated data are presented.