Multidimensional Pixel Purity Index for Convex Hull Estimation and Endmember Extraction

One of the earliest endmember extraction algorithms employed in hyperspectral image processing is the pixel purity index (PPI) algorithm. This algorithm is still popular today but suffers from several drawbacks, such as a large computational cost. Many recent papers focus on improving the speed of the PPI algorithm with high-performance computing or combinatorial methods. In this paper, we present a computationally efficient way of calculating the PPI scores, based on the geometrical interpretation of the PPI sampling process. We first demonstrate the equivalence with Monte Carlo sampling of the polar cones of the convex hull of the data set. Next, we introduce a more efficient sampling method, where we use higher dimensional subspaces to sample these polar cones instead of 1-D skewers. The resulting algorithm can be used to quickly estimate the most important convex hull vertices of the data set, determine the corresponding PPI scores, and produce a list of endmember candidates. An unweighted version of this algorithm is introduced as well, which is simpler to implement, has a higher computational performance, and yields similar endmembers. If the subspace dimension is chosen to be one, both algorithms reduce to the PPI algorithm. We demonstrate the properties of these algorithms, such as convergence speed and accuracy, on artificial and real hyperspectral data and show that the results correspond to those obtained with PPI. The proposed algorithms, however, are up to three orders of magnitude faster and can generate representative PPI scores in less than a second on real hyperspectral data sets.