Towards a statistical error estimate for convex-hull derived endmembers

Abstract

The convex hull methods for estimating spectral endmembers are subject to bias errors: mixed pixel bias - if all of the available pixels are mosaics of all m endmembers, the convex-hull derived endmember spectra are biased towards the centroid of the true endmember spectra; noise bias - additive Gaussian measurement noise inflates the convex hull away from the centroid of the noise-free convex hull. The noise bias error grows with the pixel count. This vulnerability to mixed pixel bias and noise bias prompts the following questions. Does the convex hull method throw away information by discarding the pixels lying inside the convex hull? Can bias error estimates be developed for convex-hull derived endmembers? Can bias-resistant endmember estimation methods be found? What is the gain in accuracy of the endmember estimates with increasing pixel count? What is the gain in accuracy with increasing density of pixels in the n-dimensional neighborhood of the true endmember? The following analysis focuses on these questions by omitting all sources of noise and distortion except the number and distribution of the samples in the neighborhood of the endmember.