Further results on relationship between spectral unmixing and subspace projection

A recent short communication [1] showed that an orthogonal subspace projection (OSP) classifier developed for hyperspectral image classification in [2] was equivalent to a maximum likelihood estimator (MLE) resulting from a standard method of linear unmixing. It further concluded that the MLE subsumed the OSP classifier in spite of a constant difference in their magnitudes. Coincidentally, the equivalence of the OSP approach to linear unmixing was also derived in [3] and [4] by using the least-squares estimation with the same abundance estimate given by the MLE. In this communication, we show, on the contrary, that the MLE can be viewed as an a posteriori version of the OSP classifier and, thus, belongs to a family of OSP-based classifiers. More importantly, we further show that the constant produced by the MLE determines abundance estimation and has nothing to do with classification. As a result, it only alters the abundance concentration of the classified pixels, but not classification results.