Recovery of constituent spectra using non-negative matrix factorization

In this paper a constrained non-negative matrix factorization (cNMF) algorithm for recovering constituent spectra is described together with experiments demonstrating the broad utility of the approach. The algorithm is based on the NMF algorithm of Lee and Seung, extending it to include a constraint on the minimum amplitude of the recovered spectra. This constraint enables the algorithm to deal with observations having negative values by assuming they arise from the noise distribution. The cNMF algorithm does not explicitly enforce independence or sparsity, instead only requiring the source and mixing matrices to be non-negative. The algorithm is very fast compared to other "blind" methods for recovering spectra. cNMF can be viewed as a maximum likelihood approach for finding basis vectors in a bounded subspace. In this case the optimal basis vectors are the ones that envelope the observed data with a minimum deviation from the boundaries. Results for Raman spectral data, hyperspectral images, and 31P human brain data are provided to illustrate the algorithm's performance.