We present an algorithm for blindly recovering constituent source
spectra from magnetic resonance spectroscopic imaging (MRSI) of human
brain. The algorithm is based on the non-negative matrix
factorization (NMF) algorithm, extending it to
include a constraint on the positivity of the amplitudes of the
recovered spectra and mixing matrices. This positivity constraint
enables recovery of physically meaningful spectra even in the presence
of noise that causes a significant number of the observation
amplitudes to be negative. The algorithm, which we call constrained
non-negative matrix factorization (cNMF), does not enforce
independence or sparsity, though it recovers sparse sources quite
well. It 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. We incorporate the cNMF algorithm into
a hierarchical decomposition framework, showing that it can be used to
recover tissue-specific spectra, e.g., spectra indicative of
malignant tumor. We demonstrate the hierarchical procedure on
1H long echo time (TE) brain absorption spectra and conclude that the
computational efficiency of the cNMF algorithm makes it well-suited
for use in diagnostic work-up.
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.
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