Nonnegative Matrix Factorization for Rapid Recovery of Constituent Spectra in Magnetic Resonance Chemical Shift Imaging of the Brain

Abstract

We present an algorithm for blindly recovering constituent source spectra from magnetic resonance (MR) chemical shift imaging (CSI) of the human brain. The algorithm, which we call constrained nonnegative matrix factorization (cNMF), does not enforce independence or sparsity, instead only requiring the source and mixing matrices to be nonnegative. It is based on the nonnegative matrix factorization (NMF) algorithm, extending it to include a constraint on the positivity of the amplitudes of the recovered spectra. This 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. We demonstrate and characterize the algorithm's performance using 31P volumetric brain data, comparing the results with two different blind source separation methods: Bayesian spectral decomposition (BSD) and nonnegative sparse coding (NNSC). We then incorporate the cNMF algorithm into a hierarchical decomposition framework, showing that it can be used to recover tissue-specific spectra given a processing hierarchy that proceeds coarse-to-fine. We demonstrate the hierarchical procedure on 1H brain data and conclude that the computational efficiency of the algorithm makes it well-suited for use in diagnostic work-up.

Joint Bayesian Endmember Extraction and Linear Unmixing for Hyperspectral Imagery

“…The posterior distribution of the unknown parameter vector can be computed from marginalization using the following hierarchical structure: (27) where and are defined in (9) and (26), respectively. Moreover, under the assumption of a priori independence between , and , the following result can be obtained: (28) where , and have been defined in (24), (22) and (25), respectively.…”

Reconstruction of reflectance spectra using robust nonnegative matrix factorization

“…In this section, we apply K-L decomposition [2], nonnegative matrix factorization [4], [5], nonnegative sparse coding (NNSC) [8], nonnegative matrix factorization with sparseness constraints (NMFSC) [9], constrained nonnegative matrix factorization [19], and the proposed robust nonnegative matrix factorization method to a database of reflectance spectra. Denote by S = ff f f j : j = 1; .…”

“…The constrained nonnegative matrix factorization (cNMF) is a recent algorithm that was proposed for blindly recovering constituent source spectra from magnetic resonance chemical shift imaging of human brain [19]. This algorithm is formulated as minimize kS 0 W Hk 2 subject to H 0: Fig.…”

Regularized nonnegative matrix factorization: Geometrical interpretation and application to spectral unmixing

“…Hence, it has already found diverse applications in data spectral analysis, mostly as a tool for blind unmixing or extraction of pure spectra (endmembers) from observed noisy mixtures. Examples include Raman scattering (e.g., Sajda et al, 2003;Li et al, 2007;Miron et al, 2011), hyperspectra unmixing (e.g., Miao and Qi, 2007;Zymnis et al, 2007;Zhang et al, 2008;Jia and Qian, 2009;Guo et al, 2009;Huck et al, 2010;Chan et al, 2011;Heylen et al, 2011;Qian et al, 2011;Iordache et al, 2011;2012;Plaza et al, 2012;Bioucas-Dias et al, 2012;Zdunek, 2012), spectral unmixing in microscopy (e.g., Pengo et al, 2010), chemical shift imaging (e.g., Sajda et al, 2004), reflectance spectroscopy (e.g., Pauca et al, 2006;Hamza and Brady, 2006), fluorescence spectroscopy (e.g., Gobinet et al, 2004), two-photon spectroscopic analysis (e.g., Hancewicz and Wang, 2005), astrophysical ice spectra unmixing (e.g., Igual et al, 2006;Igual and Llinares, 2008;Llinares et al, 2010), and gas chromatography-mass spectrometry (e.g., Likic, 2009). …”