Abstract
A frequent problem in analysis is the need to find two matrices, closely related to the underlying measurement process, which when multiplied together reproduce the matrix of data points. Such problems arise throughout science, for example, in imaging where both the calibration of the sensor and the true scene may be unknown and in localized spectroscopy where multiple components may be present in varying amounts in any spectrum. Since both matrices are unknown, such a decomposition is a bilinear problem. We report here a solution to this problem for the case in which the decomposition results in matrices with elements drawn from positive additive distributions. We demonstrate the power of the methodology on chemical shift images (CSI). The new method, Bayesian spectral decomposition (BSD), reduces the CSI data to a small number of basis spectra together with their localized amplitudes. We apply this new algorithm to a19F nonlocalized study of the catabolism of 5-fluorouracil in human liver,31P CSI studies of a human head and calf muscle, and simulations which show its strengths and limitations. In all cases, the dataset, viewed as a matrix with rows containing the individual NMR spectra, results from the multiplication of a matrix of generally nonorthogonal basis spectra (the spectral matrix) by a matrix of the amplitudes of each basis spectrum in the the individual voxels (the amplitude matrix). The results show that BSD can simultaneously determine both the basis spectra and their distribution. In principle, BSD should solve this bilinear problem for any dataset which results from multiplication of matrices representing positive additive distributions if the data overdetermine the solutions.
