Algorithms for multivariate image analysis and other large-scale applications of multivariate curve resolution (MCR) typically employ constrained alternating least squares (ALS) procedures in their solution. The solution to a least squares problem under general linear equality and inequality constraints can be reduced to the solution of a non-negativity-constrained least squares (NNLS) problem. Thus the efficiency of the solution to any constrained least square problem rests heavily on the underlying NNLS algorithm. We present a new NNLS solution algorithm that is appropriate to large-scale MCR and other ALS applications. Our new algorithm rearranges the calculations in the standard active set NNLS method on the basis of combinatorial reasoning. This rearrangement serves to reduce substantially the computational burden required for NNLS problems having large numbers of observation vectors.

We describe several methods of applying equality constraints while performing procedures that employ alternating least squares. Among these are mathematically rigorous methods of applying equality constraints, as well as approximate methods, commonly used in chemometrics, that are not mathematically rigorous. The rigorous methods are extensions of the methods described in detail in Lawson and Hanson's landmark text on solving least squares problems, which exhibit well-behaved least squares performance. The approximate methods tend to be easy to use and code, but they exhibit poor least squares behaviors and have properties that are not well understood. This paper explains the application of rigorous equality-constrained least squares and demonstrates the dangers of employing non-rigorous methods. We found that in some cases, upon initiating multivariate curve resolution with the exact basis vectors underlying synthetic data overlaid with noise, the approximate method actually results in an increase in the magnitude of residuals. This phenomenon indicates that the solutions for the approximate methods may actually diverge from the least squares solution.

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Hyperspectral confocal fluorescence microscopy, when combined with multivariate curve resolution (MCR), provides a powerful new tool for improved quantitative imaging of multi-fluorophore samples. Generally, fully non-negatively constrained models are used in the constrained alternating least squares MCR analyses of hyperspectral images since real emission components are expected to have non-negative pure emission spectra and concentrations. However, in this paper, we demonstrate four separate cases in which partially constrained models are preferred over the fully constrained MCR models. These partially constrained MCR models can sometimes be preferred when system artifacts are present in the data or where small perturbations of the major emission components are present due to environmental effects or small geometric changes in the fluorescing species. Here we demonstrate that in the cases of hyperspectral images obtained from multicomponent spherical beads, autofluorescence from fixed lung epithelial cells, fluorescence of quantum dots in aqueous solutions, and images of mercurochrome-stained endosperm portions of a wild-type corn seed, these alternative, partially constrained MCR analyses provide improved interpretability of the MCR solutions. Often the system artifacts or environmental effects are more readily described as first and/or second derivatives of the main emission components in these alternative MCR solutions since they indicate spectral shifts and/or spectral broadening or narrowing of the emission bands, respectively. Thus, this paper serves to demonstrate the need to test alternative partially constrained models when analyzing hyperspectral images with MCR methods.

The electrochemical reaction behavior of a commercial Li-ion battery (LiFePO4-based cathode, graphite-based anode) has been measured via in situ neutron diffraction. A multivariate analysis was successfully applied to the neutron diffraction data set facilitating in the determination of Li bearing phases participating in the electrochemical reaction in both the anode and cathode as a function of state-of-charge (SOC). The analysis resulted in quantified phase fraction values for LiFePO4 and FePO4 cathode compounds as well as the identification of staging behavior of Li6, Li12, Li24, and graphite phases in the anode. An additional Li-graphite phase has also been tentatively identified during electrochemical cycling as LiC48 at conditions of ∼5% to 15% SOC.

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The combination of hyperspectral confocal fluorescence microscopy and multivariate curve resolution (MCR) provides an ideal system for improved quantitative imaging when multiple fluorophores are present. However, the presence of multiple noise sources limits the ability of MCR to accurately extract pure-component spectra when there is high spectral and/or spatial overlap between multiple fluorophores. Previously, MCR results were improved by weighting the spectral images for Poisson-distributed noise, but additional noise sources are often present. We have identified and quantified all the major noise sources in hyperspectral fluorescence images. Two primary noise sources were found: Poisson-distributed noise and detector-read noise. We present methods to quantify detector-read noise variance and to empirically determine the electron multiplying CCD (EMCCD) gain factor required to compute the Poisson noise variance. We have found that properly weighting spectral image data to account for both noise sources improved MCR accuracy. In this paper, we demonstrate three weighting schemes applied to a real hyperspectral corn leaf image and to simulated data based upon this same image. MCR applied to both real and simulated hyperspectral images weighted to compensate for the two major noise sources greatly improved the extracted pure emission spectra and their concentrations relative to MCR with either unweighted or Poisson-only weighted data. Thus, properly identifying and accounting for the major noise sources in hyperspectral images can serve to improve the MCR results. These methods are very general and can be applied to the multivariate analysis of spectral images whenever CCD or EMCCD detectors are used.

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