A method for the separation of additive spectra of complex mixtures is developed on the basis of a linear algebra technique and nonlinear optimization algorithms. It is shown to be possible, under certain conditions, to uniquely separate a set of complex spectral curves consisting of the same components, but with different proportions, into the unknown spectra of the pure constituents and to give their respective relative concentrations. The method proposed is a variant of the self-modeling curve-resolution approach based on the singular value decomposition of the data matrix formed by the set of digitized spectra of mixtures. The spectra of components are calculated as linear combinations of left-side singular vectors of the data matrix provided that both individual spectra and their concentrations are nonnegative and the shapes of the spectra are as dissimilar as possible. The technique provides a unique decomposition if each fundamental spectrum has at least one wavelength with zero intensity and the other pure spectra are nonzero at this wavelength. The algorithm is evaluated on an artificial data set to clearly demonstrate the method. The approach described in this paper may be applied to any experiment whose outcome is a continuous curve y(x) that is a sum of unknown, nonnegative, linearly independent functions.
Vladimir V. Volkov, "Separation of Additive Mixture Spectra by a Self-Modeling Method," Appl. Spectrosc. 50, 320-326 (1996)