Discriminant image resolution: a novel multivariate image analysis method utilizing a spatial classification constraint in addition to bilinear nonnegativity

Abstract

Spectroscopic image analysis can be characterized as two different and distinct problems depending on the kind of information required from the solution. In its simplest form, the data can be decomposed into two submatrices, each of which carries different aspects of pure component information. Typically, this means information about pure spectra and pure intensities are obtained from the solution. This is the well-known and well-characterized bilinear form that by itself cannot guarantee a unique solution due to the rotational ambiguity inherent in the mathematical solution. This problem has been addressed in a number of ways by different authors using novel constraints applied to the least-squares solution. This usually takes the form of natural constraints as suggested by Tauler as the standard methodology to improve the resolution of data during alternative least squares (ALS) iterative process. A second type of multivariate image analysis problem is proposed in this paper that is quite different from the tradition methods and in some ways potentially is more useful. This involves the solution as a class problem in which the relevant information is not necessarily contained in pure component information, but rather, in unique combinations of the pure components that are allowed to be spatially collocated. This discriminant image resolution (DIR) method theoretically can be treated as a more generalized solution to the problem because the distribution of components is allowed to freely mix in simplified combinations of solutions. The result is a constrained least-squares solution where the constraints are more limited and therefore less restrictive. The constraints in this case employ the results of probabilistic class partition information by applying Bayesian discriminant clustering to the intensity submatrix. This amounts to a spatial constraint because the probability of class association is used as a way of limiting the components that are allowed to appear in a given pixel. This is a unique modification of the original modified alternating least squares (MALS) concept and to the authors' knowledge, the first time this type of constraint has been combined with an ALS algorithm for analysis of multivariate or hyperspectral data. Modified alternating least squares (MALS) is used as the computational engine to drive good convergence while imposing the constraints in this modified ALS model. This kind of analysis produces single and simple multicomponent images and spectra based on classification from spectral information. The present constraint is most useful in resolving image data when the true images are not severely overlapped but it will also perform well under conditions of more severe collinearity.