Embodiments of the present invention are directed to determining the spectral vector of electromagnetic radiation reflected from, transmitted through, or emitted from a sample using a set of n intensity measurements. In general, the spectral vector has a dimension k that is greater than the number of measured intensities n. However, in many cases, the physical and chemical constraints of a system, when properly identified and modeled, effectively reduce the number of unknowns, generally the k components of the spectral vector, to an extent that allows for the spectral vector to be characterized from a relatively small number n of measured intensities.