Pump-probe microscopy is an emerging technique that provides detailed chemical information of absorbers with sub-micrometer spatial resolution. Recent work has shown that the pump-probe signals from melanin in human skin cancers correlate well with clinical concern, but it has been difficult to infer the molecular origins of these differences. Here we develop a mathematical framework to describe the pump-probe dynamics of melanin in human pigmented tissue samples, which treats the ensemble of individual chromophores that make up melanin as Gaussian absorbers with bandwidth related via Frenkel excitons. Thus, observed signals result from an interplay between the spectral bandwidths of the individual underlying chromophores and spectral proximity of the pump and probe wavelengths. The model is tested using a dual-wavelength pump-probe approach and a novel signal processing method based on gnomonic projections. Results show signals can be described by a single linear transition path with different rates of progress for different individual pump-probe wavelength pairs. Moreover, the combined dual-wavelength data shows a nonlinear transition that supports our mathematical framework and the excitonic model to describe the optical properties of melanin. The novel gnomonic projection analysis can also be an attractive generic tool for analyzing mixing paths in biomolecular and analytical chemistry.