Gaussian process with derivative information for the analysis of the sunlight adverse effects on color of rock art paintings.

Microfading Spectrometry (MFS) is a method for assessing light sensitivity color (spectral) variations of cultural heritage objects. The MFS technique provides measurements of the surface under study, where each point of the surface gives rise to a time-series that represents potential spectral (color) changes due to sunlight exposition over time. Color fading is expected to be non-decreasing as a function of time and stabilize eventually. These properties can be expressed in terms of the partial derivatives of the functions. We propose a spatio-temporal model that takes this information into account by jointly modeling the spatio-temporal process and its derivative process using Gaussian processes (GPs). We fitted the proposed model to MFS data collected from the surface of prehistoric rock art paintings. A multivariate covariance function in a GP allows modeling trichromatic image color variables jointly with spatial distances and time points variables as inputs to evaluate the covariance structure of the data. We demonstrated that the colorimetric variables are useful for predicting the color fading time-series for new unobserved spatial locations. Furthermore, constraining the model using derivative sign observations for monotonicity was shown to be beneficial in terms of both predictive performance and application-specific interpretability.