Estimating the position of illuminants in paintings under weak model assumptions: an application to the works of two Baroque masters

The problems of estimating the position of an illuminant and the direction of illumination in realist paintings have been addressed using algorithms from computer vision. These algorithms fall into two general categories: In model-independent methods (cast-shadow analysis, occluding-contour analysis, ...), one does not need to know or assume the three-dimensional shapes of the objects in the scene. In model-dependent methods (shape-fromshading, full computer graphics synthesis, ...), one does need to know or assume the three-dimensional shapes. We explore the intermediate- or weak-model condition, where the three-dimensional object rendered is so simple one can very confidently assume its three-dimensional shape and, further, that this shape admits an analytic derivation of the appearance model. Specifically, we can assume that floors and walls are flat and that they are horizontal and vertical, respectively. We derived the maximum-likelihood estimator for the two-dimensional spatial location of a point source in an image as a function of the pattern of brightness (or grayscale value) over such a planar surface. We applied our methods to two paintings of the Baroque, paintings for which the question of the illuminant position is of interest to art historians: Georges de la Tour's Christ in the carpenter's studio (1645) and Caravaggio's The calling of St. Matthew (1599-1600). Our analyses show that a single point source (somewhat near to the depicted candle) is a slightly better explanation of the pattern of brightness on the floor in Christ than are two point sources, one in place of each of the figures. The luminance pattern on the rear wall in The calling implies the source is local, a few meters outside the picture frame-not the infinitely distant sun. Both results are consistent with previous rebuttals of the recent art historical claim that these paintings were executed by means of tracing optically projected images. Our method is the first application of such weak-model methods for inferring the location of illuminants in realist paintings and should find use in other questions in the history of art.