Improved methods for dewarping images in convex mirrors in fine art: Applications to van Eyck and Parmigianino

We derive and demonstrate new methods for dewarping images depicted in convex mirrors in artwork and for estimating the three-dimensional shapes of the mirrors themselves. Previous methods were based on the assumption that mirrors were spherical or paraboloidal, an assumption unlikely to hold for hand-blown glass spheres used in early Renaissance art, such as Johannes van Eyck's Portrait of Giovanni (?) Arnol ni and his wife (1434) and Robert Campin's Portrait of St. John the Baptist and Heinrich von Werl (1438). Our methods are more general than such previous methods in that we assume merely that the mirror is radially symmetric and that there are straight lines (or colinear points) in the actual source scene. We express the mirror's shape as a mathematical series and pose the image dewarping task as that of estimating the coe cients in the series expansion. Central to our method is the plumbline principle: that the optimal coe cients are those that dewarp the mirror image so as to straighten lines that correspond to straight lines in the source scene. We solve for these coe cients algebraically through principal component analysis, PCA. Our method relies on a global gure of merit to balance warping errors throughout the image and it thereby reduces a reliance on the somewhat subjective criterion used in earlier methods. Our estimation can be applied to separate image annuli, which is appropriate if the mirror shape is irregular. Once we have found the optimal image dewarping, we compute the mirror shape by solving a di erential equation based on the estimated dewarping function. We demonstrate our methods on the Arnol ni mirror and reveal a dewarped image superior to those found in prior work|an image noticeably more rectilinear throughout and having a more coherent geometrical perspective and vanishing points. Moreover, we and the mirror deviated from spherical and paraboloidal shape; this implies that it would have been useless as a concave projection mirror, as has been claimed. Our dewarped image can be compared to the geometry in the full Arnol ni painting; the geometrical agreement strongly suggests that van Eyck worked from an actual room, not, as has been suggested by some art historians, a \ ctive" room of his imagination. We apply our method to other mirrors depicted in art, such as Parmigianino's Self-portrait in a convex mirror and compare our results to those from earlier computer graphics simulations.