Camera orientation, calibration and inverse perspective with uncertainties: A Bayesian method applied to area estimation from diverse photographs

Large collections of images have become readily available through modern digital catalogs, from sources as diverse as historical photographs, aerial surveys, or user-contributed pictures. Exploiting the quantitative information present in such wide-ranging collections can greatly benefit studies that follow the evolution of landscape features over decades, such as measuring areas of glaciers to study their shrinking under climate change. However, many available images were taken with low-quality lenses and unknown camera parameters. Useful quantitative data may still be extracted, but it becomes important to both account for imperfect optics, and estimate the uncertainty of the derived quantities. In this paper, we present a method to address both these goals, and apply it to the estimation of the area of a landscape feature traced as a polygon on the image of interest. The technique is based on a Bayesian formulation of the camera calibration problem. First, the probability density function (PDF) of the unknown camera parameters is determined for the image, based on matches between 2D (image) and 3D (world) points together with any available prior information. In a second step, the posterior distribution of the feature area of interest is derived from the PDF of camera parameters. In this step, we also model systematic errors arising in the polygon tracing process, as well as uncertainties in the digital elevation model. The resulting area PDF therefore accounts for most sources of uncertainty. We present validation experiments, and show that the model produces accurate and consistent results. We also demonstrate that in some cases, accounting for optical lens distortions is crucial for accurate area determination with consumer-grade lenses. The technique can be applied to many other types of quantitative features to be extracted from photographs when careful error estimation is important.