3D Model Accuracy and Gauge Fixing

In computer vision we often estimate a 3D model of an object, and its uncertainty, up to an unknown scale factor. This unknown scale factor means that we cannot directly infer positions and lengths on the model, nor their uncertainties. In order to make quantitative distance measurements on this model, we must obtain the unknown scale factor. However, how we determine the scale factor will affect the accuracy of the rescaled model. In this paper we study the problem of estimating the absolute scale of an object and its uncertainty, starting from a 3D reconstruction up to a scale factor, and a reference length. The theory derived here can be used both to correctly transform a model covariance matrix under rescaling, and to select a good length on the object from which to obtain the scale, and so maximize accuracy.