There is much theory about parameter estimation, dealing with properties such as the bias and variance of the estimate. This theory is based on analysis of the probability density functions of the measurements and the parameter space. In this appendix, we discuss such topics as bias of an estimator, the variance, the Cramér-Rao lower bound on the variance, and the posterior distribution. The treatment will be largely informal, based on examples, and exploring these concepts in the context of reconstruction.

The general lesson to be learnt from this discussion is that many of these concepts depend strongly on the particular parametrization of the model. In problems such as 3D projective reconstruction, where there is no preferred parametrization, these concepts are not well defined, or depend very strongly on assumed noise models.

A simple geometric estimation problem. The problem we shall consider is related to the triangulation problem of determining a point in space from its projection into two images. To simplify this problem, however, we consider its 2-dimensional analog. In addition, we fix one of the rays reducing the problem to one of estimating the position of a point along a known line from observing it in a single image.

Thus, consider a line camera (that is, one forming a 1D image as in section 6.4.2-(p175)) observing points on a single line. Let the camera be located at the origin (0, 0) and point in the positive Y direction.