In past chapters we have given algorithms for the estimation of various quantities associated with multiple images – the projection matrix, the fundamental matrix and the trifocal tensor. In each of these cases, linear and iterative algorithms were given, but little consideration was given to the possibility that these algorithms could fail. We now consider under what conditions this might happen.

Typically, if sufficiently many point correspondences are given in some sort of “general position” then the quantities in question will be uniquely determined, and the algorithms we have given will succeed. However, if there are too few point correspondences given, or else all the points lie in certain critical configurations, then there will not be a unique solution. Sometimes there will be a finite number of different solutions, and sometimes a complete family of solutions.

This chapter will concentrate on three of the main estimation problems that we have encountered in this book, camera resectioning, reconstruction from two views and reconstruction from three views. Some of the results given here are classical, particularly the camera resectioning and two-view critical surface problems. Others are more recent results. We consider the different estimation problems in turn.

Camera resectioning

We begin by considering the problem of computing the camera projection matrix, given a set of points in space and the corresponding set of points in the image. Thus, one is given a set of points Xi in space that are mapped to points xi in the image by a camera with projection matrix P.