This paper investigates estimating exact imaging transformations accurately, reliably and efficiently. It is shown that in certain common computer vision situations the transformation required can be defined by a small number of parameters. Search is only required over these parameters, and consequently the search algorithms to estimate the transformation can be run at frame rate, without sacrificing robustness or accuracy. Performance is superior to often used approximations to these transformations. Two examples are illustrated: planar panoramic mosaicing, and augmented reality. Both applications run at frame rate on standard desktop machines, such as an SGI Indy or a PC.