Recognising two planar objects under a projective transformation

This thesis involves solving problems associated with object recognition for two dimensional images under a projective transformation. In order to recognise an object under any viewing angle requires invariant features to be identified. These invariant features can be used to match two images arising from two different views of a single object. One invariant is the projective curvature which characterises all curves up to a projective transformation. However the projective curvature depends on seventh order derivatives so is very sensitive to noise in the discretisation of the images and is of little practical use. Using links between the projective group and its subgroups, invariant points are found which depend on much lower order derivatives so are less sensitive to noise. They can be located on the images using a smoothing process then used to match the curves. However the smoothing process introduces error into the invariant points so that there will be error in the matching. This will not cause a problem if the two images have significantly different image features as then they can be detected within the wider tolerances of error. But this will cause a problem in distinguishing two images which are similar but different as it will not be known whether the error in the matching is due to error in the identification of the invariant points or not. A method, called the canonical form method, is developed incorporating an error analysis which corrects the error in the matchings of the images. This enables two similar but different two dimensional objects to be distinguished. It also provides the background knowledge to solve new problems as they arise. In addition to this practical method for two dimensional object recognition, a new characterisation of curves under the projective group and two of its subgroups is done using potentials and an alternative method for deriving and representing the projective curvature is given