Finding Edges and Lines in Images

The problem of detecting intensity changes in images is canonical in vision. Edge detection operators are typically designed to optimally estimate first or second derivative over some (usually small) support. Other criteria such as output signal to noise ratio or bandwidth have also been argued for. This thesis is an attempt to formulate a set of edge detection criteria that capture as directly as possible the desirable properties of an edge operator. Variational techniques are used to find a solution over the space of all linear shift invariant operators. The first criterion is that the detector have low probability of error i.e. failing to mark edges or falsely marking non-edges. The second is that the marked points should be as close as possible to the centre of the true edge. The third criterion is that there should be low probability of more than one response to a single edge. The technique is used to find optimal operators for step edges and for extended impulse profiles (ridges or valleys in two dimensions). The extension of the one dimensional operators to two dimentions is then discussed. The result is a set of operators of varying width, length and orientation. The problem of combining these outputs into a single description is discussed, and a set of heuristics for the integration are given.