A Variational Approach to Edge Detection

l’tlt: probIc:rn of dztc~ctin~ iri:,c:il::ity c!~arlges in images is canonical in \ isIon. lXi;e del,ectlou opc.1 :ltc,rs arc typ]Lally designed to optimally estimate first or second derival.ivr over some (usually small) support. Other criteria such as output signal to noise ratio or bandwidth have also been argued for. This paper describes an attempt to formulate a set of edge detection criteria that capture as directly as possible the desirable properties of the detector. Variational techniques are used to find 2 solution over the space of all possible functions. 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 third criterion is claimed to be new, and it became necessary when an operator designed using the first two criteria was found to have excessive multiple responses. The edge model that will be considered here is 2 one-dlmensional step edge in white Gaussian noise although the same technique has been applied to an extended impulse or ridge profile. The result is a one dimensional operator that approximates the first derivative of a Gaussian. Its extension to two dimensions is also discussed.