A ne Invariant Edge Maps and Active Contours

In this paper we undertake a systematic investigation of a ne invariant object detection. Edge detection is rst presented from the point of view of the a ne invariant scale-space obtained by curvature based motion of the image level-sets. In this case, a ne invariant edges are obtained as a weighted di erence of images at di erent scales. We then introduce the a ne gradient as the simplest possible a ne invariant di erential function which has the same qualitative behavior as the Euclidean gradient magnitude. These edge detectors are the basis both to extend the a ne invariant scalespace to a complete a ne ow for image denoising and simpli cation, and to de ne a ne invariant active contours for object detection and edge integration. The active contours are obtained as a gradient ow in a conformally Euclidean space de ned by the image on which the object is to be detected. That is, we show that objects can be segmented in an a ne invariant manner by computing a path of minimal weighted a ne distance, the weight being given by functions of the a ne edge detectors. The geodesic path is computed via an algorithm which allows to simultaneously detect any number of objects independently of the initial curve topology.