A ne Invariant Detection : Edge Maps , AnisotropicDi usion , and Active Contours

In this paper we undertake a systematic investigation of a ne invariant object detection and image denoising. 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 maps are derived as a weighted di erence of images at di erent scales. We then introduce the a ne gradient as an a ne invariant di erential function of lowest possible order with qualitative behavior similar to the Euclidean gradient magnitude. These edge detectors are the basis for the extension of the a ne invariant scale-space 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 gradient path is computed via an algorithm which allows to simultaneously detect any number of objects independently of the initial curve topology. Based on the same theory of a ne invariant gradient ows we show that the a ne geometric heat ow is minimizing, in an a ne invariant form, the area enclosed by the curve.