Regularized Laplacian Zero Crossings as Optimal Edge Integrators

We view the fundamental edge integration problem for object segmentation in a geometric variational framework. First we show that the classical zero-crossings of the image Laplacian edge detector as suggested by Marr and Hildreth, inherently provides optimal edge-integration with regard to a very natural geometric functional. This functional accumulates the inner product between the normal to the edge and the gray level image-gradient along the edge. We use this observation to derive new and highly accurate active contours based on this functional and regularized by previously proposed geodesic active contour geometric variational models. We also incorporate a 2D geometric variational explanation to the Haralick edge detector into the geometric active contour framework.