A family of second-order nonlinear differential operators that are useful in building simple edge detectors is treated. The authors provide a uniform definition for these operators and describe their internal geometry. Based on their geometric analysis, they associate symbolic descriptors with the abstract edges defined by the operators. One of them, Q, is essentially the same operator that J.F. Canny (ibid., vol.PAMI-8, p.552-7, 1986) used to define his two-dimensional simple edge detector. The authors give a parse tree for Q, which provides a complete analysis of its local geometry. The analysis shows that Q has a fondness for edges that lift to certain types of asymptotic curves. These include select straight line segments lying on the surface, inflections of mean curvature, and paths with constant rate of ascent. The methods are drawn largely from vector calculus and differential geometry. The authors illustrate their theoretical work with results obtained when Q was applied to image data that were interpolated with a B-spline.>

We use rotational and curvature properties of vector fields to identify critical features of an image. Using vector analysis and dif-ferential geometry, we establish the properties needed, and then use these properties in three ways. First, our results make it theoretically possible to identify extremal edges of an intensity function f(x, y) of two variables by considering the gradient vector field V = ¿f. There is also enough information in ¿f to find regions of high curvature (i.e., high curvature of the level paths of f). For color images, we use the vector field V = (I, Q). In application, the image is partitioned into a grid of squares. On the boundary of each square, V/|V| is sampled, and these unit vectors are used as the tangents of a curve ¿. The rotation number (or topological degree) ¿(¿) and the average curvature f|¿¿| are computed for each square. Analysis of these numbers yields infor-mation on edges and curvature. Experimental results from both simu-lated and real data are described.