USING PARTIAL DERIVATIVES OF 3D IMAGES TO EXTRACT TYPICAL SURFACE-FEATURES

Three-dimensional edge detection in voxel images is used to locate poi nts corresponding to surfaces of 3D structures. The next stage is to c haracterize the local geometry of these surfaces in order to extract p oints or lines which may be used by registration and tracking procedur es. Typically one must calculate second-order differential characteris tics of the surfaces such as the maximum, mean, and Gaussian curvature . The classical approach is to use local surface fitting, thereby conf ronting the problem of establishing links between 3D edge detection an d local surface approximation. To avoid this problem, we propose to co mpute the curvatures at locations designated as edge points using dire ctly the partial derivatives of the image. By assuming that the surfac e is defined locally by a isointensity contour (i.e., the 3D gradient at an edge point corresponds to the normal to the surface), one can ca lculate directly the curvatures and characterize the local curvature e xtrema (ridge points) from the first, second, and third derivatives of the gray level function. These partial derivatives can be computed us ing the operators of the edge detection. In the more general case wher e the contours are not isocontours (i.e., the gradient at an edge poin t only approximates the normal to the surface), the only differential invariants of the image are in R(4). This leads us to treat the 3D ima ge as a hypersurface (a three-dimensional manifold) in R(4). We give t he relationships between the curvatures of the hypersurface and the cu rvatures of the surface defined by edge points. The maximum curvature at a point on the hypersurface depends on the second partial derivativ es of the 3D image. We note that it may be more efficient to smooth th e data in R(4). Moreover, this approach could also be used to detect c orners of vertices. We present experimental results obtained using rea l data (X ray scanner data) and applying these two methods. As an exam ple of the stability, we extract ridge lines in two 3D X ray scanner d ata of a skull taken in different positions. (C) 1995 Academic Press, Inc.