Using Partial Derivatives of 3D Images to Extract Typical Surface Features

Three-dimensional edge detection in voxel images is used to locate points corresponding to surfaces of 3D structures. The next stage is to characterize the local geometry of these surfaces in order to extract points or lines which may be used by registration and tracking procedures. Typically one must calculate second-order differential characteristics of the surfaces such as the maximum, mean, and Gaussian curvature. The classical approach is to use local surface fitting, thereby confronting the problem of establishing links between 3D edge detection and local surface approximation. To avoid this problem, we propose to compute the curvatures at locations designated as edge points using directly the partial derivatives of the image. By assuming that the surface 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 calculate directly the curvatures and characterize the local curvature extrema (ridge points) from the first, second, and third derivatives of the gray level function. These partial derivatives can be computed using the operators of the edge detection. In the more general case where the contours are not isocontours (i.e., the gradient at an edge point only appoximates the normal to the surface), the only differential invariants of the image are in R4. This leads us to treat the 3D image as a hypersurface (a three-dimensional manifold) in R4. We give the relationships between the curvatures of the hypersurface and the curvatures of the surface defined by edge points. The maximum curvature at a point on the hypersurface depends on the second partial derivatives of the 3D image. We note that it may be more efficient to smooth the data in R4. Moreover, this approach could also be used to detect corners of vertices. We present experimental results obtained using real data (X ray scanner data) and applying these two methods. As an example of the stability, we extract ridge lines in two 3D X ray scanner data of a skull taken in different positions.