Using partial Derivatives of 3D images to extract typical surface features

3D edge detection in voxel images provides points corresponding to the surfaces forming the 3D structure. The next stage is to characterize the local geometry of these surfaces in order to extract points or lines on which registration and tracking procedures can rely. Typically one must calculate second order differential characteristics of the surfaces such as maximum, mean and Gaussian curvatures. The classical approach is to use local surface fitting and therefore it faces the problem of establishing links between 3D edge detection and local surface approximation. To get rid of this question, we propose to compute the curvatures on the edge points from the partial derivatives of the image. By assuming that the surface is defined by the iso-contours) 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 grey 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 iso-contours (i.e. the gradient at an edge point does not approximate 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 3D dimensional manifold) in R4. We set up the relationships existing between the curvatures of the hypersurface and the curvatures of the surface traced by the edge points. We express the maximum curvature at a point of the hypersurface with the second partial derivatives of the 3D image. We notice that it may be more efficient to smooth the data in R4. Moreover this approach could also be used to detect corners or vertices. We present experimental results obtained on real data (X-scanner images) using these two methods. As an application we extract the ridge line and show their stability using two 3D X-scanner images of a skull taken at different positions.