Using differential geometry in R/sup 4/ to extract typical features in 3D density images

Abstract

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. To avoid the need to find links between 3D edge detection and local surface approximation, the authors propose a method involving computing the curvatures on the edge points from the second partial derivatives of the image. The 3D image is treated as a hypersurface (a 3D dimensional manifold) in R/sup 4/. Relationships are established between the curvatures of the hypersurface and the curvatures of the surface traced by the edge points. The maximum curvature at a point of the hypersurface is expressed with the second partial derivatives of the 3D image. These curvatures can also be directly computed in R/sup 3/ using a realistic assumption, but it may be more efficient to smooth the data in R/sup 4/. For instance, in the 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 R/sup 4/. This approach could also be used to detect corners or vertices. Experimental results are presented.<>