ABSTRACT
Radiologists need to analyze the shapes of anatomical surfaces embedded in 1R3 given the images of their sets of singular values under the radiographic projection. They usually rely on a set of correspondence rules such as “the law of tangential incidence” (one sees a line of contrast where the X-ray has been tangent to the surface) or the “silhouette sign” and variants of it (e.g., if two contrast curves cross on the radiograph, then the pieces of surfaces which produced them do not stand at the same distance from the film). Thus using local rules to build a germ of surface embedding over the contrast curves, and given some a priori knowledge on the anatomical surface, interpreters can deduce numerous geometrical and topological facts about the surface. It is easy to find configurations where such simple rule systems are confused
(Fig. 1) or configurations which would call for new rules. However, such cases are not practically relevant because they are exceptional. The sign systems in use are found to work for most images in a simple and stable way. The generic properties of surface projections, as introduced by Whitney, are the key to the existence of such sign systems. Similar sign systems can be used in sectional imaging for the analysis and reconstruction of surfaces from their cross-sections, as a consequence of Morse theory.
