Shape from Radiological Density

detected in each slice by means of surface fitting or interpolation methods [7–15]. Surface fitting approaches can also In this paper we propose a strategy to solve the problem of recovering the 3-D shape of anatomical structures from single be used for the recovery of the approximate shape of an X-ray images, i.e., the problem of Shape from Radiological anatomical structure from a small set of tomographic or Density (SFRD). In order to overcome the noninvertibility of radiographic images when the boundaries of such a structhe process of image generation, we formulate a minimal set ture or other surface landmarks (obtained, for example, of physical assumptions that are used to constrain SFRD and by matching fiducial points in different views) are availto transform it into a well-posed problem. Our shape recovery able [16–19]. strategy requires the solution of four problems: (a) linearization The above-mentioned methods use the output data of a of the process of X-ray image generation, (b) image segmentasegmentation or a feature-matching algorithm as geometric tion, (c) estimation of a map of the local thickness of each constraints for the recovery of 3-D shape. However, in the anatomical structure of interest, and (d) recovery of the 3-D case of X-ray projective imaging, an additional, important shape of each structure from its boundaries and thickness map. In this paper we assume that problems (a) and (b) have already source of 3-D information is available: the selective absorpbeen faced, and propose a solution for problems (c) and (d). tion of X-ray photons by the different tissues being imaged. Experimental results on synthetic images, X-ray images of Such a source of information is used, for example, in phantoms, and real radiograms are reported.  1997 Academic Press computed tomography for the reconstruction of a density image from a complete set of projections on the grounds of Radon’s theorem [20]. Unfortunately, when the number