Shape from Radiological Density

In this paper we propose a strategy to solve the problem of recovering the 3-D shape of anatomical structures from single X-ray images, i.e., the problem of Shape from Radiological Density (SFRD). In order to overcome the noninvertibility of the process of image generation, we formulate a minimal set of physical assumptions that are used to constrain SFRD and to transform it into a well-posed problem. Our shape recovery strategy requires the solution of four problems: (a) linearization of the process of X-ray image generation, (b) image segmentation, (c) estimation of a map of the local thickness of each anatomical structure of interest, and (d) recovery of the 3-D shape of each structure from its boundaries and thickness map. In this paper we assume that problems (a) and (b) have already been faced, and propose a solution for problems (c) and (d). Experimental results on synthetic images, X-ray images of phantoms, and real radiograms are reported.