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Simulation by Diffusion on a Manifold with Boundary: Applications to Ultrasonic Prenatal Medical Imaging

Consider the ultrasonic imaging of a fetal head, a standard procedure for assessing the growth and development of the fetus in utero. The size and shape of certain cross sections of the skull are particularly important in such assessments. The technician/radiologist spatially moves an ultrasonic transducer over the pregnant woman's abdomen and acquires three-dimensional information from the real-time two-dimensional video images. The ability to draw conclusions from such images is built upon an acquired practical knowledge as to what is bone, what is tissue, and what is sensor noise. Can one mechanize this intuitive understanding which allows the technician/radiologist to delineate between inherent biological variation and variation due to sensor noise, and in so doing, reconstruct the fetal head? The approach of the present paper is that of a probabilistic recovery of the fetal head. The "space of realizable fetal heads" is viewed as the result of similarity (translation × scale × rotation) transformatio...