Resumen
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) transformations being applied to a given fetal head prototype. A model of the ultrasonic imaging of a fetal head is formulated and prior knowledge is incorporated, resulting in an a posteriori probability measure on the "space of realizable fetal heads." Sampling from this probability measure is achieved and justified via a time-homogeneous diffusion on a Riemannian, compact, simply connected, oriented manifold with boundary. The discretization of the diffusion is implemented into code, and the algorithm is applied to actual ultrasound prenatal images. Key words. medical image processing, stochastic differential equations, manifold with boundary, stochastic optimization