Doctors often need precise three-dimensional information about a patient's inner organs. For instance, a surgeon attaching an artificial joint needs to know the position of a screw inside bone, a radiologist looking for breast cancer benefits from knowing the three-dimensional form of clusters of microcalcifications, and a dentist can safely remove a tooth is she knows the way tooth roots embrace the nerves inside the jaw bone.
In three-dimensional medical X-ray imaging, radiographs are taken of a patient from multiple directions. X-rays attenuate inside the patient's body depending on the thickness of different tissues, resulting in a projection image. From the acquired set of projection images, three-dimensional structure of the inner organs needs to be determined. Mathematically speaking, we have available a collection of line integrals of a non-negative function f(x), the X-ray attenuation coefficient, and we need to determine f from the knowledge of those line integrals.
The most widely used three-dimensional X-ray imaging technology is Computerized Tomography (CT), where an extensive set of line integrals is measured using a large, sturdy and expensive scanner device. The result is a highly accurate three-dimensional reconstruction of f(x). However, many diagnostic and therapeutic tasks would benefit from three-dimensional information but a full CT scan is impractical. Reasons for this include the high radiation dose of a CT scan and high cost of scanners. In my research I have studied a new kind of three-dimensional medical X-ray imaging modality, where the input is a few radiographs and output is an imperfect three-dimensional reconstruction of tissue. By imperfect I mean here a reconstruction that is not necessarily a high-quality approximation of the attenuation coefficient f(x), but contains all the relevant information for the clinical task at hand. The main benefit of the new modality is that any digital X-ray device can be used for three-dimensional imaging. The main difficulty is that a small number of radiographs does not contain enough information to fully reconstruct f(x), and so the reconstruction task requires complicated mathematical algorithms.
One promising approach for reconstruction of f(x) from sparse projection data is Bayesian inversion. I have studied Bayesian inversion in a joint research project between GE Healthcare and Finnish universities. The foundations of our approach is explained in articles [Physics in Medicine and Biology 48 (2003) PDF] and [Physics in Medicine and Biology 48 (2003) PDF] by Kolehmainen, S, Järvenpää, Kaipio, Koistinen, Lassas, Pirttilä and Somersalo. See the figure below for a comparison of a classical reconstruction method called tomosynthesis (left) and our Bayesian approach (right). The data for these reconstructions was collected from a skull specimen using a dentist's standard X-ray imaging equipment. The images represent two different slices through dental structures.