New high resolution 3D structures of biological macromolecules can now be determined directly using 3D electron cryo-microscopy. These new molecular structures will underpin advances in medicine and nano-technology. See this recent review for more details:[Nature 525, 172–174 (10 September 2015) doi:10.1038/525172a]
There are further opportunities for improvement, including the development of new mathematical techniques.
The method for 3D reconstruction of 3D maps from the electron microscopy images is analogous to computed X-ray tomography in medical imaging. The images record a projected density of the structure, and a series of these can be used to compute a 3D density map (the inverse problem).
In single particle cryoEM, molecules are frozen in solution (and are therefore randomly oriented), so part of the challenge is to identify the different orientations of the particles in the images. This is different from medical tomography where the object or source is rotated about the other, giving a known geometrical relationship of the images.
Electron tomography mode is more similar to medical imaging techniques, but gives lower resolution. The specimen can be tilted, but only to a limited degree, giving incomplete data.
There is scope for more advanced mathematical methods to solve these problems in a better way. I will outline the problems in more detail, and show some specific examples related to relevant biological structures we are working on.