abstract: Already in 1968 one recognized that the transmission electron microscope could be used in a tomographic setting as a tool for structure determination of macromolecules. However, its usage in mainstream structural biology has been limited and the reason is mostly due to the devastating combination of noisy data and incomplete data problems that leads to severe ill-posedness. Despite these problems its importance is beginning to increase, especially in drug discovery.
From a mathematical point of view, the reconstruction problem in electron tomography amounts to the solution of an inverse scattering problem. To solve this inverse problem there are two major challenges that must be dealt with. The first is to develop an accurate model of the process of image formation in the transmission electron microscope, which in the terminology of inverse problems is the determination of the forward operator. The second is to choose a suitable regularization method for the inverse problem.
In the model for image formation, the electron-specimen interaction is modelled as a diffraction tomography problem and the picture is completed by adding a description of the optical system of the transmission electron microscope. We discuss various approximations of this model and the numerical problems that arise when one attempts at a numerical implementation of the forward operator. Next we turn our attention to the inverse problem which is very difficult mainly due to the devastating combination of very noisy data and the severe ill-posedness (due to the limited data problems). Restrictions in the data acquisition geometry leads to limited angle tomographic data and therefore implies that the conditions for stable reconstruction are not fulfilled. Moreover, only a subregion of the specimen is subject to electron exposure, so we are dealing with local tomographic data which leads to non-uniqueness. The severe ill-posedness means that a regularization method must used to obtain reconstructions of any practical value at all, and a good reconstruction is likely to require a carefully chosen regularization. Moreover, the non-uniqueness is best understood using microlocal analysis which allows us to explain those singularities that are stably visible from the limited data given by the data collection protocol in the electron microscope. Open mathematical problems related to regularisation theory and microlocal analysis will be mentioned.
Finally, if time permits, we provide some examples of reconstructions from electron tomography and demonstrate some of the biological interpretations that one can make.