Bayesian Tomography for Projections with an Arbitrary Transmission Function with an Application in Electron Microscopy

The vast majority of the developments in tomography assume that the transmission of the probe through the sample follows Beer's Law, i.e., the rule of exponential attenuation. However, for transmission electron microscopy of samples a few times their mean free path, Beer's Law is no longer an accurate description of the transmission of the probe as a function of the sample thickness. Recent simulations [Z. H. Levine, Appl. Phys. Lett. 82, 3943 (2003)] have demonstrated accounting for the correct transmission function leads to superior tomographic reconstructions for a photonic band gap sample 8 µm square. Those recent simulations assumed that data was available at all angles, i.e., over 180°. Here, we consider a limited-angle case by generalizing the Bayesian formalism of Bouman and Sauer to allow an arbitrary transmission function. The new formalism is identical to that of Bouman and Sauer when the transmission function obeys Beer's Law. The examples, based on 140° of data, suggest that using the physical transmission function is a requirement for performing limited angle reconstructions.