3. Bayesian Learning of Material Density Function by Multiple Sequential Inversions of 2-D Images in Electron Microscopy

We discuss a novel inverse problem in which the data is generated by the sequential contractive projections of the convolution of two unknown functions, both of which we aim to learn. The method is illustrated using an application that relates to the multiple inversions of image data recorded with a scanning electron microscope, with the aim of learning the density of a given material sample and the microscopy correction function. Given the severe logistical difficulties in this application of taking multiple images at different viewing angles, a novel imaging experiment is undertaken, resulting in expansion of information. In lieu of training data, it is noted that the highly discontinuous material density function cannot be modelled using a Gaussian process (GP) as the parametrisation of the required nonstationary covariance function of such a GP cannot be achieved without training data. Consequently, we resort to estimating values of the unknown functions at chosen locations in their domain—locations at which an image data are available. Image data across a range of resolutions lead to multiscale models which we use to estimate material densities from the micrometre to nanometre length scales. We discuss applications of the method in nondestructive learning of material density using simulated metallurgical image data, as well as perform inhomogeneity detection in multicomponent composite on nanometre scales, by inverting real image data of a brick of nanoparticles.