Bayesian Density Estimation via Multiple Sequential Inversions of 2-D Images with Application in Electron Microscopy

We present a new Bayesian methodology to learn the unknown material density of a given sample by inverting its two-dimensional images that are taken with a Scanning Electron Microscope. An image results from a sequence of projections of the convolution of the density function with the unknown microscopy correction function that we also learn from the data; thus learning of the unknowns demands multiple inversions. We invoke a novel design of experiment, involving imaging at multiple values of the parameter that controls the sub-surface depth from which information about the density structure is carried, to result in the image. Real-life material density functions are characterized by high density contrasts and are highly discontinuous, implying that they exhibit correlation structures that do not vary smoothly. In the absence of training data, modeling such correlation structures of real material density functions is not possible. So we discretize the material sample and treat values of the density function at chosen locations inside it as independent and distribution-free parameters. Resolution of the available image dictates the discretization length of the model; three models pertaining to distinct resolution classes (at μm to nano metre scale lengths) are developed. We develop priors on the material density, such that these priors adapt to the sparsity inherent in the density function. The likelihood is defined in terms of the distance between the convolution of the unknown functions and the image data. The posterior probability density of the unknowns given the data is expressed using the developed priors on the density and priors on the microscopy correction function as elicited from the Microscopy literature. We achieve posterior samples using an adaptive Metropolis-within-Gibbs inference scheme. The method is applied to learn the material density of a 3-D sample of a nano-structure, using real image data. Illustrations on simulated image data of alloy samples are also included