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We present a new Bayesian methodology to learn the unknown material densityof a given sample by inverting its two-dimensional images that are taken with aScanning Electron Microscope. An image results from a sequence of projectionsof the convolution of the density function with the unknown microscopycorrection function that we also learn from the data. We invoke a novel designof experiment, involving imaging at multiple values of the parameter thatcontrols the sub-surface depth from which information about the densitystructure is carried, to result in the image. Real-life material densityfunctions are characterised by high density contrasts and typically are highlydiscontinuous, implying that they exhibit correlation structures that do notvary smoothly. In the absence of training data, modelling such correlationstructures of real material density functions is not possible. So we discretisethe material sample and treat values of the density function at chosenlocations inside it as independent and distribution-free parameters. Resolutionof the available image dictates the discretisation length of the model; threemodels pertaining to distinct resolution classes are developed. We developpriors on the material density, such that these priors adapt to the sparsityinherent in the density function. The likelihood is defined in terms of thedistance between the convolution of the unknown functions and the image data.The posterior probability density of the unknowns given the data is expressedusing the developed priors on the density and priors on the microscopycorrection function as elicitated from the Microscopy literature. We achieveposterior 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 realnano-structure and of simulated alloy samples.

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