# Paper No. 11-29

**D Chakrabarty, N Gabrielyan, S Paul, F Rigat and R Beanland**

**Bayesian Learning of Material Density and the Blurring Function, given 2-D images taken with Bulk Microscopy Techniques**

**Abstract:** We present a new non-destructive Bayesian inverse methodology to learn the unknown material density ρ(x) ¸ 0 and the unknown blurring function η(z) ¸ 0, given 2-dimensional images of the material taken with the Scanning Electron Microscope, in Back Scattered Electrons or X-rays. Here X is the 3-dimensional spatial vector, the third component of which is Z. The novelty of the advanced methodology lies in its ability to perform the estimation of the unknown functions via multiple (¸ 2), inversions, given that the image results from the projection of the convolution of unknowns, followed by spatial-averaging over a stipulated “interaction volume” inside the material, within which the electrons of the incident beam - during an electron scattering experiment - interact atomistically with the material. We expand the data space by invoking multiple images at distinct beam energies and invoke geometric priors on the density and strong priors on η(z) using information available in existing microscopy literature. In our fully discretised model, the likelihood is defined as a function of the distance between the image data in a pixel and the contribution of the relevant voxels to the spatially-averaged projection of the convolution of the unknowns. The uniqueness of the estimates is discussed by viewing the posterior in the small noise limit. The inversion of real SEM images of a blend of Nickel and Silver nanoparticles, is included.