Werner Kühlbrandt is thanked for a critical reading of the manuscript. The PACEM is funded by Deutsche Forschungsgemeinschaft (DFG) within the Cluster of Excellence “Macromolecular Complexes”.
The field of materials science has seen a dramatic increase in the use of X-ray or electron-based tomographic studies of materials. Despite the availability of advanced materials–characterization tools, rapid and sensitive detectors, and massive computational resources, there is still a dire need for accurate physical models and the associated algorithms that can assist the user in (1) predicting what the data should look like, given a model of the material system, and (2) extracting all available information from an acquired data set. For instance, high angle annular dark field (HAADF) electron tomography is used to reconstruct nanoscale objects in 3D (e.g., ), but to-date such reconstructions are mostly qualitative instead of quantitative. In medical X-ray tomography applications (for instance, dual energy CT-scans ), tomographic reconstruction results in a quantitative 3D map of the object\’s density distribution; one can quantitatively identify bone, tissue, empty spaces, fluids, and so on. In medium resolution HAADF tomography, on the other hand, there is no clear understanding of what the quantity is that is being reconstructed; since the HAADF signal is considered to be proportional to Z2, with Z the atomic number, one should ask the question: can we actually reconstruct Zas a function of position in the sample? It appears from the recent HAADF literature (e.g., ) that the electron tomography GDC-0199 Supplier has not yet answered this apparently simple question. Some progress has been made at the atomic length scale , where 3D reconstructions of nano-particles are now within the realm of possibilities, but at the larger length scale (tens of nanometers to microns, e.g., the relevant length scales for many modern materials applications) no reports of quantitative HAADF-based reconstructions can be found. This example illustrates that today\’s modern data processing algorithms in electron microscopy are not necessarily being employed to the fullest extent. Extracting all possible information from a data set requires not only algorithms for the analysis of the reconstructions but also predictive (forward) algorithms so that microstructure models can be compared to actual data sets. In 3D TEM and SEM studies, such algorithms are still rare and in this contribution we describe a forward modeling approach for HAADF-STEM tomography that may ultimately make this powerful technique more quantitative.
Model-based iterative reconstruction (MBIR) algorithms have emerged as a mathematical and algorithmic framework for integrating physical models of materials and devices with experimentally measured data to form quantitative inversions of 3D material parameter volumes . The MBIR framework formulates the problem of data inversion as an estimation problem, in which the unknown quantity is the image or volume to be reconstructed. The MBIR problem typically then reduces to an optimization with terms representing the match of the measured data to the theoretical prediction and the known and statistical ensemble properties of the material. MBIR is a powerful framework because it allows for incorporation of general nonlinear physics-based forward models, joint estimation of unknown physical parameters (e.g., instrument calibration parameters), and both hard and soft constraints resulting from material properties and statistical material characteristics.
While conventional image reconstruction methods (e.g., filtered back projection (FBP) or simultaneous iterative reconstruction technique (SIRT) [6,7]) depend on linearity assumptions, MBIR does not require such approximations, and can produce quantitatively accurate reconstructions in a wide range of scenarios. In many cases, the use of more complex and accurate physical models leads to nonlinear forward models, e.g., surface-connected voids modify serial sectioning BSE observations due to what is essentially an occlusion process; and the attenuation of the bright-field beam changes the HAADF scattering amplitude. Both of these phenomena represent nonlinear forward dependencies, which can be fully incorporated in MBIR methods and the corresponding cost function can be solved using a range of mathematical tools, such as multiresolution/multigrid methods , adjoint differentiation , and Fréchet differentiation . A key advantage of MBIR methods is that they can accommodate limitations in experimental systems by estimating calibration (i.e., hyper-) parameters automatically as part of the reconstruction process. In addition, diverse information regarding the known physical properties of a sample, along with its ensemble statistical properties, can also be integrated into the MBIR approach. Reconstruction regularity can be imposed through Bayesian prior modeling of local and even global material statistics.