Optimization of STEM-HAADF electron tomography reconstructions by smart parameters selection in compressed-sensing based algorithms

Zusammenfassung

3D structural analysis of nanostructured materials by means of Electron Tomography is currently a well-stablished approach, which provides meaningful information out of reach for conventional 2D TEM/STEM analysis1. Most recent efforts in the field concentrate on improvements to reduce the amount of required information (tilt range, number of projections or sampled image points) while still improving the quality of thereconstructed volumes. At this respect, new methods based on Compressed-Sensing (CS) and exploiting the minimization of the Total Variation (TVM) of the whole set of images in the tilt series, have proven as quite efficient at this respect2. The implementation of CS-TVM algorithms, as e.g. in TVAL3, involves the use of apriori user-fixed parameters, which provide a balance between the level of image detail and the match between the experimental and reconstructed sinograms3. Particularly, the model used in TVAL3 can be written as follows: min ��,���� ∑(‖����‖2 + �� 2 ‖������ − ����‖2 2 − ���� ��(�� ������ − ����)) + �� 2 ‖���� − ��‖2 2 − ��(���� − ��), where x is the image we reconstruct, b is the sinogram of x, and Di calculates the gradient of x at pixel i. TVAL3 method adds new variables ���� to avoid nondifferenciability difficulties when the ��2 norm derivative is calculated to obtain an optimal value of x. Therefore, ���� variables are related to the variable x by means of the constraints ���� = ������. Finally, ���� and �� are Lagrangian penalty parameters which are updated in the lagrangian iterations usually through a gradient method and µ and β are square penalty parameters that should be fixed in advance. Note that the results of the reconstruction will depend on the values chosen for and parameters, which are routinely estimated on the basis of previous experience or, instead, following recommendations in the literature. Although CS-TVM reconstructions very often provide much better results than other alternatives (e.g. SIRT or WBP), the manual selection of parameters does not guarantee that the algorithm provides an optimum output. To overcome this limitation, we have devised and developed a smart procedure that automatically searches for the optimum combination of parameters, within reasonable execution times. This novel approach was tested on two different problems of interest in catalysis. The first, Figures 1(a-b), relates to the characterization of Au nanoparticles supported on CeO2 nanocubes. The automatically determined parameter values provides in this case reconstructed volumes, Figure 1(b), which are much more efficiently segmented into Au (yellow) and CeO2 (red) components, as well as CeO2 crystallites with better defined, not so rough, surfaces, in better agreement with 2D TEM/STEM observations. The second sample, consisting of a C-based membrane with an ordered array ofnanopores, was more challenging. In this case the reconstruction obtained using routine parameters, Figure 1(c), losses most of the pore details which, on the contrary are finely captured in the optimized reconstruction, Figure 1(d). Moreover, thesegmentation of the reconstructed volume allowed us quantifying the pore volume of the system. The value estimated from the optimum TVAL3 reconstruction (0.55 cc/g) was quite close to the experimental one, determined by physisorption techniques. Likewise, the pore size distribution computed from the 3D reconstructions matched very closely that determined at macroscopic level. The developed methodology is quite general and can be applied to a variety of samples, including those containing nanopores, which fall out of reach for “conventional” TVM approaches.