The most commonly used algorithms to reconstruct HAADF STEM data such as Filtered Back Projection (FBP) and iterative reconstruction algorithms such as ART, SART and SIRT assume a linear image formation model. However, the linearity assumption is only a crude approximation of the non-linear behaviour of the real image formation model. Moreover, the limited angular range results in smearing in the reconstruction along the missing wedge and the typically small amount of projections creates a largely undersampled problem. The reconstruction can therefore be regarded as a limited data problem for which a regular SIRT reconstruction was not designed. One way to solve this problem is to reduce the number of unknowns by adding prior knowledge to the reconstruction algorithm. We present a simulation study for the use of a 3D Gaussian atomic model as a prior in the iterative reconstruction method developed in [1] using the ASTRA toolbox [2]. Our algorithm starts from an initial SIRT reconstruction that typically does not reveal the crystal structure but does agree relatively well with the projection data. In the subsequent iterations of our algorithm we combine this initial reconstruction with a gradually refined estimation – both in image space and projection space – of the 3D atom grid. For a quantitative validation we simulated 26 projection images (pixel size 16 pm) for two Au nanoparticles consisting of 1415 and 6525 atoms respectively with a frozen phonon approach using the MULTEM software [3]. The angular range was limited to 100 degrees, resulting in a large missing wedge and Poisson noise was added to obtain a signal to noise ratio of 10. The average distance between atom positions in the phantom and the reconstruction was found to be less than 6 pm in the direction of the missing wedge for both particles and 3 to 5 pm in the other dimensions for the 1415 atom particle (Fig. 1) and the 6525 atom particle (Fig. 2) respectively, resulting in subpixel accuracy for the recovered atom positions (Fig. 3). The recovery of the 6525 atom positions took approximately 20 minutes.

References

1. B. Goris, J. De Beenhouwer, A. De Backer, D. Zanaga, K. J. Batenburg, A. Sánchez-Iglesias, L. M. Liz-Marzán, S. Van Aert, S. Bals, J. Sijbers, et al., “Measuring Lattice Strain in Three Dimensions through Electron Microscopy”, Nano Letters 2015, 15 (10), pp6996-7001. 

2. W. Van Aarle, W J. Palenstijn, J. De Beenhouwer, T. Altantzis, S. Bals, K. J. Batenburg, and J. Sijbers, “The ASTRA Toolbox: a platform for advanced algorithm development in electron tomography”, Ultramicroscopy, vol. 157, pp. 35–47, 2015.

3. I. Lobato, D. Van Dyck, MULTEM: a new multislice program to perform accu-

rate and fast electron diffraction and imaging simulations using Graphics

Processing Units with CUDA, Ultramicroscopy (2015) 9–17.

Figures:

Figure 1: Evolution of the average distance in 3 dimensions between the positions of each atom in the phantom and its corresponding atom in the reconstruction for a Au nanoparticle consisting of 1415 atoms.

Figure 2: Evolution of the average distance in 3 dimensions between the positions of each atom in the phantom and its corresponding atom in the reconstruction for a Au nanoparticle consisting of 6525 atoms.

Figure 3: Simulated projection example (left); slice through the SIRT reconstruction along the missing wedge (middle) and the recovered atomic positions for the same slice (right).

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EMC Abstracts - https://emc-proceedings.com/abstract/direct-estimation-of-3d-atom-positions-of-simulated-au-nanoparticles-in-haadf-stem/