Improved 3D Resolution of Electron Tomograms 1129
result of shifting the global center-of-mass to the origin for each projection, as done in some instances (Scott et al., 2012), is shown in (f). As is apparent, the result of this approach for this data set is practically useless. From the reconstructions and sinogram visuals, it is
apparent that our current approach yields suitably good align- ment and reconstruction. Cross-correlation performs second best, although not particularly well. Our original center- of-mass approach, that uses only the maximal sequence length, shown in (g) and (l), does not produce useful result in this case.
SUMMARY
In summary, we have proposed new mathematical data- processing techniques for the alignment of the set of images acquired in a tilt series, and before the 3D reconstruction step. The image alignment is a crucial component of improving the resolution of tomograms. This is performed through an improved center-of-mass alignment that over- comes changes in total mass in the imaging area throughout a tilt series. We have found this to be the most robust and generally applicable method for image alignment of the techniques available today. We demonstrate the technique through application to a zeolite cluster with Pt nanoparticles, and show that the combination of this image alignment technique together with compressed sensing approaches for 3D reconstruction yield accurate and high-resolution 3D tomograms. The improvements with our new approach are also justified in a challenging simulation example. Our algorithm, which is openly available online (Sanders), is applicable to tilt series for any material, thereby providing a route for higher-resolution 3D imaging across the physical, biological, and medical sciences. However, further results and studies on additional data sets may need to be carried out to understand the full extent of our approach. Finally, we have also demonstrated the effectiveness of our TV algo- rithms for the zeolite data set, and these algorithms and corresponding documentation are also available in Sanders (2017), along with more general higher-order TV methods (Sanders et al., 2017).
ACKNOWLEDGMENTS
This work is supported in part by the grants NSF-DMS 1502640 and AFOSR FA9550-15-1-0152. A portion of the research was conducted under the Laboratory Directed Research and Development Program at the Pacific North- west National Laboratory. The authors wish to thank Johannes A. Lercher and Guoju Yang for providing the zeolite samples.
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