**Matlab
Source Codes for Projection Alignment **

**in
Atomic Electron Tomography (AET)**

*Posted on Sept. 27,
2017*

**I.
Overview**

To achieve atomic resolution reconstruction using AET
[1], the projections in a tilt series have to be aligned to a common axis (not necessarily
the true tilt axis) with atomic level precision along both the x and y axes. We
have implemented two different alignment procedures: center of mass (CM) and
common line methods [2]. These methods have been successful for achieving high-precision
alignment of electron tomography tilt series. Combined with powerful
reconstruction algorithms such as EST
and GENFIRE
[3-5], they produce atomic resolution reconstructions to reveal crystal defects
and disorder in materials with unprecedented 3D detail [1,2,4-7].

**II.
Center of Mass Alignment**

To align the projections along the direction
perpendicular to the tilt axis, we developed a method based on the CM [2]. When
a 3D object is tilted around the *y*
axis from 0° to 360°, the CM of the object forms a circle. However, in the
special geometry where the center of mass coincides with the origin of the x
axis, this circle becomes a point. To determine the center of mass in this
special geometry, one can project each 2D projection onto the x axis, chose a
pixel as the origin and calculated the center of mass (CM) along the *x* axis, *x _{CM}*

**III.
Common Line alignment**

Common line alignment method has been developed in the
electron tomography community since 1970, especially for accurate spatial
alignment along the tilt axis [8,9]. According to the Fourier slice theorem,
each measured 2D projection image corresponds to a slice through the origin of
3D Fourier space. Therefore, for a single tilt tomography dataset, all projections
(*i.e.*, Fourier slices) will intersect
on a single line along the tilt axis in Fourier space. In real space, this
means that if we further project the measured 2D projection images onto the
tilt axis to obtain 1D curves, they all should exactly overlap each other. By
using cross-correlation, we can align all 1D projected curves together to
achieve accurate alignment along the tilt axis.

To facilitate those who are interested in our method,
we make the alignment method codes in Matlab freely available below.

Click
here to download a .zip archive including MATLAB codes for the
CM and common line alignment methods with an example dataset.

To achieve the best result, we suggest that you
combine the CM and common line alignment methods with angular refinement using GENFIRE. For experimental
tilt series, the background of each projection must be carefully subtracted
before the alignment.

If you use the above
source codes in your publications and/or presentations, we request you cite our
paper: M.
C. Scott, C. C. Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius, U. Dahmen, B.
C. Regan and J. Miao, "Electron tomography at 2.4-ångström resolution",
*Nature* **483**, 444–447 (2012).

This
document was prepared by Yongsoo Yang, AJ Pryor, and John Miao in the
Department of Physics & Astronomy and California NanoSystems Institute,
University of California, Los Angeles, California, 90095, USA. Email: miao@physics.ucla.edu.

1.
J. Miao, P. Ercius and S. J. L. Billinge,
"Atomic electron tomography: 3D structures without crystals", *Science* **353**, aaf2157 (2016).

2.
M. C. Scott, C. C. Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius,
U. Dahmen, B. C. Regan and J. Miao, "Electron
tomography at 2.4-ångström resolution", *Nature* **483**, 444–447
(2012).

3.
J. Miao, F. Förster and O. Levi, "Equally Sloped
Tomography with Oversampling Reconstruction", *Phys. Rev. B*. **72**, 052103
(2005).

4.
Y. Yang, C.-C. Chen, M. C. Scott, C. Ophus, R. Xu, A.
Pryor Jr., L. Wu, F. Sun, W. Theis, J. Zhou, M. Eisenbach, P. R. C. Kent, R. F. Sabirianov,
H. Zeng, P. Ercius and J. Miao, "Deciphering chemical
order/disorder and material properties at the single-atom level", *Nature* **542**, 75-79 (2017).

5.
A. Pryor Jr., Y. Yang, A. Rana, M. Gallagher-Jones, J. Zhou, Y. H. Lo, G.
Melinte, W. Chiu, J. A. Rodriguez and J. Miao, "GENFIRE: A generalized
Fourier iterative reconstruction algorithm for high-resolution 3D
imaging", *Sci. Rep*. **7**, 10409 (2017).

6.
C. C. Chen, C. Zhu, E. R. White, C.-Y. Chiu, M. C. Scott, B. C. Regan, L. D.
Marks, Y. Huang and J. Miao, “Three-dimensional imaging of dislocations in a
nanoparticle at atomic resolution”, *Nature*
**496**, 74–77 (2013).

7.
R. Xu, C.-C. Chen, L. Wu, M. C. Scott, W. Theis, C. Ophus, M. Bartels, Y. Yang, H. Ramezani-Dakhel,
M. R. Sawaya, H. Heinz, L. D. Marks, P. Ercius and J. Miao, "Three-Dimensional Coordinates of
Individual Atoms in Materials Revealed by Electron Tomography", *Nature Mater.* **14**, 1099-1103 (2015).

8.
R. A. Crowther, L. A. Amos, J. T. Finch, D. J. De Rosier, and A. Klug, “Three
dimensional reconstructions of spherical viruses by Fourier synthesis from electron
micrographs.”, *Nature* **226**, 421–425 (1970).

9.
Y. Liu, P. A. Penczek, B. McEwen, and J. Frank, “A
marker-free alignment method for electron tomography”, *Ultramicroscopy** ***58**,** **393–402 (1995).