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A python package collecting an assortment of geometric phase analysis algorithms and tools to analyse regular (moiré) lattices.

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# pyGPA

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A python package collecting an assortment of Geometric Phase Analysis algorithms and tools to analyse regular (moiré) lattices.

Geometric Phase Analysis enables the extraction of a _displacement field_ from a distorted lattice, by comparing to reference waves. As such, it is a spatial variant of lock-in amplification. This package implements several specializations of GPA. In particular: 
- It extends the original GPA algorithm by optimizing over a range of reference vectors, (as previously described by\[[1](https://doi.org/10.1016/j.optlaseng.2005.10.012)\]) to achieve accurate characterization of larger distortions. This is especially useful when analysing (atomic) small angle moiré patterns.
- From the displacement field, it is possible to extract local properties of the underlying lattice.
- It is also possible from this displacement field, to correct for the distortion, a process known in STM as the Lawler-Fujita algorithm \[[2](https://doi.org/10.1038/nature09169)\].
- Once the displacement field is known, it is also possible to perform unit cell averaging directly from the distorted image, creating low-noise or higher magnification (depending on the PSF of the instrument) images of the unit cell.
- This unit cell average can be extended back to the full image to create a low noise version, or explore difference from mean unit cell.

## Installation

Installation is somewhat non-trivial, due to dependency on a [non-packaged implementation](https://github.com/TAdeJong/moisan2011) of the smooth + periodic decomposition. Furthermore, to use [CuPy](https://docs.cupy.dev/en/stable/install.html) acceleration, a NVIDIA graphics card and working CUDA installation are needed.

First clone the repository. In the `pyGPA` folder, run the following:

```bash
$ pip install -e .
$ pip install git+https://github.com/TAdeJong/moisan2011.git#egg=moisan2011
```

For CUDA acceleration, follow installation instructions of CuPy https://docs.cupy.dev/en/stable/install.html.

## Testing

This project uses ``pytest`` and ``hypothesis`` to run tests. The test coverage is far from complete and can be extented though! To test:

Install the test dependencies.

```bash
$ pip install -r requirements_test.txt
```

Run the tests.

```bash
$ pytest
========================== 20 passed in 42.28s ============================
```

Note: The `cuGPA` tests will be skipped automatically if no working `CuPy` installation is detected.

## See also

https://github.com/TAdeJong/moire-lattice-generator

[T.A. de Jong _et al._Imaging moiré deformation and dynamics in twisted bilayer graphene. Nat Commun **13**, 70 (2022)](https://doi.org/10.1038/s41467-021-27646-1) ([arxiv](https://arxiv.org/abs/2107.14716))

[T. Benschop, T.A. de Jong _et al._ Phys. Rev. Research *3*, 013153](https://doi.org/10.1103/PhysRevResearch.3.013153) (Code not actually used for that work, but builds upon the ideas)

https://github.com/TAdeJong/weighed_phase_unwrap

Kemao, Qian. "Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations." [Optics and Lasers in Engineering 45.2 (2007): 304-317.](https://doi.org/10.1016/j.optlaseng.2005.10.012)

# Acknowledgement

This work was financially supported by the [Netherlands Organisation for Scientific Research (NWO/OCW)](https://www.nwo.nl/en/science-enw) as part of the [Frontiers of Nanoscience (NanoFront)](https://www.universiteitleiden.nl/en/research/research-projects/science/frontiers-of-nanoscience-nanofront) program.

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