QFTPy

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Quaternion Fourier Transform and Convolution

<img src="https://latex.codecogs.com/svg.image?\mathcal{X}^{\rm&space;D}(u,v)=\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}W_M^{mu}x(m,n)W_N^{nv},\quad&space;W_N={\rm&space;exp}(-{\bf&space;\mu}\frac{2\pi}{N}),\quad\mu=\frac{1}{\sqrt{3}}(\rm&space;{i&plus;j&plus;k})" title="\mathcal{X}^{\rm D}(u,v)=\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}W_M^{mu}x(m,n)W_N^{nv},\quad W_N={\rm exp}(-{\bf \mu}\frac{2\pi}{N}),\quad\mu=\frac{1}{\sqrt{3}}(\rm {i+j+k})" />

Quick Start

• Install QFTPy using pip from PyPI

  pip install QFTPy
  

  or via wheel file in Releases

  pip install /path/to/latest/release.whl
  

• Dependencies include numpy and numpy-quaternion. The latter one enables quaternion in numpy like:

  import numpy as np
  import quaternion
  
  q = np.quaternion(w, x, y, z)
  

  QFTPy internal uses it as quaternion support.

• QFTPy provides more than

  • conv, conv2 for quaternion convolution (qftpy.conv)
  • qfft, qfft2 and shift and inverse for Quaternion Fourier Transform (qftpy.fft)
  • q2im, im2q for color image I/O (qftpy.io)
  • And more in qftpy.utils.

Documentation

• README as a quick start guide.
• API References
• Examples in Jupyter Notebook.
• And it couldn't be better if you star or cite this repository when it helps to your works.

Road Map

I'm going to enhance data validation (mainly shape and dtype) in the near future. Contributions are always welcome.

References

Convolution and Fourier Transform definitions follow paper works and software from Sangwine.

[1] Todd A. Ell, Nicolas Le Bihan, Stephen J. Sangwine. Quaternion Fourier Transforms for Signal and Image Processing[M]. WILEY. 2014.
[2] Salem Said, Nicolas Le Bihan, Stephen J. Sangwine. Fast complexified quaternion Fourier transform[J]. arXiv:math/0603578. 2008.
[3] S.J. Sangwine. Colour image edge detector based on quaternion convolution[J]. Electronics Letters Online. 1998.
[4] n-le_bihan, sangwine. Quaternion toolbox for Matlab[CP/OL]. 2021. https://sourceforge.net/projects/qtfm.

License

MIT