Fourrier Transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. This transformation is widely used in imaging because it allows to see signals at regular frequencies.
Theory
Fourier transform
The Fourier transform is an analysis process, decomposing a complex-valued function \(f(x)\) into its constituent frequencies and their amplitudes:
\(\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i \xi x} \, dx\).
Inverse transform
The inverse process is synthesis, which recreates \(f(x)\) from its transform:
\(f(x) = \int_{-\infty}^{\infty} \hat{f}(\xi) e^{2\pi i \xi x} \, d\xi\).
Siril allows to transform an image in the frequency space thanks to a Fast Fourier Transform algorithm. The result is in the form of two images. The first one, automatically loaded, contains the magnitude (or modulus) of the transform, the second one contains the phase. The location of the two images must be entered in the Direct Transform tab (see illustration below) of the dialog. It is then possible to modify the modulus image by removing frequency peaks corresponding to unwanted signals. It is important not to forget to save the changes.
The Centered option, when checked, centers the origin of the Direct Fourier Transform. If not, the origin is at the top-left corner.
Direct Transform tab.
To reconstruct the image, click on the Inverse Transform tab and enter the filepath of the modulus and phase images.
Inverse Transform tab.
Siril command line
fftd modulus phase
Siril command line
ffti modulus phase