2D Fourier transform and MRI image reconstruction
To decompose a 2D image, we need to perform a 2D Fourier transform. The first step consists in performing a 1D Fourier transform in one direction (for example in the row direction Ox). In the following example, we can see:
- the original image that will be decomposed row by row
- the gray level intensities of the choosen line
- the spectrum obtained after 1D Fourier transform
Note that low spatial frequencies are prevailing. Low spatial frequencies have the greatest change in intensity. On the contrary, high spatial frequencies have lower amplitudes. General shape of the image is described by low spatial frequencies: this is also true with MRI images.
The second step of 2D Fourier transform is a second 1D Fourier transform in the orthogonal direction (column by column, Oy), performed on the result of the first one.
The final result is called Fourier plane that can be represented by an image.
In this example, here is how to read the Fourier plane:
- Horizontal and vertical axis correspond to horizontal and vertical spatial frequencies
- Pixel intensity corresponds to the amplitude (or magnitude) of frequency component
- Color corresponds to the phase of frequency component.
The image of the Fourier plane is often a magnitude image (gray levels), but you must not forget that the amplitude is always associated with a phase information (in color in our example).