7. Properties of the 2D Fourier Transform

The Fourier Theorem explains that signals can be expressed in terms of sinusoids. The signal is transformed from  the spatial (X) domain to the spatial frequency domain (1/X).

FT of different patterns

The Fourier Transform of some of the most basic shapes and patterns were taken.

Fourier transform (bottom row) of some basic shapes (top row) -annulus, sqaure and square annulus (from L to R)
Fourier Transform (bottom row) of slits and dots (top row)

Anamorphic Property of the Fourier Transform
Using the image of a sinusoid, we study some of the properties of the FT. In the following image, we have a sinusoid and its corresponding FT. Further addition of bias or a constant integer to the intensity does not change the rendered image or the Fourier Transform. One, in the rendering of the image, SIP normalizes the values of the image matrix. Secondly, the Fourier transform shows the image in the spatial frequency domain. Addition of bias does not change the frequency of the signal. Hence, the similar FTs for all biased images.

Image of a sinusoid (left), addition of bias yields the same image and FT (right)

Changing the frequency of the sinusoid however directly changes its FT.

Sinusoids with increasing frequency (top row), the FTs change accordingly (bottom row)

Rotation of the sinusoid translates to a rotation of the FT. Initially, the sinusoid has a frequency along the y-axis. As the image is rotated, it changes from having a frequency along the y-axis to having frequency components also along the x. This changes the corresponding FT. Without the x-axis frequency, the FT is limited to y-axis-centered dots that are off from the x-axis. With new frequencies along the x, these dots also gain values along the x-axis. Here we see the FT of a combination of sinusoids as dots forming the corners of a square.

(Top row, from L to R) rotated sinusoid, combination of sinusoids along the x and y axes and a combination of rotated sinusoids along the x and y axes, corresponding FTs are included (bottom row).

Thanks to CPU, McDo Katips and my neighbors from Sunrise Bldg for sharing their wifi connection. 🙂


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