Calculating 2D FTs with a handicap

Given an image, we can visualize its Fourier transform, shifting the representation from spatial to frequency domain. For example:


https://preview.redd.it/zrr568r3lykc1.png?width=659&format=png&auto=webp&s=c91472d4a811c27a668a005651aeeda46a5359be

Furthermore, the Inverse Fourier Transform allows us to reconstruct the image by summing the sinusoidal waves :

​

https://reddit.com/link/1b0m5xn/video/e0uat6e5lykc1/player

I am also familiar with the mathematical details:

​

https://preview.redd.it/0bxr5ahelykc1.png?width=1082&format=png&auto=webp&s=04ee871aac6e2f0961dd57365cc4befb808a9772

Where inner integral stands for the rows and the outer stands for the columns.

My question is this: How is a 2D Fourier transform calculated given a limitation? If I have …

computervision domain example fourier image integral representation spatial