The more I read, the more it confuses me. For example, I know that when you transform an image from spatial space to frequency space. It gives a plot and you can filter some stuff out and convert back and it solves problem. How to understand how it works? How do you know what frequecies to filter out? and how to interpret fourier space? Thanks in advance!
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Just as a [sound wave can be broken down into a combination of frequencies](https://upload.wikimedia.org/wikipedia/commons/5/50/Fourier_transform_time_and_frequency_domains.gif), an image can also be broken down into frequencies. For sound waves we have only one dimension. Images have two dimensions, so the frequency representation needs be two-dimensional as well. That’s why we end up with a Fourier *image*. This is [easy to understand for periodic images](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcT7i-9IFyhN8liGRw_kYusz8RZ9kn6nCvPUC420_bOItPW2zOFdy0d2LRH01IFZoiosq3Y&usqp=CAU). But the same is true for *any* image. It just takes more frequencies.
Now I think Fourier space (or k-space as it is also called) can be difficult to understand because it’s visualized as an image. It’s a spatial representation of frequency space, which confuses things.
What we’re interested in in a Fourier space image is the pixel *values*. This is represented as pixel brightness. A white pixel means a high value, and a dark pixel means a dark value.
The center of the image is the 0 frequency. As we go further away along the x-axis, [we increase the frequency](https://miro.medium.com/v2/resize:fit:1400/0*7FHYkFctaCvE5x0_.png). Bright pixels here mean that the original image has a high intensity of high frequency imagery.
It’s easier to understand if we look at the [Fourier image of a spatial image with very defined frequencies](https://slideplayer.com/slide/9349847/28/images/22/2-D+discrete+Fourier+transform+Simple+examples.jpg).
We can use this knowledge to our advantage. [If a picture has periodic noise](https://cdnintech.com/media/chapter/66464/1512345123/media/F2.png), we can convert the image to Fourier space, look for the corresponding pixels, blacken them, and convert the image back.
We can also use this to compress images. Most useful image data is in the low-frequency range. [We can cut away a sizeable portion of the high-frequency Fourier space and the image will still look fine](https://demofox2.files.wordpress.com/2020/11/lokialancompress.png). This is the basis of jpeg compression.
The actual conversion between image space and Fourier space is done mathematically.
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