I've been investigating about the discrete cosine transform. I think I understand the practical applications it has and how it is used in image/audio compression. I also know it is related with the Fourier transform. However, what I still don't know is why or how the information or energy is all compacted in "the low frequencies".
I have read that DFT and DCT map functions from the time-space domain to the "frequency domain". How can I think of the frequency domain?
When compressing images, why do we care more about the lower frequencies and what exactly are they?
Any answer will be appreciated, but if you can focus your answer to linear algebra it will be even better.
From a linear algebra point of view, one can think of the problem as approximating a matrix with another that is low rank. The natural way in which an image matrix is low rank is that there are portions of the image that do not have a high level of detail. For example, a picture of a sunny day may have a large portion of the image being a nearly-uniform blue sky.
A high level of detail corresponds to having rapidly changing image features within a small spatial region. Features in small spatial regions can be encoded in high frequencies, which have short wavelengths, whereas rough features on large spatial regions can be encoded in low frequencies, which have long wavelengths.
Therefore, assuming we do not care about or have high detail in our image, we only want to encapsulate the lower frequencies.