I need to understand JPG decompression so that I don't need other libraries that just do it.
After being able to identify the different parts of a JPG file in terms of file format, what do I need to do, understand or learn first in mathematical or algorithmic terms so I can start implementing decoding primitives?
Look at this answer to find all the specifications you need to read, and then read them. Several times. Front to back. Then start to implement, testing along the way often with many example jpeg files.
It wouldn't hurt to know a little bit about fourier transforms and then the discrete cosine transform, and also how Huffman codes work. Though you could pick up much of what you need from the specifications.
