How are quantized DCT coeffiecients serialised in JPEG?

Question

I've read in dozens of articles, scientific papers, and toy implementations that the steps in JPEG compression are roughly as follows

1. Take 8x8 DCT
2. Divide by quantization matrix
3. Round to integers
4. Run-length & Hufmann

And then the inverse is pretty much the same. What is left out in everything on the topic I've found so far is the magnitude of the data and the corresponding serialization.

It appears implicitly assumed that all the coefficients are stored as unsigned bytes. However, as I understand it, the DC coefficient is in the range 0-255, while the AC coefficients can be negative. Are the AC coefficients in the range ±255, or ±127, or something else?

What is the common way to store these coefficients in a compact way?

Answer 1

The first-hand source to read is of course the ITU-T T.81 standard document. Looks like the first Google link leads to a paywall.. it's on the w3 site, though: https://www.w3.org/Graphics/JPEG/itu-t81.pdf

1. Take 8-bit input samples (0..255)
2. Subtract 128 (-128..127)
3. Do N*N fDCT, where N=8
4. Output can have log2(N)+8 bits = 11 bits (-1024..1023)

DC coefficients are stored as a difference, so they can have 12 bits.

Answer 2

The encoding process depends upon whether you have a sequential scan or a progressive scan. The details of the encoding process are too complicated to fit within an answer here.

I highly recommend this book:

https://www.amazon.com/Compressed-Image-File-Formats-JPEG/dp/0201604434/ref=sr_1_2?ie=UTF8&qid=1531091178&sr=8-2&keywords=JPEG&dpID=5168QFRTslL&preST=_SX258_BO1,204,203,200_QL70_&dpSrc=srch

It is the only source I know of that explains JPEG end-to-end in plain English.