3.4 Jpeg entropy coding

The entropy plots of the Quantisation of the DCT coefficients show the theoretical entropies of each DCT sub-band. In practise this would be a poor way to code thedata because:

• 64 separate entropy codes would be required (each requiring many extra states to represent run-length coding of zeros).
• The statistics for each code are likely to vary significantly from image to image.
• To transmit the code table for each sub-band as header information would involve a large coding overhead (many extra bits).
• Coding the sub-bands separately does not take account of the correlations which exist between the positions of the non-zero coefsin one sub-band with those of nearby sub-bands (see subfigures (c) and (d) from a previous module).

The code is applied to each block of $8\times 8$ quantised DCT coefs from a single $8\times 8$ pel region. The blocks are the coefs before reordering as shown in subfigure (b) of a previous module and comprise one coef from each of the 64 sub-bands.

Each block of $8\times 8$ quantised coefs is formed into a 1-D vector by zig-zag scanning in the sequence: $$\begin{pmatrix}0 & 1 & 5 & 6 & 14 & 15 & 27 & 28\\ 2 & 4 & 7 & 13 & 16 & 26 & 29 & 42\\ 3 & 8 & 12 & 17 & 25 & 30 & 41 & 43\\ 9 & 11 & 18 & 24 & 31 & 40 & 44 & 53\\ 10 & 19 & 23 & 32 & 39 & 45 & 52 & 54\\ 20 & 22 & 33 & 38 & 46 & 51 & 55 & 60\\ 21 & 34 & 37 & 47 & 50 & 56 & 59 & 61\\ 35 & 36 & 48 & 49 & 57 & 58 & 62 & 63\\ \end{pmatrix}$$

The jpeg code for dc coefs

The first coefficient (0) of each block (vector) is the DC coef, which represents the mean value of the pels in the block (see the top leftbasis function in subfigure (a) from previous discussion).

The DC coefs still exhibit significant local correlations (top left of subfigure (b) ), so differential coding is used in which the value to be codedis the difference between the current DC coef and that of the previous block - the blocks are scanned from left toright, row by row. The first block in each row is coded with respect to zero.

The histogram of entropies of the DC coef differences is compared in with that of the raw DC coefs from this previous figure . We note the histogram peak around zero and see that the entropy is reduced from 6.42 bits to 6.07 bits.

The size of the differences can in theory be up to $\pm (255\times 8)=\pm (2040)$ if the input pels occupy the range $-128$ to $+(127)$ (the DCT has a gain of 8 at very low frequencies). Hence the Huffman code table would have to bequite large. JPEG adopts a much smaller code by using a form of floating-point representation, where Size is the base-2 exponent and Additional Bits are used to code the polarity and precise amplitude as follows: