3.2 Quantisation of DCT coefficients

3.2 Quantisation of DCT coefficients

The domain transformation of the pixels does not actually yield any compression. A block of 64 pixels is transformed into 64 coefficients. Due to the orthonormality of the transformation, the energy in both the pixel and the transform domains are equal, hence no compression is achieved. However, transformation causes the significant part of the image energy to be concentrated at the lower frequency components, with the majority of the coefficients having little energy. It is the quantisation and variable length coding of the DCT coefficients that lead to bit rate reduction. Moreover, by exploiting the human eye's characteristics, which are less sensitive to picture distortions at higher frequencies, one can apply even coarser quantisation at these frequencies, to give greater compression. Coarser quantisation step sizes force more coefficients to zero and as a result more compression is gained, but of course the picture quality deteriorates accordingly.

The class of quantiser that has been used in all standard video codecs is based around the so-called uniform threshold quantiser (UTQ). It has equal step sizes with reconstruction values pegged to the centriod of the steps. This is illustrated in Figure 3.4.

Figure 3.4: Quantisation characteristics

The two key parameters that define a UTQ are the threshold value, th, and the step size, q. The centroid value is typically defined mid way between quantisation intervals. Note that, although AC transform coefficients have nonuniform characteristics, and hence can be better quantised with nonuniform quantiser step sizes (the DC coefficient has a fairly uniform distribution), bit rate control would be easier if they were quantised linearly. Hence, a key property of UTQ is that the step sizes can be easily adapted to facilitate rate control.

A further two subclasses of UTQ can be identified within the standard codecs, namely those with and without a dead zone. These are illustrated in Figure 3.5 and will be hereafter abbreviated as UTQ-DZ and UTQ, respectively. The term dead zone commonly refers to the central region of the quantiser, whereby the coefficients are quantised to zero.

Figure 3.5: Uniform quantisers (a with dead zone) (b without dead zone)

Typically, UTQ is used for quantising intraframe DC, F(0, 0), coefficients, and UTQ-DZ is used for the AC and the DC coefficients of interframe prediction error. This is intended primarily to cause more nonsignificant AC coefficients to become zero, so increasing the compression. Both quantisers are derived from the generic quantiser of Figure 3.4, where in UTQ th is set to zero but in UTQ-DZ it is set to q/2 and in the most inner region it is allowed to vary between q/2 and q just to increase the number of zero-valued outputs, as shown in Figure 3.5. Thus the dead zone length can be from q to 2q. In some implementations (e.g. H.263 or MPEG-4), the decision and/or the reconstruction levels of the UTQ-DZ quantiser might be shifted by q/4 or q/2.

In practice, rather than transmitting a quantised coefficient to the decoder, its ratio to the quantiser step size, called the quantisation index, I

(3.6) 

is transmitted. (In eqn. 3.6 the symbol ⌊.⌋ stands for rounding to the nearest integer.) The reason for defining the quantisation index is that it has a much smaller entropy than the quantised coefficient. At the decoder, the reconstructed coefficients, F^q(u, v), after inverse quantisation are given by:

(3.7) 

If required, depending on the polarity of the index, an addition or subtraction of half the quantisation step is required to deliver the centroid representation, reflecting the quantisation characteristics of Figure 3.5.

It is worth noting that, for the standard codecs, the quantiser step size q is fixed at 8 for UTQ but varies from 2–62, in even step sizes, for the UTQ-DZ. Hence the entire quantiser range, or the quantiser parameter Q_p (half the quantiser step size), can be defined with five bits (1–31).

Uniform quantisers with and without a dead zone can also be used in DPCM coding of pixels (section 3.1). Here, the threshold is set to zero, th = 0, and the quantisers are usually identified with even and odd numbers of levels, respectively.

One of the main problems of linear quantisers in DPCM is that for lower bit rates the number of quantisation levels is limited and hence the quantiser step size is large. In coding of plain areas of the picture, if a quantiser with an even number of levels is used, then the reconstructed pixels oscillate between -q/2 and q/2. This type of noise at these areas, in particular at low luminance levels, is visible and is called granular noise.

Larger quantiser step sizes with an odd number of levels (dead zone) reduce the granular noise, but cause loss of pixel resolution at the plain areas. This type of noise when the quantiser step size is relatively large is annoying and is called contouring noise.

To reduce granular and contouring noises, the quantiser step size should be reduced. This of course for a limited number of quantisation levels (low bit rate) reduces the outmost reconstruction level. In this case large pixel transitions such as sharp edges cannot be coded with good fidelity. It might take several cycles for the encoder to code one large sharp edge. Hence, edges appear smeared and this type of noise is known as slope overload noise.

In order to reduce the slope overload noise without increasing the granular or contouring noise, the quantiser step size can change adaptively. For example, a lower step size quantiser is used at the plain areas and a larger step size is employed at the edges and high texture areas. Note that the overhead of adaptation can be very costly (e.g. one bit per pixel).

The other method is to use a nonlinear quantiser with small step sizes at the inner levels and larger step sizes at the outer levels. This suits DPCM video better than the linear quantiser. Nonlinear quantisers reduce the entropy of the data more than linear quantisers. Hence data is less dependent on the variable length codes (VLC), increasing the robustness of the DPCM video to channel errors.