Non Uniform Quantization

Although uniform quantization is straight forward and appears to be a natural approach it may not be optimal.

Suppose is much more likely to be in one region than in others. It is reasonable to assign more reconstruction levels to that region.

If falls rarely between and , the reconstruction level is rarely used.

Rearranging reconstruction levels so that they all lie between and makes more sense. Quantizers in which reconstruction and transition levels do not have even spacing is called non-uniform quantization.

The notion that uniform quantizer is the optimal MMSE when is uniform suggests another approach. Specifically we can map f to g in such a way that is uniform.

Quantize g with a uniform quantization and then perform the inverse nonlinearity.

The non linearity is called companding. One choice of nonlinearity or companding is given by

The resulting is uniform in the interval . The nonuniform quantization by companding minimizes the distortion.

Image Quantization:

In case of image coding, one has to quantize many scalars. One approach is to quantize each scalar independently. This approach is called scalar quantization of a vector source . Suppose we have N scalars called , . Each scalar is quantized to reconstruction levels. If can be expressed as a power of 2 and each reconstruction level is coded with an equal number of bits, will be related to required number of bits by

Total number of bits B required in coding N Scalars is

The total number of reconstruction levels L is

If we have a fixed number of bits B to code all N scalars using scalar quantization of a vector source, B has to be divided among N scalars. The optimal strategy for this bit allocation depends on error criterion used and the probability density functions of the scalar. The optimal strategy typically involves assigning more bits to scalars with larger variance and fewer bits to scalars with small variance. As an example suppose we minimize the mean square error wrt for where is the result of quantization of . If the probability density functions are the same for all scalars except for their variance, and we use the same quantization method for each of the scalars, an approximate solution to the bit allocation problem is:

where

i.e. number of reconstruction levels for is proportional to the standard deviation of .

Although the above equation is an approximate solution obtained under certain specific conditions, it is useful as a reference in other bit allocation problems.

As in the equation can be - ve and is not in general an integer, a constraint has to be imposed in solving the bit allocation problem such that is a non - ve integer.

(continued inthe next slide)