Relationship between transformation stage and quantisation stage
The above discussion brings into focus the close relationship between the transformation stage and the quantisation stage in image coding.
If the transformation stage eliminates the linear nor non linear dependence among scalars to be coded, then the potential performance improvement of vector quantisation over scalar quantization in the quantisation stage diminishes, and vector quantisation becomes less attractive
This partially explains why the performance improvement by vector quantisation is more pronounced for waveform coders than for transform coders. The scalars such as image intensities tend to be more correlated when used in waveform coders compared to the scalars such as DCT coefficient in a transform coder.
The improvement in performance by vector quantizaition allows us to code a scalar with less than one bit in some cases. If we code each scalar independently and allow 2 reconstruction levels per scalar, the minimum bit rate is 1 bit per scalar.
With vector quantization it is possible to allow each scalar to have 2 or more reconstruction levels when viewed individually, with a bit rate greater than 1 bit per scalar when viewed jointly.
For the example (1) considered, scalar quantization assigns 2 reconstruction levels per scalar i.e. a total of 4 reconstruction levels for 2 scalars. Therefore bit rate is one bit per scalar. With vector quantization we have 2 reconstruction levels for 2 scalars
1/2bit/scalar. However this performance improvement comes at a price. The costs associated with vector quantisation in terms of computational and storage requirements are far greater than with scalar quantisation. Most of the cost is associated with design of codebook, its storage and the need to search codebook.
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