3.4.3 Quantisation

Figure 3.34 Two-stage wavelet decomposition of image

Figure 3.35 Five-stage wavelet decomposition of image

A more general example of a uniform quantiser is:

Y = F Q.Q P

where QP is a quantisation ‘step size’. The quantised output levels are spaced at uniform intervals of QP (as shown in the following example).

Example Y = QP.round(X/QP)

Figure 3.36 shows two examples of scalar quantisers, a linear quantiser (with a linear mapping between input and output values) and a nonlinear quantiser that has a ‘dead zone’ about zero (in which small-valued inputs are mapped to zero).

Figure 3.36 Scalar quantisers: linear; nonlinear with dead zone

In image and video compression CODECs, the quantisation operation is usually made up of two parts: a forward quantiser FQ in the encoder and an ‘inverse quantiser’ or (IQ) in the decoder (in fact quantization is not reversible and so a more accurate term is ‘scaler’ or ‘rescaler’). A critical parameter is the step size QP between successive re-scaled values. If the step size is large, the range of quantised values is small and can therefore be efficiently represented (highly compressed) during transmission, but the re-scaled values are a crude approximation to the original signal. If the step size is small, the re-scaled values match the original signal more closely but the larger range of quantised values reduces compression efficiency.

Quantisation may be used to reduce the precision of image data after applying a transform such as the DCT or wavelet transform removing remove insignificant values such as near-zero DCT or wavelet coefficients. The forward quantiser in an image or video encoder is designed to map insignificant coefficient values to zero whilst retaining a reduced number of significant, nonzero coefficients. The output of a forward quantiser is typically a ‘sparse’ array of quantised coefficients, mainly containing zeros.

3.4.3.2 Vector Quantisation

A vector quantiser maps a set of input data (such as a block of image samples) to a single value (codeword) and, at the decoder, each codeword maps to an approximation to the original set of input data (a ‘vector’). The set of vectors are stored at the encoder and decoder in a codebook. A typical application of vector quantisation to image compression [5] is as follows:

1.Partition the original image into regions (e.g. M × N pixel blocks).

2.Choose a vector from the codebook that matches the current region as closely as possible.

3.Transmit an index that identifies the chosen vector to the decoder.

4.At the decoder, reconstruct an approximate copy of the region using the selected vector.

A basic system is illustrated in Figure 3.37. Here, quantisation is applied in the spatial domain (i.e. groups of image samples are quantised as vectors) but it could equally be applied to