Image-processing - Dequantizer

Question

I begin to study how codec works, and I found a "summary-diagram" for DCT-codec (DCT stands for Discrete Cosine Transform). On this diagram, there are of course two steps: the encoding step and the decoding one. I try to make a reproduction of this diagram herebelow:

ENCODING STEP Image (8*8 blocks) -->| FDCT |-->| Quantizer |-->| Entropy Encoder |--> Compressed data

DECODING STEP Compressed data -->| Entropy Decoder |-->| Dequantizer |-->| IDCT |--> Image (8*8 blocks)

The diagram seems clear to me, except the "Dequantizer". I don't understand how the quantification step can be reversed, since it is this step that causes information loss. What does this "block" do? Can someone give me more precisions?

Thanks'.

Answer 1

Formally, the quantization is a mapping from a partitioned interval $[-A, A] = \cup_{k=1}^N I_k$ to the set of integers $\{1, \ldots, k\}$. The dequantization is the mapping from the quantization indices to a set of values $x_k$ chosen from each $I_k$.

When the quantizer used is uniform without any scaling (that is to say, when it is a simple rounding), the dequantizer is the identity. But when the quantizer is non-uniform, the integer passed into the entropy coder is not just a "rounded" value of the coefficient - an exponent, scaling factor, or any other monotonous transformation might have been applied to it. The dequantizer will have to reverse all the reversible steps in the quantification operation.

For example, in the AAC codec, the quantized value sent to the entropy coder is:

$$Q(x_i) = \text{sgn}(x) \left[\left(\frac{|x_i|}{s_i}\right)^\frac{3}{4} + 0.4054\right]$$

So the dequantizer will have to apply a $\frac{4}{3}$ exponent and multiply by the scale factor $s_i$ to recover the value.