One of the method of quantization is to quantize pixels in spatial domain (i.e. without transforming).  For example, HEVC supports so called “transform skipping mode“, i.e. the transform can be skipped fully  quantization is performed in spatial domain.

If  quantization is uniform  with a step size b, the quantization error for a digital image may be approximated as a uniformly distributed signal with zero mean and a variance of  b2/12

For example, if the step-size b=2 then the variance of the quantization error is 4/12=0.3

Let’s consider the uniform quantization of an input image pixel-by-pixel by the step-size b=2. According to Chebyshev’s inequality (this inequality does not require any assumption on the distribution of a random variable):

 

the probability of quantization error exceeding the magnitude 2: 

P(|QError| ≥ 2)  ≤ = 0.075

 

In other words, with  quantization with the step-size b=2,  not more than 7.5% of pixels in an image would exceed their original values by 2 or higher after the quantization. 

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