138 
expression of the noise power generated by the uniform quantizer, i.e. 
: 02 
mean squared error = = 
3542 
where Q is the quantization step. 
The quantization error on each Cosine Transformed data sample are 
independent and uniformaly distributed between -Q/2 and +Q/2. The mean 
image reconstruction error is zero, so that no biasing is introduced by 
the compression, 
The distribution of the errors on the image pixels reconstructed by 
Inverse Cosine Transformation is given by the Central Limit Theorem. This 
theorem states that, under rather general conditions, the linear 
combination of a large number of random variables has, at the limit a 
Gaussian distribution, 
The Inverse Cosine Transformation performed by the compressor constituting 
a linear operation performed on a block of 256 quantized data samples, the 
error on the reconstructed image can be approximated as Gaussian and the 
probability of an error of X grey tone units will be given by the 
expression: 
  
V12 
p(X) = — | exp (-6x2/Q2)dx 
Q/2* 
| X-3 
The peak image coding error can be considered, for all practical purposes, 
as, say, the error not exceeded by more than one pixel every million. The 
expression of the peak error, derived from the cumulative Gaussian 
distribution, is then: peak error -» [20] where [x] denotes the nearest 
integer value to x. 
The reconstruction error is thus controlable by setting the corresponding 
value of Q. When Q is selected, the image reconstruction distortion is 
fixed. The compressor is then operating in "constant distortion mode". 
The distortion can be made as low as desirable, and in particular, 
inferior to the sensor quantization noise level. Only the arithmetic 
noise generated by the digital machine performing the Cosine 
Transformation prevents the compression with full reversibility. 
  
By varying the image reconstruction error from block to block, the 
compressor can operate in "fixed compression mode". This implies 
compressing the image block in two steps. In the first step the 
compression ratio obtained withe the current value of Q is calculated. In 
the second step, a new value of Q is determined, using the logarithmic 
relationship between data rate and distortion error, allowing to encode 
the data with the number of bits available. 
  
The minimum number of bits R required to transmit uniformly quantized 
single sample Gaussian random data with a mean squared distortion D is 
given by the expression [4]: 
R(D) = 0.25 + 0.5 log 
where S is the standard deviation of the data.