Quantization

The continuous range of light intensity $0\le x \le G$ received by the digital image acquisition system need be quantized to $L$ gray levels (e.g., $L=256$).

• Uniform distribution Define $L+1$ boundaries
  
  $$t_k=kG/L=k\Delta,\;\;\;(k=0,1,\cdots,L)$$
  
  
  where $\Delta\stackrel{\triangle}{=}G/L$. And define the $L$ discrete gray levels to represent the L intervals:
  
  $$r_k=t_k+\Delta/2=k\Delta+\Delta/2=(k\Delta+k\Delta+\Delta)/2 =(t_k+t_{k+1})/2$$
  
  

  Then the quantization can be defined as a function
  

  $$x'=f(x)=r_k,\;\;\;\;\mbox{iff}\;\;t_k \le x \le t_{k+1}$$
  
  

  

• Mean square error optimization Define mean square error of the quantization process as
  
  $${\cal E} = E[ (x-x')^2]=\int_{-\infty}^\infty (x-x')^2 p(x)dx= \sum_{i=0}^{L-1} \int_{t_i}^{t_{i+1}} (x-r_i)^2 p(x)dx$$
  
  
  where $p(x)$ is distribution of input intensify $x$. The optimal quantization in terms of $t_k$ and $r_k$ can be found by minimizing $\cal{E}$, by solving
  
  $$\frac{\partial {\cal E}}{\partial t_k}=\frac{\partial {\cal E}}{\partial r_k}=0$$
  
  
  This method requires $p(x)$ to be known. The previous quantization is optimal when $p(x)$ is a uniform distribution.

• Contrast equalization The perceived contrast is a function of the intensity. Specifically, we perceive the same contrast between the object and its surrounding if
  
  $$\frac{df}{f}=d(ln f)=constant$$
  
  
  i.e., we are less sensitive to contrast when the intensity is high. (As an example, consider the perceived brightness of a 3-way bulb with 50, 100 and 150 Watts.) Consequently, the contrast can be defined as a logarithmic function of the intensity:
  
  $$y=f(x)=a\;log_e(1+x)$$
  
  
  As shown in the figure, to perceive the same contrast, larger intensity difference is needed for higher intensity regions than lower ones.

  

  To most efficiently use the limited number of gray levels available, we can allocate more gray levels in the low intensity region where our eye is more sensitive to contrast) than in high intensity region.