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., $L=2^8=256$). The numbers of gray levels of the following eight images are respectively 256, 128, 64, 32, 16, 8, 4, and 2, respectively.

• Uniform distribution

  Define $L+1$ boundaries

  $$t_k=kG/L=k\Delta,\;\;\;(k=0,1,\cdots,L)$$ (1)

  where $\Delta\stackrel{\triangle}{=}G/L$. And define the $L$ discrete gray levels to represent the L intervals:

  $$r_k=t_k+\Delta/2=k\Delta+\Delta/2=(k\Delta+k\Delta+\Delta)/2 =(t_k+t_{k+1})/2$$ (2)

  Then the quantization can be defined as a function

  $$x'=f(x)=r_k,\;\;\;\; \;\;t_k \le x \le t_{k+1}$$ (3)

  

• Mean square error optimization

  Define mean square error of the quantization process as

  $${\cal E} = E[ (x-x')^2]=\int_{-\infty}^\infty (x-x')^2 p(x)dx= \sum_{i=0}^{L-1} \int_{t_i}^{t_{i+1}} (x-r_i)^2 p(x)dx$$ (4)

  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

  $$\frac{\partial {\cal E}}{\partial t_k}=\frac{\partial {\cal E}}{\partial r_k}=0$$ (5)

  This method requires $p(x)$ to be known. The previous quantization is optimal when $p(x)$ is a uniform distribution. When $p(x)$ is not uniform, more gray levels will be assigned to the gray scale regions corresponding to higher $p(x)$.

• Contrast equalization

  The perceived contrast is a function of the intensity. Specifically, we perceive the same contrast between the object and its surrounding if

  $$\frac{\Delta f}{f} \approx \frac{df}{f}=d(ln\,f)=constant$$ (6)

  where $f$ is the intensity and $\Delta f \approx df$ is the intensity difference, the absolute contrast (Weber's law). For example,

  $$\frac{20-10}{10}=\frac{200-100}{100}$$ (7)

  i.e., a high contrast of $\Delta f=200-100=100$ at a high absolute intensity $f=100$ is perceived the same as a much lower contrast of $\Delta f=20-10=10$ at a low absolute intensity $f=10$. In other words„ we are less sensitive to contrast when the intensity $f$ is high. As another example, consider the perceived brightness of a 3-way light bulb with 50, 100 and 150 Watts (with the assumption that the brightness is proportional to the power consumption). The perceived contrast between 50 and 100 is higher than that between 100 and 150 as $(100-50)/50 > (150-100)/100$. Consequently, the perceived contrast can be defined as a logarithmic function of the intensity:

  $$y=f(x)=a\;log_e(1+x)$$ (8)

  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.

  

  Weber's law describes a general phenominon in human perception. Another example is the difference between different sound frequencies. The difference between $C_4$ (middle C, 261.63 Hz) and $C_5$ (523.25 Hz) is an octave, perceived the same as the difference between $C_5$ and $C_6$ (1046.5 Hz), although the frequency differences between the two pairs are quite different (261.63 Hz. vs. 523.25 Hz).

• Gamma correction

  In the image acquisition process, nonlinear mapping may occur in various stages. For example, in the camera system, the in-coming light intensity may be nonlinearly mapped to the film or digital recording sensors, in the cathode ray tube (CRT), the applied voltage may be nonlinearly mapped to the brightness of the CRT display, and in the biological visual system, the in-coming light intensity is nonlinearly perceived by retina and the visual cortex of the brain. The goal of gamma correction is to compensate for all such nonlinear mappings, the following power function that relates the input $x$ to the output $y$ can be considered:

  $$y=A x^\gamma$$ (9)

  where the ranges of both the input and output are normalized so that $0\le x,y,\le 1$. Here $A$ is a constant scaling factor, and $\gamma$ is a parameter that characterizes the nonlinearity. Obviously when $\gamma=1$, $y$ is linearly related to $x$. Otherwise, we have a nonlinear mapping. As an example, the nonlinear CRT mapping modeled by $y=x^\gamma=2^{2.2}$ can be corrected by another nonlinear mapping $z=y^{1/\gamma}=y^{1/2.2}=(x^{2.2})^{1/2.2}=x$, as shown below: