Reduction

Image size can be easily reduced by subsampling, e.g., getting rid of every other pixel in each row and column:

$$f_{4 \times 4}=\left[ \begin{array}{cccc} 1 & 2 & 4 & 3 2 ... ... \left[ \begin{array}{cc} 99 & 8 2 & 4 \end{array} \right]$$

In any of the four possible subsampling cases, three fourths of the information contained in the original image is lost. A better way is to find the average of a $2 \times 2$ neighborhood as the resulting pixel:

$$\left[ \begin{array}{cccc} 1 & 2 & 4 & 3 2 & 99 & 3 & 8 \\... ...t[ \begin{array}{cc} 26 & 4.5 2.5 & 2.5 \end{array} \right]$$

Again, this operation can be implemented in a two-step process:

• Regional averaging by convolving with
  
  $$H_{2 \times 2}=\frac{1}{4}\left[ \begin{array}{cc} 1 & 1 1 & 1 \end{array} \right]$$
  
  
  to get
  
  $$f_{4 \times 4} * H_{2 \times 2}=f'_{4 \times 4}=\left[ \begin... ....5 & 2.5 & 1.5 1.25 & 0.75 & 1.25 & 1.0 \end{array} \right]$$
  
  

• Subsampling $f'_{4 \times 4}$ to get
  
  $$\left[ \begin{array}{cc} 26 & 4.5 2.5 & 2.5 \end{array} \right]$$