Digital Gradient

For discrete digital images, the derivative in gradient operation

$$D_x[f(x)]=\frac{d}{dx}f(x)=\lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$$

becomes difference

$$D_n[f[n]]=f[n+1]-f[n],\;\;\;\;\mbox{or}\;\;\;\;\frac{f[n+1]-f[n-1]}{2}$$

Two steps for finding discrete gradient of a digital image:

• Find the difference: in the two directions:
  
  $$g_m[m,n]=D_m[f[m,n]]=f[m+1,n]-f[m,n]$$
  
  
  
  
  $$g_n[m,n]=D_n[f[m,n]]=f[m,n+1]-f[m,n]$$
  
  

• Find the magnitude and direction of the gradient vector:
  
  $$\vert g[m,n]\vert=\sqrt{g^2_m[m,n]+g^2_n[m,n]},\;\;\;\; \mbox{or}\;\;\;\;\vert g[m,n]\vert=\vert g_m\vert+\vert g_n\vert$$
  
  
  
  
  $$\angle g[m,n]=tan^{-1} \frac{g_n[m,n]}{g_m[m,n]}$$
  
  

The differences in two directions $g_m$ and $g_n$ can be obtained by convolution with the following kernels:

• Roberts
  
  $$\left[ \begin{array}{rr} -1 & 1 0 & 0 \end{array} \right],... ... \left[ \begin{array}{rr} -1 & 0 1 & 0 \end{array} \right]$$
  
  
  or
  
  $$\left[ \begin{array}{rr} 0 & 1 -1 & 0 \end{array} \right],... ... \left[ \begin{array}{rr} 1 & 0 0 & -1 \end{array} \right]$$
  
  

• Prewitt (3x3)
  
  $$\left[ \begin{array}{rrr} -1 & 0 & 1 -1 & 0 & 1 -1 & 0 ... ...} -1 & -1 & -1 0 & 0 & 0 1 & 1 & 1 \end{array} \right]$$
  
  

• Sobel
  
  $$\left[ \begin{array}{rrr} -1 & 0 & 1 -2 & 0 & 2 -1 & 0 ... ...} -1 & -2 & -1 0 & 0 & 0 1 & 2 & 1 \end{array} \right]$$
  
  

• Prewitt (4x4)
  
  $$\left[ \begin{array}{rrrr} -3 & -1 & 1 & 3 -3 & -1 & 1 & 3... ...1 & -1 \\ 1 & 1 & 1 & 1 3 & 3 & 3 & 3 \end{array} \right]$$
  
  

Note the Prewitt and Sobel operators first find the averages of one direction and then find the difference of these averages in the another direction.