Digital Gradient

For discrete digital images, the derivative in gradient operation

$$D_x[f(x)]=\frac{d}{dx}f(x)=\lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$$ (20)

becomes the difference

$$D_n[f[n]]=f[n+1]-f[n],\;\;\;\; \;\;\;\; \frac{f[n+1]-f[n-1]}{2}$$ (21)

Two steps for finding discrete gradient of a digital image:

• Find the difference: in the two directions:

  $$g_m[m,n]=D_m[f[m,n]]=f[m+1,n]-f[m,n]$$ (22)

  $$g_n[m,n]=D_n[f[m,n]]=f[m,n+1]-f[m,n]$$ (23)

• Find the magnitude and direction of the gradient vector:

  $$\vert\vert g[m,n]\vert\vert=\sqrt{g^2_m[m,n]+g^2_n[m,n]},\;\;\;\; \;\;\;\;\vert\vert g[m,n]\vert\vert=\vert\vert g_m\vert\vert+\vert\vert g_n\vert\vert$$ (24)

  $$\angle g[m,n]=\tan^{-1} \left(\frac{g_n[m,n]}{g_m[m,n]}\right)$$ (25)

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

• Roberts

  $$\left[ \begin{array}{rr} -1 & 1 \\ 0 & 0 \end{array} \right], \;\;\;\; \left[ \begin{array}{rr} -1 & 0 \\ 1 & 0 \end{array} \right]$$ (26)

  or

  $$\left[ \begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array} \right], \;\;\;\; \left[ \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array} \right]$$ (27)

• Sobel (3x3)

  $$\left[ \begin{array}{rrr} -1 & 0 & 1 \\ -2 & 0 & 2 \\ -1 & 0 & 1 ... ...[ \begin{array}{rrr} -1 & -2 & -1 \\ 0 & 0 & 0 \\ 1 & 2 & 1 \end{array} \right]$$ (28)

• Prewitt (3x3)

  $$\left[ \begin{array}{rrr} -1 & 0 & 1 \\ -1 & 0 & 1 \\ -1 & 0 & 1 ... ...[ \begin{array}{rrr} -1 & -1 & -1 \\ 0 & 0 & 0 \\ 1 & 1 & 1 \end{array} \right]$$ (29)

• Prewitt (4x4)

  $$\left[ \begin{array}{rrrr} -3 & -1 & 1 & 3 \\ -3 & -1 & 1 & 3 \\ ... ... -3 \\ -1 & -1 & -1 & -1 \\ 1 & 1 & 1 & 1 \\ 3 & 3 & 3 & 3 \end{array} \right]$$ (30)

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