Digital Gradient

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For discrete digital images, the derivative in gradient operation

$egin{displaymath}D_x[f(x)]=frac{d}{dx}f(x)=lim_{Delta x ightarrow 0}frac{f(x+Delta x)-f(x)}{Delta x} end{displaymath}$

becomes the difference

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

Two steps for finding discrete gradient of a digital image:

Find the difference: in the two directions:
$egin{displaymath}g_m[m,n]=D_m[f[m,n]]=f[m+1,n]-f[m,n] end{displaymath}$

$egin{displaymath}g_n[m,n]=D_n[f[m,n]]=f[m,n+1]-f[m,n] end{displaymath}$
Find the magnitude and direction of the gradient vector:
$egin{displaymath}vertvert g[m,n]vertvert=sqrt{g^2_m[m,n]+g^2_n[m,n]},;;......n]vertvert=vertvert g_mvertvert+vertvert g_nvertvert end{displaymath}$

$egin{displaymath}angle g[m,n]= an^{-1} left(frac{g_n[m,n]}{g_m[m,n]} ight) end{displaymath}$

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

Roberts

or
Sobel (3x3)
Prewitt (3x3)
Prewitt (4x4)
$egin{displaymath}left[ egin{array}{rrrr} -3 & -1 & 1 & 3 -3 & -1 & 1 & 3......1 & -1 \1 & 1 & 1 & 1 3 & 3 & 3 & 3 end{array} ight]end{displaymath}$

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

Next: Compass Gradient Operations Up: gradient Previous: The Gradient Operator

Ruye Wang 2013-11-08