Low-level feature extraction (including edge detection) - Feature Extraction and Image Processing - page 108

for increasing template, or window, size. The coefficients of smoothing within the Sobel
operator, Figure 4.10 , are those for a window size of 3. In Mathcad, by specifying the size
of the smoothing window as winsize , then the template coefficients smooth x_win can
be calculated at each window point x_win according to Code 4.3 .

1 0 -1 1 2 1

2 0 -2 0 0 0

-1 -2 -1

1 0 -1

(a) Mx (b) My

Figure 4.10 Templates for Sobel operator

(winsize-1)!
(winsize-1-x_win)! x_win!

smooth :=

x_win

Code 4.3 Smoothing function

The differencing coefficients are given by Pascal's triangle for subtraction:

Window size
2 1 -1

3 1 0 -1

4 1 1 -1 -1

5 1 2 0 -2 -1

This can be implemented by subtracting the templates derived from two adjacent expansions
for a smaller window size. Accordingly, we require an operator which can provide the
coefficients of Pascal's triangle for arguments which are a window size n and a position k .
The operator is the Pascal(k,n) operator in Code 4.4 .

n!
(n-k)!k!

if (k 0)(k n)

Pascal (k,n:=

0 otherwise

Code 4.4 Pascal's triangle

The differencing template, diff x_win , is then given by the difference between two
Pascal expansions, as given in Code 4.5 .
These give the coefficients of optimal differencing and optimal smoothing. This general
form of the Sobel operator combines optimal smoothing along one axis, with optimal
differencing along the other. This general form of the Sobel operator is then given in Code