Low-level feature extraction (including edge detection) - Feature Extraction and Image Processing - page 122

When the horizontal second-order operator is combined with a vertical second-order
difference we obtain the full Laplacian template, given in Figure 4.24 .

0 -1 0

-1 4 -1

0 -1 0

Figure 4.24 Laplacian edge detection operator

Application of the Laplacian operator to the image of the square is given in Figure 4.25 .
The original image is provided in numeric form in Figure 4.25 (a). The detected edges are
the zero-crossings in Figure 4.25 (b) and can be seen to lie between the edge of the square
and its background.

12 341121
22 301221
3 0 38 39 37 36 3 0
4 1 40 44 41 42 2 1
1 2 43 44 40 39 1 3
2 0 39 41 42 40 2 0
12 022311
02 131042

0 0 000000
0 1 -31 -47 -36 -32 0 0
0 -44 70 37 31 60 -28 0
0 -42 34 12 1 50 -39 0
0 -37 47 8 -6 3-42 0
0 -45 72 37 45 74 -34 0
0 5 -44 -38 -40 -31 -60
0 0 000000

p = L =

(a) Image data (b) After Laplacian operator

Figure 4.25 Edge detection via the Laplacian operator

An alternative structure to the template in Figure 4.24 is one where the central weighting
is 8 and the neighbours are all weighted as -1. Naturally, this includes a different form of
image information, so the effects are slightly different. (In both structures, the central
weighting can be negative and that of the four or the eight neighbours can be positive,
without loss of generality.) Actually, it is important to ensure that the sum of template
coefficients is zero, so that edges are not detected in areas of uniform brightness. One
advantage of the Laplacian operator is that it is isotropic (like the Gaussian operator): it has
the same properties in each direction. However, as yet it contains no smoothing and will
again respond to noise, more so than a first-order operator since it is differentiation of a
higher order. As such, the Laplacian operator is rarely used in its basic form. Smoothing
can use the averaging operator described earlier but a more optimal form is Gaussian
smoothing. When this is incorporated with the Laplacian we obtain a Laplacian of Gaussian
(LoG) operator which is the basis of the Marr-Hildreth approach, to be considered next. A
clear disadvantage with the Laplacian operator is that edge direction is not available. It
does, however, impose low computational cost, which is its main advantage. Though