Image Processing Reference
In-Depth Information
d 2 f /d x 2 in Figure 4.22 (c), is greatest where the rate of change of the signal is greatest and
zero when the rate of change is constant. The rate of change is constant at the peak of the
first-order derivative. This is where there is a zero-crossing in the second-order derivative,
where it changes sign. Accordingly, an alternative to first-order differentiation is to apply
second-order differentiation and then find zero-crossings in the second-order information.
2
1
f(x)
0
2
4
6
-1
-2
x
(a) Cross-section through image data
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2
2
d
d
d
d
1
()
fx
()
fx
0
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4
6
x
x
2
-1
0
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x
(c) Second-order edge detection
(b) First-order edge detection
Figure 4.22
First- and second-order edge detection
4.3.2
Basic operators: the Laplacian
The Laplacian operator is a template which implements second-order differencing. The
second-order differential can be approximated by the difference between two adjacent
first-order differences:
f ″ ( x ) f ′ ( x ) - f ′ ( x + 1)
(4.22)
Which, by Equation 4.6, gives
f ″ ( x ) - f ( x ) + 2 f ( x + 1) - f ( x + 2)
(4.23)
This gives a horizontal second-order template as given in Figure 4.23 .
-1
2
-1
Figure 4.23
Horizontal second-order template
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