Image Processing Reference
In-Depth Information
C 2 = 2.7953, C 3 = 0.0542 and C 4 = -3.7953. Spacek also showed how it was possible to
derive operators which optimise filter performance for different combinations of the
performance factors. In particular, an operator with the best possible noise suppression
formulated by optimising the noise suppression performance alone, without the other two
measures, is given by:
2 sin(
r
) - cos(
(4.27)
fc r
() =
r
) + 2 + 1
r
Spacek then showed how these operators could give better performance than Canny's
formulation, as such challenging the optimality of the Gaussian operator for noise smoothing
(in step edge detection). In application, such an advantage can be assessed only by
experimentation. For example, one study (Jia, 1995) found the Spacek operator to be
advantageous in automatic face recognition by its ability to retain a greater proportion of
feature points to edge points than found by the Canny operator.
One difficulty with optimal smoothing functionals expressed in one-dimensional form
is their extension to become a two-dimensional image operator. For the Spacek operator,
one approach is to consider Equation 4.26 as a circularly symmetric functional expressed
in terms of radius r and to generate the coefficients of a template-smoothing operator in
this manner. For the Spacek operator, this is followed by Sobel edge detection and then by
non-maximum suppression and hysteresis thresholding. The application of the Spacek
operator is shown in Figure 4.30 (b) in comparison with the result achieved by the Canny
operator, in Figure 4.30 (a). Clearly, there are differences between these images, the crease
in the skin below the eye has appeared, as has some more detail. Clearly, the thresholds
could be altered on the Canny operator to reveal different edge regions. However, some of
these differences can be critical in particular applications, motivating choice of the appropriate
operator.
(a) Canny
(b) Spacek
(c) Petrou
Figure 4.30
Comparison of advanced first-order edge detection operators
4.4.2
Petrou operator
Petrou questioned the validity of the step edge model for real images (Petrou, 1991). Given
that the composite performance of an image acquisition system can be considered to be
that of a low-pass filter, any step-changes in the image will be smoothed to become a ramp.
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