Figure 9.40 Illustrative window.

And

It can easily be seen that Eq. (9.62) gives the difference of the averages of pixels in two consecutive columns. Similarly, Eq. (9.63) gives the difference of the averages of pixels in two consecutive rows. The value of с given by Eq. (9.64) is more complicated, but this value is not needed in the edge detection. The gradient of the plane can then be found as

or

which can further be approximated as

or

This gradient is usually called a digital gradient. Computation of the digital gradient is more complicated than that of the Roberts gradient. But it is less sensitive to noise, because in this process, averaging is done prior to differencing.

Algorithm 7: edge detection via zero crossing. Edges in image in­tensities can be located through detection of the zero crossing of the second derivative of the edges. This approach applies especially well when the gray-level transition region broadens gradually rather than when there is an abrupt change. Figure 9.42a shows the case when the intensity of an edge appears as a ramp function. Parts (b) and (c) of the same figure show its first and second derivatives. Note that the second derivative crosses zero if an edge exists. Similar observations are obtained for a signal with smooth intensity change at the edge (see Fig. 9.43).

As mentioned before, the Laplacian operator is more sensitive to noise. Any small ripple in f(x) is enough to generate an edge point and therefore a lot of

Figure 9.41 Window used for least-square edge detection operator.

Figure 9.42 Edge detection via zero crossing.

Figure 9.43 Edge detection via zero crossing.

Figure 9.44 Image processed with Laplacian operator alone: (a) oroginal image; (b) processed image

artifact noises will be introduced when the Laplacian operator is used alone (see Figure 9.44b). Due to this noise sensitivity the application of noise-reduction processing prior to edge detection is desirable when images with noisy back­ground are processed. Notice that an edge point is different from a noise point in that at an edge point the local variance is sufficient large. With this property in mind, the "false" edge points can be identified and discarded. Using a window (2M + 1) X (2M + 1), with M chosen around 2 or 3, the local variance ) can be estimated by

Figure 9.45 Result obtained after Laplacian operator associated with local v evaluation approach is applied to the image shown in Fig. 9.44a.

where

Comparing the local variance, , for the point(i,j), i, j - 1, 2, .... N -- I, which are zero-crossing points of the Laplacian with an ap-proximately chosen threshold will eliminate the "false" edge points accordingly.

Figure 9.45 shows the results when a Laplacian operator associated with the local variance evaluation approach is applied to the image shown in Fig. 9.44. Figure 9.46 is the block diagram of a Laplacian operator associated with local variance evaluation for use with the window shown in Figure 9.47. Figure 9.48 is another example that illustrates this method.

Line and spot detection: It is clear that lines can be viewed as extended edges, and spots as isolated edges. Isolated edges can be detected by comparing the pixel value with the average or median of its neighborhood pixels. A 3X3 mask such as

Figure 9.46 Block diagram of the Laplacian operator associated with local variance evaluation.

Figure 9.47 (2M + 1) X (2M + 1) window for local variance implementation.

can make W^Tx substantially greater than гею at the isolated points. For line detection, the compass gradient operators shown in Fig. 9.49 will respond to lines of various orientations with for allwhen x is closest to the ith mask. For the detection of combinations of isolated points and lines of various orientations, conceptually, we can use W_1, W₂, W₃, and W₄ as the four masks for edge detection, and W_s, W₆, W₇, and W_B as the four masks for line detection. By comparing the angle of the pixel vector x, with its projections onto the "edge" subspace and that onto the "line" subspace, we can then decide to which subspace (edge or line subspace) the pixel x belongs, based on which of the angles is smaller. Obviously, magnitude of the projection of x onto the edge subspace is

where andrepresent, respectively, the projections of x onto the vectors

W_1, W₂, W₃, and W₄. Similar arguments apply to W₅, W₆, W₇, and W_s. The magnitude of projection of x onto the line subspace is .

The angle between the pixel vector x with its projection onto the edge subspace is

and that between the vector x with its projection onto the line subspace is

Figure 9.48 Edge detection by Laplacian operator associated with local variance evaluation, : (a) original image, (b) processed with Laplacian alone; (e) processed with Laplacian operator taking the local variance into consideration.

where

Figure 9.48 Continued