R?w1z1?w2z2?...?w9z9??wizii?19（）
W1 W2 W5 W3 W6 W4 W7 W8 W9 FIGURE A general 3 x 3 mask.
where z; is the gray level of the pixel associated with mask coefficient Wi. As usual, the response of the mask is defined with respect to its center location. The details for implementing mask operations are discussed in Section . Point Detection
The detection of isolated points in an image. is straightforward in principle. Using the mask shown in Fig. (a), we say that a point has been detected at the location on which the mask is centered if
|R| ≥ T T is a nonnegative threshold and R is given by Eq. . Basically,this formulation measures the weighted differences between the center point and its neighbors. The idea is that an isolated point (a point whose gray level is significantly different from its background and which is located in a homogeneous or nearly homogeneous area) will be quite different from its surroundings, and thus be easily detectable by this type of mask. Note that the mask in Fig. (a) is the same as the mask shown in Fig. (d) in connection with Laplacian operations. However, the emphasis here is strictly on the detection of points. That is, the only differences that are considered of interest are those large enough (as determined by T, to be considered isolated points. Note that the mask coefficients sum to zero, indicating that the mask response will be zero in areas of constant gray level.
-1 -1 -1 -1 8 -1 （a）
-1 -1 -1
（b） （c） （d）
FIGURE (a) Pointdetection mask. (b) X-ray image of a turbine blade with a porosity. (c) Result of point detection. (d) Result of using Eq. .(Original image courtesy of X-TEK Systems Ltd.)
EXAMPLE :Detection of isolated points in an image.
We illustrate segmentation of isolated points from an image with the aid of Fig. (6), which shows an X-ray image of a jet-engine turbine blade with a porosity in the upper, right quadrant of the image. There is a single black pixel embedded within the porosity. Figure (c) is the result of applying the point detector mask to the X-ray image, and Fig. (d) shows the result of using Eq. with T equal to 90% of the highest absolute pixel value of the image in Fig. (c). (Threshold selection is discussed in detail in Section The single pixel is clearly visible in this image (the pixel was enlarged manually so that it would be visible after printing). This -type of detection process is rather specialized because it is based on single-pixel discontinuities that have a homogeneous background in the area of the detector mask. When this condition is not satisfied, other methods discussed in this chapter are more suitable for detecting gray-level discontinuities.
Line Detection
The next level of complexity is line detection. Consider the masks shown in Fig. . If the first mask were moved around an image, it would respond more strongly to lines (one pixel thick) oriented horizontally. With a constant background, the maximum response would result when the line passed through the middle row of the mask. This is easily verified by sketching a simple array of 1's with a line of a different gray level (say, 5's) running horizontally through the array. A similar experiment would reveal that the second mask in Fig. responds best to lines oriented at +450; the third mask to vertical lines; and the fourth mask to lines in the -450 direction . These directions can be established also by noting that the preferred direction of each mask is weighted with a larger coefficient ., 2) than other possible directions. Note that the coefficients in each mask sum to zero, indicating a zero response from the masks in areas of constant gray level.
-1 2 -1 -1 2 -1 -1 2 -1 -1 -1 2 -1 2 -1 2 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 2 -1 -1 -1 2 -1 -1 -1 2 Horizontal +45° Vertical -45°
FIGURE Line masks.
Let R1, R2, R3, and R4 denote the responses of the masks in Fig. , from left to right, where the R's are given by Eq. . Suppose that the four masks are run individually through an image. If, at a certain point in the image, |Ri| > |Rj|, for all j ≠ i, that point is said to be more likely associated with a line in the direction of mask i. For example, if at a point in the image, |Ri|>|Rj|, for j = 2, 3. 4, that particular point is said to be more likely associated with a horizontal line. Alternatively, we may be interested in detecting lines in a specified direction. In this case, we would use the mask associated with that direction and threshold its output, as in Eq . In other words, if we are interested in detecting all the lines in an image in the direction defined by a given mask, we simply run the mask through the image and threshold the absolute value of the result. The points that are left are the strongest responses, which, for lines one pixel thick, correspond closest to the direction defined by the
mask. The following example illustrates this procedure. EXAMPLE :Detection of lines in a specified direction
Figure (a) shows a digitized (binary) portion of a wire-bond mask for an electronic circuit. Suppose that we are interested in finding all the lines that are one pixel thick and are oriented at-45\Fig. absolute value of the result is shown in Fig. (b). Note that all vertical and horizontal components of the image were eliminated, and that the components of the original image that tend toward a -45° direction
（a）
（b） （c） FIGURE Illustration of line detection (a) Binary wirebond mask.
(b) Absolute value of result after processing with -45° line detector. (c) Result of thresholding image. (b) produced the strongest responses in Fig. (b).
In order to determine which lines best fit the mask, we simply threshold this image. The result of using a threshold equal to the maximum value in the image is shown in Fig. (c).The maximum value is a good choice for a threshold in applications
such as this because the input image is binary and we are looking for the strongest responses. Figure (c) shows in white all points that passed the threshold test. In this case, the procedure extracted the only line segment that was one pixel thick and oriented at -450 (the other component of the image oriented in this direction in the top, left quadrant is not one pixel thick). The isolated points shown in Fig. (c) are points that also had similarly strong responses to the mask. In the original image, these points and their immediate neighbors are oriented in such as way that the mask produced a maximum response at those isolated locations. These isolated points can be detected using the mask in Fig. (a) and then deleted, or they could be deleted using morphological erosion, as discussed in the last chapter. Edge Detection
Although point and line detection certainly are important in any discussion on segmentation, edge detection is by far the most common approach for detecting meaningful discontinuities in gray level. In this section we discuss approaches for implementing first- and second-order digital derivatives for the detection of edges in an image. We introduced these derivatives in Section in the context of image enhancement. The focus in this section is on their properties for edge detection. Some of the concepts previously introduced are restated briefly here for the sake continuity in the discussion. Basic formulation
Edges were introduced informally in Section In this section we look at the concept of a digital edge a little closer. Intuitively, an edge is a set of connected pixels that lie on the boundary between two regions. However, we already went through some length in Section to explain the difference between an edge and a boundary. Fundamentally, as we shall see shortly, an edge is a \boundary, owing to the way it is defined, is a more global idea. A reasonable definition of \requires the ability to measure gray-level transitions in a meaningful way.
We start by modeling an edge intuitively. This will lead us to a formalism to which \transitions in gray levels can be measured. Intuitively, an ideal
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