Figure 9.33 Digital implementation of Laplacian operator.

where is a constant. Equation(9.41) can be interpreted as

In other words, we can simply use a subtractive linear combination of the blurred image and its Laplacian to restore the unblurred image. Reference to Fig. 9.33 gives

and

The Laplacian of the image function f(x,y) will then be

or

(a) (b) (c)

Figure 9.34 Laplacian masks.

If both andare chosen to be unity,image processing, we have

where

is the average of the four neighbors of the image point under consideration. Equations (9.47) and (9.48) can further be put in a form as the convolution of a mask (called a Laplacian mask) with the image function window

The Laplacian mask can be used to detect lines, line ends, and points over edges. By convolution of an image with a Laplacian mask, we can obtain edges sharpening without regard to edge direction. The mask shown in part (b) of Fig. 9.34 is obtained by adding the mask shown in part (a) to the result obtained when part (a) is rotated by 45°. The mask shown in part (c) is obtained by subtracting the mask in part (a), after it has been rotated by 45°, from twice the mask shown in part (a). The Laplacian of pixels at the edge is high, but it is not as high as that at the noise point. This is because the edge is directional, whereas for the noise point, both andare high. With this property in mind, the

noise points can be distinguished from the edge pixels. Some other measures will be provided for the detection of edges only.

Algorithm 5. Nonlinear edge operators. Among the nonlinear edge oper­ators are the Sobel operator, Kirsch operator, and Wallis operator.

Sobel operator: This operator is a 3 X 3 window centered at the point (j,k) as shown in Fig. 9.35. The intensity gradient at the point (j,k) is defined as either

or

Figure 9.35 a 3x3 window for edge detection.

Where s_x and s_y are respectively, computed from its neighbor according to

or the Sobel high-pass weighting masks are, respectively andfor computation of the horizontal and vertical components of the gradient vector in the x and y direction at the center point 3X3 window inFig. 9.35

and

Convolving these masks with an image f(x,y) over all the points on the image gives the gradient image. Figures 9.36 to 9.39 show the original images, the images comprising all the horizontal edge elements, those comprising all the vertical edge elements, as well as the complete images which combine all the edge elements responding to various directional masks.

Kirsh edge operator: Another 3x3 nonlinear edge enhancement algorithm was suggested by Kirsch (see Fig. 9.35). The subscripts of it made such that they are labeled in ascending order. Modulo 8 arithmetic in this computation. Then the enhancement value of the pixel is given as

where S_i, and T_i are computed, respectively, from

(a)

(b)

Figure 9.36 Gradient image obtained by applying Sobel edge operator: (a) original image; (b) response of all horizontal edges; (c) response of all vertical edge elements; (d) response of all 45° edge elements; (e) response of all 135° edge elements; ff) response of all (s_x + s_y) edge elements; (g) response of (s₄₅+s₁₃₅) edge elements; (h) complete gradient image

(a)

(b)