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(a) Delta lter: h i, j
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(b) Lowpass lter: h i, j
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(c) Highpass lter: h i, j
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(d) Sobel lter: h i, j
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(e) Laplacian lter: h i, j
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(f) Emboss lter: h i, j
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A 3 3 lowpass lter (b) is called mean lter. An important note for a smoothing lter is that the lter kernel must have only one peak value and must be symmetric in both horizontal and vertical directions. A highpass lter (c) can be obtained by subtracting the lowpass lter kernel from the delta lter kernel. In real applications, highpass lters are often used for image sharpening. Edges have sharp changes of local intensity level. Edge detection is the rst and the key step in image analysis and recovery. For machine vision, edge detection is used to determine the objects. Sobel lter kernel is widely used as an edge lter. Similar to the Sobel kernel, Prewitt lter kernel can be used for edge ltering. However, both operators can apply only to one direction at a time. The horizontal Sobel kernel (d) can be rotated by 90o to obtain the vertical kernel. A vertical kernel using the Prewitt operator is given in MATLAB script example15_7.m. Laplacian kernel (e) is a 2-D operator that checks the numbers of zero crossing.
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Example 15.7: MATLAB Image Processing Toolbox provides the function filter2 to implement a 2-D ltering using 2-D correlation method. The examples of using filter2 are:
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R = filter2(coeff, RGB(:,:,1)); G = filter2(coeff, RGB(:,:,2)); B = filter2(coeff, RGB(:,:,3));
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where coeff is the 2-D lter kernel, RGB is the input image matrix, and R, G, and B are the arrays of ltered output of the RGB components. MATLAB also provides the built-in function imfilter for ltering images. The imfilter can replace the filter2 function with the following syntax: newRGB = imfilter(RGB, coeff); The 2-D ltering results are shown in Figure 15.13. Figure 15.13(a) shows the image after applying delta function. The output image remains unchanged when it uses delta lter kernel. The lowpass lter averages the image pixels so the ltered image is smoothed as shown in Figure 15.13(b). The lowpass lter is often used to reduce the noise in images. Figure 15.13(c) is the result of highpass lter, which emphasizes the high-frequency components to increase its sharpness. The Laplacian lter sharpens the edges as shown in Figure 15.13(d), where the edges of output image are highlighted. Figure 15.13(e) shows the Emboss lter creating a 3-D like image. A Sobel lter kernel emphasizes the image s horizontal edge as shown in Figure 15.13(f), but the edges in vertical direction are not being emphasized. Since the sum of the lter kernel equals to zero, the output image is very dark with only the edges being highlighted. When placing the lter coef cients in the
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Figure 15.13 Results of 2-D image ltering using different 3 3 lter kernels: (a) the result using delta kernel; (b) the result using lowpass kernel; (c) the result using highpass kernel; (d) the result using Laplacian kernel; (e) the result using Emboss kernel; (f) the result using Sobel kernel; (g) the result using Prewitt kernel; and (h) the result using blur kernel
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vertical direction, the lter can be used to emphasize the vertical edges of the image. Figure 15.13(g) shows that the vertical edges are emphasized using a Prewitt lter. Figure 15.13(h) is the result of using a blur lter kernel, which blurs the image by averaging the four neighboring pixels with the center pixel. When implementing a xed-point lter for image processing, the over ow problem must be carefully handled. For example, when an image with 8-bit data is processed, the pixel value must be limited to 255 if the ltered output is greater than 255. The data must be set to zero when the lter output is negative. The implementation of a 2-D lter requires four nested loops on each pixel for every coef cient of the lter kernel. Therefore, the image ltering is a very computational intensive process in image applications. For many real-time image and video applications, it is limited to low-order lter kernels such as 3 3 or 5 5.
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