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武汉理工大学

毕业设计（论文）

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ID Spatial Filtering

neighborhood processing consists of (1) selecting a center point, (x, y); (2) performing an operation that involves only the pixels in a predefined neighborhood about (x , y); (3) letting the

Nonlinear spatial filtering result of that operation be the 'response' of the process at that point; and (4) repeating the process for every point in the image.The process of moving the center point creates new neighborhoods, one for each pixel in the input im-age. The two principal terms used to identify this operation are neighborhood processing and spatial filtering, with the second term being more prevalent. As explained in the ollowing section, if the computations performed on the pixels of the neighborhoods are linear, the operation is called linear spatial filtering (the term spatial convolution also used) ; otherwise it is called nonlinear spatial filtering.

Linear Spatial Filtering

The concept of linear filtering has its roots in the use of the Fourier transform

for signal processing in the frequency domain, a topic discussed in detail in Chapter 4 n the present chapter, we are interested in filtering operations that are performed directly on the pixels of an image. Use of the term linear spatial filtering differentiates this type of process from frequency domain filtering.

The linear operations of interest in this chapter consist of multiplying each

pixel in the neighborhood by a corresponding coefficient and summing the re-sults to obtain the response at each point (x , y) . If the neighborhood is of size m X n , mn coefficients are required. The coefficients are arranged as a matrix, called a filter, mask,filter mask, kernel, template, or window, with the first three terms being the most prevalent. For reasons that will become obvious shortly, the terms convolution filter, convolution mask, or convolution kernel, also are used.

Figure 3.13 illustrates the mechanics of linear spatial filtering. The process consists of moving the center of the filter mask, w, from point to point in an image, f. At each point (x, y), the response of the filter at that point is the sum of products of the filter coefficients and the corresponding neighborhood pixels in the area spanned by the filter mask. For a mask of size m X n we assume typically that m = 2a 1 and n = 2b 1 where a and b are nonnega-tive integers. All this says is that our principal focus is on masks of odd sizes, with the smallest meaningful size being 3 X 3. Although it certainly is not a requirement, working with odd-size masks is more intuitive because they have an unambiguous center point.

There are two closely related concepts that must be understood clearly when performing linear spatial filtering. One is correlation; the other is convolution. Correlation is the process of passing the mask W by the image array f in the manner described in Fig. 3.13. Mechanically, convolution is the same process, except that W is rotated by 180⁰ prior to passing it by f. These two concepts are best explained by some examples.

Figure 3.14(a) shows a one-dimensional function,f, and a mask , w. The ori-gin of f is assumed to be its leftmost point. To perform the correlation of the two functions, we move w so that its rightmost point coincides with the origin of f, as Fig. 3.14(b) shows. Note that there are points between the two func-tions that do not overlap. The most common way to handle this problem is to padfwith as many Os as are necessary to guarantee that there will always be corresponding points for the full excursion of w past f. This situation is illus-trated in Fig. 3.14(c).

We are now ready to perform the correlation. The first value of correlation is the sum of products of the two functions in the position shown in Fig. 3.14(c). The sum of products is 0 in this case. Next, we move w one location to the right and repeat the process [Fig. 3.14(d)]. The sum of products again is O. After four shifts [Fig. 3.14(e)], we encounter the first nonzero value of the correlation, which is (2)(1) = 2. If we proceed in this manner until w moves completely pastf [the ending geometry is shown in Fig. 3.14(f)] we would get the result in Fig. 3.14(g). This set of values is the correlation of wand f. If we had padded w, aligned the rightmost element of fwith the leftmost element of the padded w, and performed correlation in the manner just expla

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