Smoothing Spatial Filters:
Smoothing filters are used for blurring and for noise reduction. Blurring is used in preprocessing steps, such as removal of small details from an image prior to (large) object extraction, and bridging of small gaps in lines or curves. Noise reduction can be accomplished by blurring with a linear filter and also by non-linear filtering.
(1) Smoothing Linear Filters:
The output (response) of a smoothing, linear spatial filter is simply the average of the pixels contained in the neighborhood of the filter mask. These filters sometimes are called averaging filters. The idea behind smoothing filters is straightforward.By replacing the value of every pixel
in an image by the average of the gray levels in the neighborhood defined by the filter mask, this process results in an image with reduced ―sharp‖ transitions in gray levels. Because random noise typically consists of sharp transitions in gray levels, the most obvious application of smoothing is noise reduction.However, edges (which almost always are desirable features of an image) also are characterized by sharp transitions in gray levels, so averaging filters have the undesirable side effect that they blur edges. Another application of this type of process includes the smoothing of false contours that result from using an insufficient number of gray levels.
Fig.10.1 Two 3 x 3 smoothing (averaging) filter masks.The constant multiplier in front of each mask is equal to the sum of the values of its coefficients, as is required to compute an average.
A major use of averaging filters is in the reduction of ―irrelevant‖ detail in an image. By
―irrelevant‖we mean pixel regions that are small with respect to the size of the filter mask.
Figure 10.1 shows two 3 x 3 smoothing filters. Use of the first filter yields the standard average of the pixels under the mask.This can best be seen by substituting the coefficients of the mask in
which is the average of the gray levels of the pixels in the 3 x 3 neighborhood defined by the mask.Note that, instead of being 1/9, the coefficients of the filter are all 1’s.The idea here is that it is computationally more efficient to have coefficients valued 1. At the end of the filtering process the entire image is divided by 9. An m x n mask would have a normalizing constant equal to 1/mn.
A spatial averaging filter in which all coefficients are equal is sometimes called a box filter
The second mask shown in Fig.10.1 is a little more interesting. This mask yields a so-called weighted average, terminology used to indicate that pixels are multiplied by different coefficients, thus giving more importance (weight) to some pixels at the expense of others. In the mask shown in Fig. 10.1(b) the pixel at the center of the mask is multiplied by a higher value than any other, thus giving this pixel more importance in the calculation of the average.The other pixels are inversely weighted as a function of their distance from the center of the mask. The diagonal terms are further away from the center than the orthogonal neighbors (by a factor of √2) and, thus, are weighed less than these immediate neighbors of the center pixel. The basic strategy behind weighing the center point the highest and then reducing the value of the coefficients as a function of increasing distance from the origin is simply an attempt to reduce blurring in the smoothing process. We could have picked other weights to accomplish the same general objective. However, the sum of all the coefficients in the mask of Fig. 10.1(b) is equal to 16, an attractive feature for computer implementation because it has an integer power of 2. In practice, it is difficult in general to see differences between images smoothed by using either of the masks in Fig. 10.1, or similar arrangements, because the area these masks span at any one location in an image is so small.
The general implementation for filtering an M x N image with a weighted averaging filter of size m x n (m and n odd) is given by the expression
(2) Order-Statistics Filters:
Order-statistics filters are nonlinear spatial filters whose response is based on ordering (ranking) the pixels contained in the image area encompassed by the filter, and then replacing the value of the center pixel with the value determined by the ranking result. The best-known example in this category is the median filter, which, as its name implies, replaces the value of a pixel by the median of the gray levels in the neighborhood of that pixel (the original value of the pixel is included in the computation of the median). Median filters are quite popular because, for certain types of random noise, they provide excellent noise-reduction capabilities, with considerably less blurring than linear smoothing filters of similar size. Median filters are particularly effective in the presence of impulse noise, also called salt-and-pepper noise because of its appearance as white and black dots superimposed on an image.
The median, ε, of a set of values is such that half the values in the set are less than or equal to ε, and half are greater than or equal to ε. In order to perform median filtering at a point in an image,
we first sort the values of the pixel in question and its neighbors, determine their median, and
assign this value to that pixel. For example, in a 3 x 3 neighborhood the median is the 5th largest value, in a 5 x 5 neighborhood the 13th largest value, and so on. When several values in a neighborhood are the same, all equal values are grouped. For example, suppose that a 3 x 3 neighborhood has values (10, 20, 20, 20, 15, 20, 20, 25, 100). These values are sorted as (10, 15,
20, 20, 20, 20, 20, 25, 100), which results in a median of 20. Thus, the principal function of median filters is to force points with distinct gray levels to be more like their neighbors. In fact, isolated clusters of pixels that are light or dark with respect to their neighbors, and whose area is less than n2 / 2 (one-half the filter area), are eliminated by an n x n median filter. In this case
―eliminated‖ means forced to the median intensity of the neighbors. Larger clusters are affected considerably less.