4.5 Non-Linear Filtering of Images

Linear filters have the property that their output is a linear combination of their input. Filters that do not satisfy this property are non-linear. Nonlinear filters are not employed in image processing as frequently as linear filters, because in many cases they are based on heuristics closely tied to the particular application. The general idea in non-linear image filtering is that instead of using the spatial mask in a convolution process, the mask is used to obtain the neighboring pixel values, and then ordering mechanisms produce the output pixel. That is, as the mask is shifted about the image, the order of the pixels in the windowed section of the image is rearranged, and the output pixel is generated from these rearranged input pixels. Figure 4-13 illustrates this procedure, for three types of non-linear filters.

Figure 4-13: Examples of non-linear filters. g(i, j) = 50 for T ₂ and is referred to as a median filter, and is one of the more common non-linear filters used in image processing. T ₁ and T ₃ correspond to the minimum and maximum intensity in the 3x3 neighborhood surrounding f(i, j). Minimum filters form the basis of image erosion, and maximum filtering the basis of image dilation.

The primary application with which we shall explore the use of nonlinear filters will be in the enhancement of noisy images. While non-linear filters are sometimes used to perform other tasks, removal of noise...