Conventional (nonadaptive) filters were introduced in Section 2.3. These are operators that act equally on all areas of an image. Conversely, adaptive filters change their behavior with the properties of the local neighborhood of the image to be filtered. One example of an adaptive filter is local contrast enhancement by histogram equalization. Histogram equalization is a process by which the image’s gray values are modified so that the cumulative histogram follows a straight line as closely as possible (Section 2.2). Linear histogram equalization is based on a mapping function of any image intensity I to a new intensity I’:
For each pixel with intensity I, the area under the histogram from the lowest intensity Imin to the pixel intensity I is computed and normalized by the total histogram area. This is the new intensity I’. For images with integer elements (e.g., unsigned bytes with a value range from 0 to 255), Equation (5.1) simplifies to
where N is the total number of pixels and N(i) is the histogram value at intensity i (i.e., the number of pixels of intensity i). For global histogram equalization, the total number of pixels N and the histogram N(i) refer to the entire image. For a locally adaptive variation of this process, either a neighborhood of size m x m or a circular neighborhood of radius r of each pixel is selected, and N and N(i) are restricted to this neighborhood. The locally restricted histogram equalization process adapts to the statistical properties of the neighborhood region, and its exact action depends on the neighborhood rather than on the entire image.
The effect of locally adaptive contrast enhancement is demonstrated in Figure 5.1. A scanning electron microscope (SEM) image of grown collagen fibers exhibits a histogram with few bright values—the cumulative probability reaches values close to unity at relatively low gray levels. Histogram stretching (global, nonadaptive histogram equalization) leads to a more uniform gray value distribution, but many gray levels are missing, as evidenced by the gaps in the histogram, predominantly in the center region. The enhanced image does not make use of the value range available and does not have more unique gray levels than the unprocessed image. Locally adaptive contrast enhancement operating on an area of about 1/100 of the image area expands contrast within the local area. Gray values are now distributed uniformly. Strong noise amplification can be seen in the dark gaps between the collagen fibers. When local histogram equalization is performed in regions with a very narrow gray-value distribution, noise (a significant source of contrast in flat regions) is strongly amplified.
The size of the neighborhood is variable and depends on the size of the features and the size of the noise clusters. Generally, it is possible to construct adaptive methods by subdividing an image into smaller subimages and applying a global operator on the subimage. However, such a process would lead to unacceptable boundaries between the subimages. For this reason, adaptive methods use sliding (or moving) windows centered on the pixel under observation. The shape of the neighborhood also influences the result. Although a square neighborhood is easier to implement, a circular neighborhood shows less directionality. In a square neighborhood, more diagonal pixels influence the result than horizontal or vertical pixels. The treatment of image edges also needs to be considered. Most commonly, four different methods of edge handling are implemented: (1) pixels outside the image area are considered to have zero value, (2) pixels outside the image area retain the nearest edge value, (3) the image is tiled, or (4) the image is mirrored. In the context of adaptive methods, avoiding discontinuities at the edge is of critical importance, and the second and fourth methods are acceptable.
Locally adaptive algorithms can be used for noise reduction, intensity-based segmentation, and shape classification. Most locally adaptive algorithms are based on the statistical properties of the neighborhood, but more advanced filters may employ artificial intelligence methods such as fuzzy logic and neural networks.
