Top-Hat and Bottom-Hat Filtering
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The top-hat and bottom-hat filtering techniques are useful for enhancing details in an image if shading is present. On the other hand, it is important that these filtering transformations can be used to correct the effects of non-uniform illumination since proper illumination plays a major role in the process of extracting objects from the background.
The top-hat filtering, also called top-hat transformation operator or white-top-hat filtering, in the mathematical morphology estimates the trend via morphological opening and then removes
this trend from an image . The morphological opening, and the thus top-hat filter,
works in the following steps:
The top-hat filter has the property of enhancing "peaks" by applying the opening operator. It returns the difference between the result of morphological opening operation and the original data f. Figure 1004a shows a schematic of top-hat filtering. Such top hat filter is a ranking filter that compares the brightness of pixels in two different neighborhoods around each location. The central region in blue is large enough to correspond to the spike, while the outer region in red is an annulus surrounding the central one, like the brim of a top hat (where the name is from). If the maximum value in the central blue region exceeds that in the brim by more than a threshold, namely the height of the hat, then a spike has been found. The inner and outer radii of the brim and the height of the central blue region are all adjustable parameters . The filter can be used to locate either bright or dark spots, as desired.
Figure 1004b shows a simple example of top-hat filtering. The resulted image in Figure 1004b (c) is obtained by subtracting the image in Figure 1004b (b) from the original image in Figure 1004b (a), which is the opening operation process.
Figure 1004c shows an example of top-hat filtering applied to the pixel values in an more complicated one dimension (1-D) profile. The solid grey line represents the raw pixel values obtained from an experiment, while the grey-filled shape shows the top-hat filtered values. The filtered values are obtained by subtracting the dotted black line from the original grey line.
It was suggested that choosing the window size appropriately for
top-hat filtering helps to reduce the across-array variance.  If the side width of the window is about (d is the diameter of the spot), then the window will only
just cover the spot. In this case, very few background pixels will be sampled
to choose the minimum pixel value, and thus the filter is less stable. Figure 1004d shows the visualization of changes in across-array standard deviation of top-hat
filtering as the filter window increases. At position C, the filtering becomes equivalent to constant background subtraction, and thus the standard deviation of the array remains unchanged. The optimum window size occurs at B, where
the filter reduces the standard deviation of spots across the array. Therefore, the choice of the window size is really important:
Figure 1004e shows the effect of the filter window (brim size, r) of top-hat filtering applied to an electron diffraction pattern. It can be seen that the background and noise are better removed with large brim sizes. However, the filtered diffraction pattern will not be correct if the brim size is too large.
The top-hat filter technique has the following properties:
The top-hat filter allows for the robust removal of slowly changing background on an array and has become an important tool in many image analysis applications, for instance:
The bottom-hat filtering, or called black-top-hat filtering, is given by,
The bottom-hat filter has the property of enhancing "valleys" by applying the closing operator. The resulted image in Figure 1004f (c) is obtained by subtracting the image in Figure 1004f (b) from the original image in Figure 1004f (a), which is the opening operation process.
The bottom-hat filter has many applications as well, for instance:
 C. Glasbey, Image analysis of a genotyping microarray experiment. Technical Report
01-07, Biomathematics and Statistics, Scotland, 2001.