Top-hat transform

In mathematical morphology and , a top-hat transform is an operation that extracts small elements and details from given s. There exist two types of top-hat transform: the white top-hat transform is defined as the difference between the input image and its opening by some structuring element, while the black top-hat transform is defined dually as the difference between the closing and the input image. Top-hat transforms are used for various image processing tasks, such as feature extraction, background equalization, , and others. Let be a grayscale image, mapping points from a Euclidean space or discrete grid E (such as or ) into the real line. Let be a structuring element of grayscale. Then, the white top-hat transform of f is given by: where denotes the opening operation. The black top-hat transform of f (sometimes called the bottom-hat transform ) is given by: where is the closing operation. The white top-hat transform returns an image, containing those "objects" or "elements" of an input image that: Are "smaller" than the structuring element (i.e., places where the structuring element does not fit in), and are brighter than their surroundings. The black top-hat returns an image, containing the "objects" or "elements" that: Are "smaller" than the structuring element, and are darker than their surroundings. The size, or width, of the elements that are extracted by the top-hat transforms can be controlled by the choice of the structuring element . The bigger the latter, the larger the elements extracted. Both top-hat transforms are images that contain only non-negative values at all pixels. One of its most important uses in is to adjust nonuniform lighting conditions on an image and provide a better threshold value for separating objects. Assume that the user is only interested in small blobs on the image and wants to remove the larger bright objects. In this case, the white top-hat transform can remove larger bright objects and retain small blobs by selecting the size of the structuring element that is between removed objects and objects of interest.