Top-hat transform

In mathematical morphology and digital image processing, top-hat transform is an operation that extracts small elements and details from given images. 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, image enhancement, and others. In mathematical morphology and digital image processing, top-hat transform is an operation that extracts small elements and details from given images. 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, image enhancement, and others. Let f : E ↦ R {displaystyle f:Emapsto R} be a grayscale image, mapping points from a Euclidean space or discrete grid E (such as R2 or Z2) into the real line. Let b ( x ) {displaystyle b(x)} be a grayscale structuring element. Then, the white top-hat transform of f is given by: where ∘ {displaystyle circ } denotes the opening operation. The black top-hat transform of f (sometimes called the bottom-hat transform) is given by: where ∙ {displaystyle ullet } is the closing operation.