Top-hat transform

In mathematical morphology and digital image processing, a 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, a 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.

Mathematical definitions

Let f:E\mapsto \mathbb{R} be a grayscale image, mapping points from a Euclidean space or discrete grid E (such as \mathbb{R}^2 or \mathbb{Z}^2 ) into the real line. Let b(x) be a structuring element of grayscale. Then, the white top-hat transform of f is given by: T_w(f)=f-f \circ b , where \circ denotes the opening operation. The black top-hat transform of f (sometimes called the bottom-hat transform ) is given by: T_b(f)=f\bullet b-f , where \bullet is the closing operation.

Properties

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 b . 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 image segmentation is to adjust nonuniform lighting conditions on an image and provide a better threshold value for separating objects.

References

Image Analysis and Mathematical Morphology by Jean Serra, (1982) Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances by Jean Serra, (1988) An Introduction to Morphological Image Processing by Edward R. Dougherty, (1992) Hands-on Morphological Image Processing by Edward R. Dougherty and R. Lotufo, (2003) Digital Image Processing (Third Edition) by Rafael C. Gonzalez and Richard E. Woods, (2008)

Text derived from https://en.wikipedia.org/w/index.php?oldid=1319047607 under CC BY-SA 4.0; modifications have been made.