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; 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 $f:Emapsto\; R$ be a grayscale image, mapping points from an Euclidean space or discrete grid "E" (such as "R"² or "Z"²) into the real line. Let $b(x)$ be a grayscale structuring element.

Then, the white top-hat transform of "f" is given by:

:$T\_w(f)=f-fcirc\; b$,

where $circ$ denotes the opening operation.

The black top-hat transform of "f" is given by:

:,

where 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. The bigger the former, the larger the elements extracted.

Both top-hat transforms are images that contain only non-negative values at all pixels.

References

* "Image Analysis and Mathematical Morphology" by Jean Serra, ISBN 0126372403 (1982)
* "Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances" by Jean Serra, ISBN 0-12-637241-1 (1988)
* "An Introduction to Morphological Image Processing" by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)