In mathematical morphology

Mathematical morphology

Mathematical morphology is a theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions...

 and digital image processing

Digital image processing

Digital image processing is the use of computer algorithms to perform image processing on digital images. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing...

, top-hat transform is an operation that extracts small elements and details from given image

Image

An image is an artifact, for example a two-dimensional picture, that has a similar appearance to some subject—usually a physical object or a person.-Characteristics:...

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

Opening (morphology)

In mathematical morphology, opening is the dilation of the erosion of a set A by a structuring element B:A\circ B = \oplus B, \, where \ominus and \oplus denote erosion and dilation, respectively....

 by some structuring element

Structuring element

In mathematical morphology, a structuring element is a shape, used to probe or interact with a given image, with the purpose of drawing conclusions on how this shape fits or misses the shapes in the image...

; The black top-hat transform is defined dually as the difference between the closing

Closing (morphology)

In mathematical morphology, the closing of a set A by a structuring element B is the erosion of the dilation of that set,A\bullet B = \ominus B, \, where \oplus and \ominus denote the dilation and erosion, respectively....

 and the input image. Top-hat transforms are used for various image processing tasks, such as feature extraction

Feature extraction

In pattern recognition and in image processing, feature extraction is a special form of dimensionality reduction.When the input data to an algorithm is too large to be processed and it is suspected to be notoriously redundant then the input data will be transformed into a reduced representation...

, background equalization, image enhancement, and others.

Mathematical definitions

Let be a grayscale

Grayscale

In photography and computing, a grayscale or greyscale digital image is an image in which the value of each pixel is a single sample, that is, it carries only intensity information...

 image, mapping points from an Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 or discrete grid

Lattice graph

The terms lattice graph, mesh graph, or grid graph refer to a number of categories of graphs whose drawing corresponds to some grid/mesh/lattice, i.e., its vertices correspond to the nodes of the mesh and its edges correspond to the ties between the nodes.-Square grid graph:A common type of a...

E (such as R² or Z²) into the real line. Let be a grayscale structuring element.

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

where 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 latter, the larger the elements extracted.

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