Structuring element

Last updated February 25, 2019

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. It is typically used in morphological operations, such as dilation, erosion, opening, and closing, as well as the hit-or-miss transform.

Mathematical morphology theory and technique for the analysis and processing of geometrical structures

Mathematical morphology (MM) is a theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions. MM is most commonly applied to digital images, but it can be employed as well on graphs, surface meshes, solids, and many other spatial structures.

Dilation is one of the basic operations in mathematical morphology. Originally developed for binary images, it has been expanded first to grayscale images, and then to complete lattices. The dilation operation usually uses a structuring element for probing and expanding the shapes contained in the input image.

Erosion (morphology) basic operation in mathematical morphology

Erosion is one of two fundamental operations in morphological image processing from which all other morphological operations are based. It was originally defined for binary images, later being extended to grayscale images, and subsequently to complete lattices.

According to Georges Matheron, knowledge about an object (e.g., an image) depends on the manner in which we probe (observe) it.^[1] In particular, the choice of a certain structuring element for a particular morphological operation influences the information one can obtain. There are two main characteristics that are directly related to structuring elements:

Georges François Paul Marie Matheron was a French mathematician and geologist, known as the founder of geostatistics and a co-founder of mathematical morphology. In 1968, he created the Centre de Géostatistique et de Morphologie Mathématique at the Paris School of Mines in Fontainebleau. He is known for his contributions on Kriging and mathematical morphology. His seminal work is posted for study and review to the Online Library of the Centre de Géostatistique, Fontainebleau, France.

Shape. For example, the structuring element can be a "ball" or a line; convex or a ring, etc. By choosing a particular structuring element, one sets a way of differentiating some objects (or parts of objects) from others, according to their shape or spatial orientation.
Size. For example, one structuring element can be a $3\times 3$ square or a $21\times 21$ square. Setting the size of the structuring element is similar to setting the observation scale, and setting the criterion to differentiate image objects or features according to size.

Mathematical particulars and examples

Structuring elements are particular cases of binary images, usually being small and simple. In mathematical morphology, binary images are subsets of a Euclidean space R^d or the integer grid Z^d, for some dimension d. Here are some examples of widely used structuring elements (denoted by B):

A binary image is a digital image that has only two possible values for each pixel. Typically, the two colors used for a binary image are black and white. The color used for the object(s) in the image is the foreground color while the rest of the image is the background color. In the document-scanning industry, this is often referred to as "bi-tonal".

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. A is a subset of B may also be expressed as B includes A; or A is included in B.

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and similar spaces of higher dimension. It is named after the Ancient Greek mathematician Euclid of Alexandria. The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions.

Let E=R²; B is an open disk of radius r, centered at the origin.
Let E=Z²; B is a 3x3 square, that is, B={(-1,-1),(-1,0),(-1,1),(0,-1),(0,0),(0,1),(1,-1),(1,0),(1,1)}.
Let E=Z²; B is the "cross" given by: B={(-1,0),(0,-1),(0,0),(0,1),(1,0)}.

In the discrete case, a structuring element can also be represented as a set of pixels on a grid, assuming the values 1 (if the pixel belongs to the structuring element) or 0 (otherwise).

In digital imaging, a pixel, pel, dots, or picture element is a physical point in a raster image, or the smallest addressable element in an all points addressable display device; so it is the smallest controllable element of a picture represented on the screen.

In the context of a spatial index, a grid or mesh is a regular tessellation of a manifold or 2-D surface that divides it into a series of contiguous cells, which can then be assigned unique identifiers and used for spatial indexing purposes. A wide variety of such grids have been proposed or are currently in use, including grids based on "square" or "rectangular" cells, triangular grids or meshes, hexagonal grids and grids based on diamond-shaped cells.also "global grid" if it covers the entire surface of the globe)

When used by a hit-or-miss transform, usually the structuring element is a composite of two disjoint sets (two simple structuring elements), one associated to the foreground, and one associated to the background of the image to be probed. In this case, an alternative representation of the composite structuring element is as a set of pixels which are either set (1, associated to the foreground), not set (0, associated to the background) or "don't care".

Notes

↑ See (Dougherty 1992), chapter 1, page 1.

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In computer graphics, alpha compositing is the process of combining an image with a background to create the appearance of partial or full transparency. It is often useful to render image elements in separate passes, and then combine the resulting multiple 2D images into a single, final image called the composite. For example, compositing is used extensively when combining computer-rendered image elements with live footage.

Digital compositing is the process of digitally assembling multiple images to make a final image, typically for print, motion pictures or screen display. It is the digital analogue of optical film compositing.

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Opening (morphology) basic operation in mathematical morphology

In mathematical morphology, opening is the dilation of the erosion of a set A by a structuring element B:

Closing (morphology) basic operation in mathematical morphology

In mathematical morphology, the closing of a set A by a structuring element B is the erosion of the dilation of that set,

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Connected-component labeling is an algorithmic application of graph theory, where subsets of connected components are uniquely labeled based on a given heuristic. Connected-component labeling is not to be confused with segmentation.

In mathematical morphology, hit-or-miss transform is an operation that detects a given configuration in a binary image, using the morphological erosion operator and a pair of disjoint structuring elements. The result of the hit-or-miss transform is the set of positions where the first structuring element fits in the foreground of the input image, and the second structuring element misses it completely.

In digital image processing, morphological skeleton is a skeleton representation of a shape or binary image, computed by means of morphological operators.

In mathematical morphology and digital image processing, a morphological gradient is the difference between the dilation and the erosion of a given image. It is an image where each pixel value indicates the contrast intensity in the close neighborhood of that pixel. It is useful for edge detection and segmentation applications.

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.

In mathematical morphology, granulometry is an approach to compute a size distribution of grains in binary images, using a series of morphological opening operations. It was introduced by Georges Matheron in the 1960s, and is the basis for the characterization of the concept of size in mathematical morphology.

References

Edward R. Dougherty, An Introduction to Morphological Image Processing, ISBN 0-8194-0845-X (1992)
Jean Serra, Image Analysis and Mathematical Morphology, Volume 1, ISBN 0-12-637241-1 (1982)

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[1] See (Dougherty 1992), chapter 1, page 1.

Structuring element

Contents

Mathematical particulars and examples

Notes

Related Research Articles

References