mathematical morphology 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 em ...

, 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 Dilation (or dilatation) may refer to: Physiology or medicine * Cervical dilation, the widening of the cervix in childbirth, miscarriage etc. * Coronary dilation, or coronary reflex * Dilation and curettage, the opening of the cervix and surgic ...

erosion Erosion is the action of surface processes (such as water flow or wind) that removes soil, rock, or dissolved material from one location on the Earth's crust, and then transports it to another location where it is deposited. Erosion is distin ...

opening Opening may refer to: * Al-Fatiha, "The Opening", the first chapter of the Qur'an * The Opening (album), live album by Mal Waldron * Backgammon opening * Chess opening * A title sequence or opening credits * , a term from contract bridge * , ...

, and

closing Closing may refer to: Business and law * Closing (law), a closing argument, a summation * Closing (real estate), the final step in executing a real estate transaction * Closing (sales), the process of making a sale * Closure (business), Closing a ...

, as well as the

hit-or-miss transform In mathematical morphology, hit-or-miss transform is an operation that detects a given configuration (or pattern) in a binary image, using the morphological erosion operator and a pair of disjoint structuring elements. The result of the hit-or- ...

. According to

Georges Matheron Georges François Paul Marie Matheron (2 December 1930 – 7 August 2000) was a French mathematician and civil engineer of mines, known as the founder of geostatistics and a co-founder (together with Jean Serra) of mathematical morphology. In 1968, ...

, knowledge about an object (e.g., an image) depends on the manner in which we probe (observe) it.See ( Dougherty 1992), chapter 1, page 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: * 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 image A binary image is one that consists of pixels that can have one of exactly two colors, usually black and white. Binary images are also called ''bi-level'' or ''two-level'', Pixelart made of two colours is often referred to as ''1-Bit'' or ''1b ...

s are

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

s of a

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

''R''^''d'' or the integer grid ''Z''^''d'', for some dimension ''d''. Here are some examples of widely used structuring elements (denoted by ''B''): * Let ''E''=''R''²; ''B'' is an open disk of radius ''r'', centered at the origin. * Let ''E''=''Z''²; ''B'' is a 3x3 square, that is, ''B''=. * Let ''E''=''Z''²; ''B'' is the "cross" given by: ''B''=. In the discrete case, a structuring element can also be represented as a set of

pixel In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device. In most digital display devices, pixels are the smal ...

s on a

grid Grid, The Grid, or GRID may refer to: Common usage * Cattle grid or stock grid, a type of obstacle is used to prevent livestock from crossing the road * Grid reference, used to define a location on a map Arts, entertainment, and media * News g ...

, assuming the values 1 (if the pixel belongs to the structuring element) or 0 (otherwise). 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

s which are either set (1, associated to the foreground), not set (0, associated to the background) or "

don't care In digital logic, a don't-care term (abbreviated DC, historically also known as ''redundancies'', ''irrelevancies'', ''optional entries'', ''invalid combinations'', ''vacuous combinations'', ''forbidden combinations'', ''unused states'' or ''l ...

Notes

References

* Edward R. Dougherty, ''An Introduction to Morphological Image Processing'', (1992) * Jean Serra, ''Image Analysis and Mathematical Morphology, Volume 1'', (1982) {{DEFAULTSORT:Structuring Element Mathematical morphology