In
mathematical morphology
Mathematical morphology (MM) is a theory and technique for the analysis and processing of Geometry, geometrical structures, based on set theory, lattice theory, topology, and random functions. MM is most commonly applied to digital images, but it ...
, 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
wiktionary: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 ...
,
erosion
Erosion is the action of surface processes (such as Surface runoff, water flow or wind) that removes soil, Rock (geology), rock, or dissolved material from one location on the Earth's crust#Crust, Earth's crust and then sediment transport, tran ...
,
opening
Opening may refer to:
Types of openings
* Hole
* A title sequence or opening credits
* Grand opening of a business or other institution
* Inauguration
* Keynote
* Opening sentence
* Opening sequence
* Opening statement, a beginning statemen ...
, and
closing, as well as the
hit-or-miss transform.
According to
Georges Matheron, 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
square or a
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 a digital image that consists of pixels that can have one of exactly two colors, usually black and white. Each pixel is stored as a single bit — i.e. either a 0 or 1.
A binary image can be stored in memory as a bitmap: a p ...
s are
subset
In mathematics, a 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 a ...
s of a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
''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 graphics, raster image, or the smallest addressable element in a dot matrix display device. In most digital display devices, p ...
s on a
grid, 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
pixel
In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a Raster graphics, raster image, or the smallest addressable element in a dot matrix display device. In most digital display devices, p ...
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)
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Mathematical morphology