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Erosion (usually represented by ⊖) is one of two fundamental operations (the other being
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 ...
) in
morphological image processing 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 empl ...
from which all other morphological operations are based. It was originally defined for
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, later being extended to
grayscale In digital photography, computer-generated imagery, and colorimetry, a grayscale image is one in which the value of each pixel is a single sample representing only an ''amount'' of light; that is, it carries only intensity information. Graysca ...
images, and subsequently to
complete lattice In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' ...
s. The erosion operation usually uses a structuring element for probing and reducing the shapes contained in the input image.


Binary erosion

In binary morphology, an image is viewed as a
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 ...
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 ...
\mathbb^d or the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
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 ...
\mathbb^d, for some dimension ''d''. The basic idea in binary morphology is to probe an image with a simple, pre-defined shape, drawing conclusions on how this shape fits or misses the shapes in the image. This simple "probe" is called structuring element, and is itself a binary image (i.e., a subset of the space or grid). Let ''E'' be a Euclidean space or an integer grid, and ''A'' a binary image in ''E''. The erosion of the binary image ''A'' by the structuring element ''B'' is defined by: ::A \ominus B = \, where ''B''''z'' is the translation of ''B'' by the vector z, i.e., B_z = \, \forall z\in E. When the structuring element ''B'' has a center (e.g., a disk or a square), and this center is located on the origin of ''E'', then the erosion of ''A'' by ''B'' can be understood as the locus of points reached by the center of ''B'' when ''B'' moves inside ''A''. For example, the erosion of a square of side 10, centered at the origin, by a disc of radius 2, also centered at the origin, is a square of side 6 centered at the origin. The erosion of ''A'' by ''B'' is also given by the expression: A \ominus B = \bigcap_ A_, where ''A−b'' denotes the translation of ''A'' by ''-b''.


Example

Suppose A is a 13 x 13 matrix and B is a 3 x 3 matrix: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Assuming that the origin B is at its center, for each pixel in A superimpose the origin of B, if B is completely contained by A the pixel is retained, else deleted. Therefore the Erosion of A by B is given by this 13 x 13 matrix. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 This means that only when B is completely contained inside A that the pixels values are retained, otherwise it gets deleted or eroded.


Properties

* The erosion is
translation invariant In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equat ...
. * It is
increasing In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
, that is, if A\subseteq C, then A\ominus B \subseteq C\ominus B. * If the origin of ''E'' belongs to the structuring element ''B'', then the erosion is ''anti-extensive'', i.e., A\ominus B\subseteq A. * The erosion satisfies (A\ominus B)\ominus C = A\ominus (B\oplus C), where \oplus denotes the morphological dilation. * The erosion is distributive over
set intersection In set theory, the intersection of two sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A. Notation and terminology Intersection is writt ...


Grayscale erosion

In
grayscale In digital photography, computer-generated imagery, and colorimetry, a grayscale image is one in which the value of each pixel is a single sample representing only an ''amount'' of light; that is, it carries only intensity information. Graysca ...
morphology, images are functions mapping 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 ...
or
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 ...
''E'' into \mathbb\cup\, where \mathbb is the set of reals, \infty is an element larger than any real number, and -\infty is an element smaller than any real number. Denoting an image by ''f(x)'' and the grayscale structuring element by ''b(x)'', where B is the space that b(x) is defined, the grayscale erosion of ''f'' by ''b'' is given by ::(f\ominus b)(x)=\inf_ (x+y)-b(y)/math>, where "inf" denotes the
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
. In other words the erosion of a point is the minimum of the points in its neighborhood, with that neighborhood defined by the structuring element. In this way it is similar to many other kinds of image filters like the
median filter The median filter is a non-linear digital filtering technique, often used to remove noise from an image or signal. Such noise reduction is a typical pre-processing step to improve the results of later processing (for example, edge detection on an ...
and the
gaussian filter In electronics and signal processing mainly in digital signal processing, a Gaussian filter is a filter whose impulse response is a Gaussian function (or an approximation to it, since a true Gaussian response would have infinite impulse response) ...
.


Erosions on complete lattices

Complete lattice In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' ...
s are
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
s, where every subset has an
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
and a
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
. In particular, it contains a
least element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an eleme ...
and a
greatest element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an eleme ...
(also denoted "universe"). Let (L,\leq) be a complete lattice, with infimum and supremum symbolized by \wedge and \vee, respectively. Its universe and least element are symbolized by ''U'' and \emptyset, respectively. Moreover, let \ be a collection of elements from ''L''. An erosion in (L,\leq) is any operator \varepsilon: L\rightarrow L that distributes over the infimum, and preserves the universe. I.e.: * \bigwedge_\varepsilon(X_i)=\varepsilon\left(\bigwedge_ X_i\right), * \varepsilon(U)=U.


See also

*
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 ...
*
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 ...
*
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 * , ...
*
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 ...


References

* ''Image Analysis and Mathematical Morphology'' by Jean Serra, (1982) * ''Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances'' by Jean Serra, (1988) * ''An Introduction to Morphological Image Processing'' by Edward R. Dougherty, (1992) * ''Morphological Image Analysis; Principles and Applications'' by Pierre Soille, {{ISBN, 3-540-65671-5 (1999) * R. C. Gonzalez and R. E. Woods, ''Digital image processing'', 2nd ed. Upper Saddle River, N.J.: Prentice Hall, 2002. Digital geometry Mathematical morphology