.gif)
Erosion (morphology)
Encyclopedia
For use of "Erosion" in dermatopathology, see Erosion (dermatopathology)
Erosion is one of two fundamental operations (the other being dilation
) in Morphological image processing from which all other morphological operations are based. It was originally defined for binary image
s, later being extended to grayscale
images, and subsequently to complete lattice
s.
of a Euclidean space
or the integer
grid
, 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:
where Bz is the translation of B by the vector z, i.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:
.
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 1 1 1 1 1 1 0 0 0 1
0 0 0 1 1 1 1 1 1 1 0 0 0 1
0 0 0 1 1 1 1 1 1 1 0 0 0 1
0 0 0 1 1 1 1 1 1 1 0 0 0 1
0 0 0 1 1 1 1 1 1 1 0 0 0 1
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
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.
The Erosion of A by B is given by
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 1 1 1 1 1 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 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, else it gets deleted or in other words it gets eroded.
morphology, images are functions
mapping a Euclidean space
or grid
E into
, where 
is the set of reals, 
is an element larger than any real number, and 
is an element smaller than any real number.
Denoting an image by f(x) and the grayscale structuring element by b(x), the grayscale erosion of f by b is given by
where "inf" denotes the infimum
.
s are partially ordered set
s, where every subset has an infimum
and a supremum
. In particular, it contains a least element and a greatest element
(also denoted "universe").
Let
be a complete lattice, with infimum and minimum symbolized by 
and 
, respectively. Its universe and least element are symbolized by U and 
, respectively. Moreover, let 
be a collection of elements from L.
An erosion in
is any operator 
that distributes over the infimum, and preserves the universe. I.e.:
CAMILA ZAPATA
ROMINA MANAY
Dilation (morphology)
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...
) in Morphological image processing from which all other morphological operations are based. It was originally defined for binary image
Binary image
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 though any two colors can be used. The color used for the object in the image is the foreground color while the rest of the image is the...
s, later being extended to 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...
images, and subsequently to complete lattice
Complete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...
s.
Binary erosion
In binary morphology, an image is viewed as a subsetSubset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of a 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 the integerInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
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...
, 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
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...
, 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:
-
,
where Bz is the translation of B by the vector z, i.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:
.Example
Suppose A is a 13 * 13 matrix and B is a 5 * 1 matrix:0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 1 1 1 1 1 1 0 0 0 1
0 0 0 1 1 1 1 1 1 1 0 0 0 1
0 0 0 1 1 1 1 1 1 1 0 0 0 1
0 0 0 1 1 1 1 1 1 1 0 0 0 1
0 0 0 1 1 1 1 1 1 1 0 0 0 1
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
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.
The Erosion of A by B is given by
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 1 1 1 1 1 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 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, else it gets deleted or in other words it gets eroded.
Properties
- The erosion is translation invariant.
- It is increasing, that is, if
, then 
. - If the origin of E belongs to the structuring element B, then the erosion is anti-extensive, i.e.,
. - The erosion satisfies
, where 
denotes the morphological dilationDilation (morphology)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 erosion is distributive over set intersection
Grayscale erosion
In grayscaleGrayscale
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...
morphology, images are functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
mapping a 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 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 into
, where is the set of reals, is an element larger than any real number, and is an element smaller than any real number.Denoting an image by f(x) and the grayscale structuring element by b(x), the grayscale erosion of f by b is given by
-
,
where "inf" denotes the infimum
Infimum
In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...
.
Erosions on complete lattices
Complete latticeComplete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...
s are partially ordered set
Partially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
s, where every subset has an infimum
Infimum
In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...
and a supremum
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
. In particular, it contains a least element and a greatest element
Greatest element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually...
(also denoted "universe").
Let
be a complete lattice, with infimum and minimum symbolized by and , respectively. Its universe and least element are symbolized by U and , respectively. Moreover, let be a collection of elements from L.An erosion in
is any operator that distributes over the infimum, and preserves the universe. I.e.:
-
, -
.
See also
- Mathematical morphologyMathematical morphologyMathematical morphology is a theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions...
- DilationDilation (morphology)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...
- OpeningOpening (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....
- ClosingClosing (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....
CAMILA ZAPATA
ROMINA MANAY