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 ...
, opening is the
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 ...
of the
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 ...
of a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
A by a
structuring element B:
:
where
and
denote erosion and dilation, respectively.
Together with
closing, the opening serves in
computer vision
Computer vision tasks include methods for image sensor, acquiring, Image processing, processing, Image analysis, analyzing, and understanding digital images, and extraction of high-dimensional data from the real world in order to produce numerical ...
and
image processing
An image or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a pr ...
as a basic workhorse of morphological noise removal. Opening removes small objects from the foreground (usually taken as the bright pixels) of an image, placing them in the background, while closing removes small holes in the foreground, changing small islands of background into foreground. These techniques can also be used to find specific shapes in an image. Opening can be used to find things into which a specific structuring element can fit (edges, corners, ...).
One can think of ''B'' sweeping around the inside of the boundary of ''A'', so that it does not extend beyond the boundary, and shaping the ''A'' boundary around the boundary of the element.
Properties
* Opening is
idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
, that is,
.
* Opening is
increasing, that is, if
, then
.
* Opening is
anti-extensive, i.e.,
.
* Opening is
translation invariant
In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation (without rotation). Discrete translational symmetry is invariant under discrete translation.
Analogously, an operato ...
.
* Opening and closing satisfy the duality
, where
denotes closing.
Example
Perform
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 ...
:
Suppose A is the following 16 x 15 matrix and B is the following 3 x 3 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 1 1 1 0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 1 0 0 0 0 1 1 1 0 0
0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 1 1 1
0 0 1 1 1 1 0 0 0 1 1 1 1 1 0 0 1 1 1
0 0 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1
0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0
0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0
0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0
0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0
0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0
0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0
0 0 0 0 0 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
First, perform
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 ...
on A by B
):
Assuming that the origin of 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 16 x 15 matrix.
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 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0
0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 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
Then, perform
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 ...
on the result of Erosion by B:
:
For each pixel in
that has a value of 1, superimpose B, with the center of B aligned with the corresponding pixel in
.
Each pixel of every superimposed B is included in the dilation of A by B.
The dilation of
by B is given by this 16 x 15 matrix.
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 1 1 1 0 0 0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0 0 0 0 1 1 1 0 0
0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0
0 0 1 1 1 1 0 0 0 1 1 1 1 1 0 0
0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0
0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0
0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0
0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0
0 0 0 0 1 1 1 1 1 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 1 1 1 1 1 1 1 0 0 0 0 0
0 0 0 0 0 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
Therefore, the opening operation
removes small protrusions from the boundary of the object represented by A, while preserving the overall shape and size of the larger components.
Extension: Opening by reconstruction
In morphological opening
, the erosion operation removes objects that are smaller than
structuring element B and the dilation operation (approximately) restores the size and shape of the remaining objects. However, restoration accuracy in the dilation operation depends highly on the type of structuring element and the shape of the restoring objects. The opening by reconstruction method is able to restore the objects more completely after erosion has been applied. It is defined as the reconstruction by geodesic dilation of
erosions of
by
with respect to
:
where
denotes a marker image and
is a mask image in morphological reconstruction by dilation.
denotes geodesic dilation with
iterations until stability, i.e., such that
Since
,
the marker image is limited in the growth region by the mask image, so the dilation operation on the marker image will not expand beyond the mask image. As a result, the marker image is a subset of the mask image
(Strictly, this holds for binary masks only. However, similar statements hold when the mask is not binary.)
The images below present a simple opening-by-reconstruction example which extracts the vertical strokes from an input text image. Since the original image is converted from grayscale to binary image, it has a few distortions in some characters so that same characters might have different vertical lengths. In this case, the structuring element is an 8-pixel vertical line which is applied in the erosion operation in order to find objects of interest. Moreover, morphological reconstruction by dilation,
iterates
times until the resulting image converges.
See also
*
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 ...
*
Closing
*
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 ...
Bibliography
* ''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)
External links
* http://homepages.inf.ed.ac.uk/rbf/HIPR2/open.htm - Morphological Opening
References
* ''Digital Image Processing'' (''Third Edition'') by Rafael C. Gonzalez and Richard E. Woods, {{ISBN, 978-93-325-7032-0(2008)
Mathematical morphology