In
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 ...
, opening is the
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 ...
of the
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 ...
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
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 ...
, the opening serves in
computer vision
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the hum ...
and
image processing
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
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
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
, then
.
* Opening is
anti-extensive, i.e.,
.
* Opening 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 ...
.
* Opening and closing satisfy the duality
, where
denotes closing.
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 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 ...
*
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 ...
*
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 ...
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