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 empl ...

, the closing of a set ( binary image) ''A'' by a structuring element ''B'' is 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 dis ...

of the dilation of that set, :

A\bullet B = (A\oplus B)\ominus B, \,

where

\oplus

and

\ominus

denote the dilation and erosion, respectively. In image processing, closing is, together with

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 * , ...

, the basic workhorse of morphological

noise Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference aris ...

removal. Opening removes small objects, while closing removes small holes.

Properties

* It 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,

(A\bullet B)\bullet B=A\bullet B

. * 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\bullet B \subseteq C\bullet B

. * It is ''extensive'', i.e.,

A\subseteq A\bullet B

. * It 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 ...

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, {{ISBN, 0-8194-0845-X (1992)

External links

Introduction to mathematical morphology
Mathematical morphology Digital geometry

Properties

See also

Bibliography

External links