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Mathematical morphology

From Wikipedia, the free encyclopedia

A shape (in blue) and its morphological dilation (in green) and erosion (in yellow) by

a diamond-shape structuring element.

Mathematical morphology (MM) is a theory and technique for the analysis and processing of geometricalstructures, based on set theory,lattice theory,topology, and random functions. MM is most commonly

applied to digital images, but it can be employed as well ongraphs,surface meshes,solids, and many other

spatial structures.

Topological and geometricalcontinuous-space concepts such as size, shape, convexity, connectivity,

andgeodesic distance, can be characterized by MM on both continuous anddiscrete spaces. MM is also

the foundation of morphological image processing, which consists of a set of operators that transform

images according to the above characterizations.

MM was originally developed forbinary images, and was later extended tograyscalefunctions and images.

The subsequent generalization tocomplete latticesis widely accepted today as MM's theoretical foundation.
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Contents

[hide]

1 History

1.1 References

2 Binary morphology

2.1 Structuring element

2.2 Basic operators

2.2.1 Erosion

2.2.2 Dilation

2.2.3 Opening

2.2.4 Closing

2.2.5 Properties of the basic

operators

2.3 Other operators and tools

3 Grayscale morphology

3.1 Flat structuring functions

3.2 Other operators and tools

4 Mathematical morphology on complete lattices

4.1 Adjunctions (Dilation and Erosion)

4.2 Opening and Closing

4.3 Particular cases

5 References

6 External links

[edit]History

Mathematical Morphology was born in 1964 from the collaborative work ofGeorges MatheronandJean

Serra, at the cole des Mines de Paris,France. Matheron supervised thePhDthesis of Serra, devoted to

the quantification of mineral characteristics from thin cross sections, and this work resulted in a novel

practical approach, as well as theoretical advancements inintegral geometry and topology.

In 1968, the Centre de Morphologie Mathmatiquewas founded by the cole des Mines de Paris

in Fontainebleau, France, lead by Matheron and Serra.
http://en.wikipedia.org/wiki/Mathematical_morphologyhttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=1http://en.wikipedia.org/wiki/Georges_Matheronhttp://en.wikipedia.org/wiki/Georges_Matheronhttp://en.wikipedia.org/wiki/Georges_Matheronhttp://en.wikipedia.org/wiki/Jean_Serrahttp://en.wikipedia.org/wiki/Jean_Serrahttp://en.wikipedia.org/wiki/Jean_Serrahttp://en.wikipedia.org/wiki/Jean_Serrahttp://en.wikipedia.org/wiki/%C3%89cole_des_Mines_de_Parishttp://en.wikipedia.org/wiki/%C3%89cole_des_Mines_de_Parishttp://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/PhDhttp://en.wikipedia.org/wiki/PhDhttp://en.wikipedia.org/wiki/Thesishttp://en.wikipedia.org/wiki/Cross_section_(geometry)http://en.wikipedia.org/wiki/Integral_geometryhttp://en.wikipedia.org/wiki/Integral_geometryhttp://en.wikipedia.org/wiki/Topologyhttp://en.wikipedia.org/wiki/Centre_de_Morphologie_Math%C3%A9matiquehttp://en.wikipedia.org/wiki/Centre_de_Morphologie_Math%C3%A9matiquehttp://en.wikipedia.org/wiki/Fontainebleauhttp://en.wikipedia.org/wiki/Fontainebleauhttp://en.wikipedia.org/wiki/Mathematical_morphologyhttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=1http://en.wikipedia.org/wiki/Georges_Matheronhttp://en.wikipedia.org/wiki/Jean_Serrahttp://en.wikipedia.org/wiki/Jean_Serrahttp://en.wikipedia.org/wiki/%C3%89cole_des_Mines_de_Parishttp://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/PhDhttp://en.wikipedia.org/wiki/Thesishttp://en.wikipedia.org/wiki/Cross_section_(geometry)http://en.wikipedia.org/wiki/Integral_geometryhttp://en.wikipedia.org/wiki/Topologyhttp://en.wikipedia.org/wiki/Centre_de_Morphologie_Math%C3%A9matiquehttp://en.wikipedia.org/wiki/Fontainebleau


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During the rest of the 1960's and most of the 1970's, MM dealt essentially with binary images, treated

as sets, and generated a large number ofbinary operators and techniques:Hit-or-miss

transform,dilation, erosion,opening,closing,granulometry, thinning,skeletonization,ultimate

erosion,conditional bisector, and others. A random approach was also developed, based on novel image

models. Most of the work in that period was developed in Fontainebleau.

From mid-1970's to mid-1980's, MM was generalized to grayscale functions andimages as well. Besides

extending the main concepts (such as dilation, erosion, etc...) to functions, this generalization yielded new

operators, such as morphological gradients, top-hat transformand theWatershed(MM's

mainsegmentationapproach).

In the 1980's and 1990's, MM gained a wider recognition, as research centers in several countries began to

adopt and investigate the method. MM started to be applied to a large number of imaging problems and

applications.

In 1986, Jean Serra further generalized MM, this time to a theoretical framework based on complete

lattices. This generalization brought flexibility to the theory, enabling its application to a much larger number

of structures, including color images, video, graphs,meshes, etc... At the same time, Matheron and Serra

also formulated a theory for morphological filtering, based on the new lattice framework.

The 1990's and 2000's also saw further theoretical advancements, including the concepts

ofconnectionsandlevelings.

In 1993, the first International Symposium on Mathematical Morphology (ISMM) took place

in Barcelona, Spain. Since then, ISMMs are organized every 2-3 years, each time in a different part of the

world: Fontainebleau,France (1994);Atlanta,USA (1996);Amsterdam,Netherlands(1998); Palo

Alto,CA, USA (2000); Sydney, Australia (2002);Paris,France (2004);Rio de Janeiro, Brazil (2007);

andGroningen, Netherlands(2009).

[edit]References

"Introduction" by Pierre Soille, in (Serra et al. (Eds.) 1994), pgs. 1-4.

"Appendix A: The 'Centre de Morphologie Mathmatique', an overview" by

Jean Serra, in (Serra et al. (Eds.) 1994), pgs. 369-374.

"Foreword" in (Ronse et al. (Eds.) 2005)

[edit]Binary morphology

In binary morphology, an image is viewed as a subset of anEuclidean space or the integer grid ,

for some dimension d.

[edit]Structuring element
http://en.wikipedia.org/wiki/Binary_imagehttp://en.wikipedia.org/wiki/Binary_imagehttp://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/wiki/Binary_operatorhttp://en.wikipedia.org/wiki/Binary_operatorhttp://en.wikipedia.org/wiki/Hit-or-miss_transformhttp://en.wikipedia.org/wiki/Hit-or-miss_transformhttp://en.wikipedia.org/wiki/Hit-or-miss_transformhttp://en.wikipedia.org/wiki/Dilationhttp://en.wikipedia.org/wiki/Dilationhttp://en.wikipedia.org/wiki/Erosion_(morphology)http://en.wikipedia.org/wiki/Opening_(morphology)http://en.wikipedia.org/wiki/Opening_(morphology)http://en.wikipedia.org/wiki/Closing_(morphology)http://en.wikipedia.org/wiki/Closing_(morphology)http://en.wikipedia.org/wiki/Granulometry_(morphology)http://en.wikipedia.org/wiki/Granulometry_(morphology)http://en.wikipedia.org/wiki/Topological_skeletonhttp://en.wikipedia.org/wiki/Topological_skeletonhttp://en.wikipedia.org/wiki/Topological_skeletonhttp://en.wikipedia.org/w/index.php?title=Ultimate_erosion&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Ultimate_erosion&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Conditional_bisector&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Conditional_bisector&action=edit&redlink=1http://en.wikipedia.org/wiki/Grayscalehttp://en.wikipedia.org/wiki/Imagehttp://en.wikipedia.org/wiki/Imagehttp://en.wikipedia.org/wiki/Morphological_Gradienthttp://en.wikipedia.org/wiki/Top-hat_transformhttp://en.wikipedia.org/wiki/Top-hat_transformhttp://en.wikipedia.org/wiki/Watershed_(algorithm)http://en.wikipedia.org/wiki/Watershed_(algorithm)http://en.wikipedia.org/wiki/Watershed_(algorithm)http://en.wikipedia.org/wiki/Segmentationhttp://en.wikipedia.org/wiki/Segmentationhttp://en.wikipedia.org/wiki/Segmentationhttp://en.wikipedia.org/wiki/Complete_latticehttp://en.wikipedia.org/wiki/Complete_latticehttp://en.wikipedia.org/wiki/Graphhttp://en.wikipedia.org/wiki/Meshhttp://en.wikipedia.org/wiki/Meshhttp://en.wikipedia.org/wiki/Meshhttp://en.wikipedia.org/wiki/Filter_(mathematics)http://en.wikipedia.org/w/index.php?title=Connection_(morphology)&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Connection_(morphology)&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Connection_(morphology)&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Leveling_(morphology)&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Leveling_(morphology)&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Leveling_(morphology)&action=edit&redlink=1http://en.wikipedia.org/wiki/Barcelonahttp://en.wikipedia.org/wiki/Spainhttp://en.wikipedia.org/wiki/Spainhttp://en.wikipedia.org/wiki/Fontainebleauhttp://en.wikipedia.org/wiki/Fontainebleauhttp://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/Atlantahttp://en.wikipedia.org/wiki/Atlantahttp://en.wikipedia.org/wiki/United_States_of_Americahttp://en.wikipedia.org/wiki/United_States_of_Americahttp://en.wikipedia.org/wiki/Amsterdamhttp://en.wikipedia.org/wiki/Amsterdamhttp://en.wikipedia.org/wiki/Netherlandshttp://en.wikipedia.org/wiki/Netherlandshttp://en.wikipedia.org/wiki/Netherlandshttp://en.wikipedia.org/wiki/Palo_Altohttp://en.wikipedia.org/wiki/Palo_Altohttp://en.wikipedia.org/wiki/Palo_Altohttp://en.wikipedia.org/wiki/Californiahttp://en.wikipedia.org/wiki/United_States_of_Americahttp://en.wikipedia.org/wiki/Sydneyhttp://en.wikipedia.org/wiki/Australiahttp://en.wikipedia.org/wiki/Parishttp://en.wikipedia.org/wiki/Parishttp://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/Francehttp://en.wikipedia.org/wiki/Rio_de_Janeirohttp://en.wikipedia.org/wiki/Rio_de_Janeirohttp://en.wikipedia.org/wiki/Brazilhttp://en.wikipedia.org/wiki/Groningen_(city)http://en.wikipedia.org/wiki/Groningen_(city)http://en.wikipedia.org/wiki/Netherlandshttp://en.wikipedia.org/wiki/Netherlandshttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=2http://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=3http://en.wikipedia.org/wiki/Subsethttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=4http://en.wikipedia.org/wiki/Structuring_elementhttp://en.wikipedia.org/wiki/Binary_imagehttp://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/wiki/Binary_operatorhttp://en.wikipedia.org/wiki/Hit-or-miss_transformhttp://en.wikipedia.org/wiki/Hit-or-miss_transformhttp://en.wikipedia.org/wiki/Dilationhttp://en.wikipedia.org/wiki/Erosion_(morphology)http://en.wikipedia.org/wiki/Opening_(morphology)http://en.wikipedia.org/wiki/Closing_(morphology)http://en.wikipedia.org/wiki/Granulometry_(morphology)http://en.wikipedia.org/wiki/Topological_skeletonhttp://en.wikipedia.org/w/index.php?title=Ultimate_erosion&action=edit&redlink=1http:/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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

calledstructuring element, and is itself a binary image (i.e., a subset of the space or grid).

Here are some examples of widely used structuring elements (denoted by B):

Let ; B is an open disk of radius r, centered at the origin.

Let ; B is a 3x3 square, that is, B={(-1,-1), (-1,0), (-1,1), (0,-1),

(0,0), (0,1), (1,-1), (1,0), (1,1)}.

Let ; B is the "cross" given by: B={(-1,0), (0,-1), (0,0), (0,1),

(1,0)}.

[edit]Basic operatorsThe basic operations are shift-invariant (translation invariant) operators strongly related to Minkowski

addition.

Let Ebe a Euclidean space or an integer grid, andA a binary image in E.

[edit]Erosion

The erosion of the dark-blue square by a disk, resulting in the light-blue square.

The erosion of the binary imageA by the structuring element B is defined by:

,

where Bz is the translation ofB by the vector z,

i.e., , .

When the structuring element B has a center (e.g., B is a disk or a

square), and this center is located on the origin ofE, then the erosion

ofAby B can be understood as the locus of points reached by the center
http://en.wikipedia.org/wiki/Structuring_elementhttp://en.wikipedia.org/wiki/Structuring_elementhttp://en.wikipedia.org/wiki/Structuring_elementhttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=5http://en.wikipedia.org/wiki/Translational_invariancehttp://en.wikipedia.org/wiki/Translational_invariancehttp://en.wikipedia.org/wiki/Minkowski_additionhttp://en.wikipedia.org/wiki/Minkowski_additionhttp://en.wikipedia.org/wiki/Minkowski_additionhttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=6http://en.wikipedia.org/wiki/Erosion_(morphology)http://en.wikipedia.org/wiki/File:Erosion.pnghttp://en.wikipedia.org/wiki/File:Erosion.pnghttp://en.wikipedia.org/wiki/Structuring_elementhttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=5http://en.wikipedia.org/wiki/Translational_invariancehttp://en.wikipedia.org/wiki/Minkowski_additionhttp://en.wikipedia.org/wiki/Minkowski_additionhttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=6http://en.wikipedia.org/wiki/Erosion_(morphology)


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ofB when B moves insideA. 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 ofA by B is also given by the

expression: .

Example application: Assume we have received a fax of a dark

photocopy. Everything looks like it was written with a pen that is

bleeding. Erosion process will allow thicker lines to get skinny and detect

the hole inside the letter "o".

[edit]Dilation

The dilation of the dark-blue square by a disk, resulting in the light-blue

square with rounded corners.

The dilation ofA by the structuring element B is defined by:

.

The dilation is commutative, also given

by: .

IfB has a center on the origin, as before, then the dilation

ofA by B can be understood as thelocusof the points covered

by B when the center ofB moves insideA. In the above example,

the dilation of the square of side 10 by the disk of radius 2 is a
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square of side 14, with rounded corners, centered at the origin. The

radius of the rounded corners is 2.

The dilation can also be obtained

by: ,

where Bs denotes thesymmetric ofB, that

is, .

Example application: Dilation is the opposite of the erosion. Figures

that are very lightly drawn get thick when "dilated". Easiest way to

describe it is to imagine the same fax/text is written with a thicker

pen.

[edit]Opening

The opening of the dark-blue square by a disk, resulting in the light-

blue square with round corners.

The opening ofA by B is obtained by the erosion ofA by B,

followed by dilation of the resulting image by B:

.

The opening is also given by , which

means that it is the locus of translations of the structuring

element B inside the imageA. In the case of the square of

radius 10, and a disc of radius 2 as the structuring element,

the opening is a square of radius 10 with rounded corners,

where the corner radius is 2.
http://en.wikipedia.org/wiki/Rotational_symmetryhttp://en.wikipedia.org/wiki/Rotational_symmetryhttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=8http://en.wikipedia.org/wiki/Opening_(morphology)http://en.wikipedia.org/wiki/File:Opening.pnghttp://en.wikipedia.org/wiki/File:Opening.pnghttp://en.wikipedia.org/wiki/Rotational_symmetryhttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=8http://en.wikipedia.org/wiki/Opening_(morphology)


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Example application: Let's assume someone has written a

note on a non-soaking paper that writing looks like it is

growing tiny hairy roots all over. Opening essentially removes

the outer tiny "hairline" leaks and restores the text. The side

effect is that it rounds off things. The sharp edges start to

disappear.

[edit]Closing

The closing of the dark-blue shape (union of two squares) by a

disk, resulting in the union of the dark-blue shape and the light-

blue areas.

The closing ofA by B is obtained by the dilation ofA by B,

followed by erosion of the resulting structure by B:

.

The closing can also be obtained

by , whereXc denotes

the complement ofXrelative to E(that

is, ). The above means

that the closing is the complement of the locus of

translations of the symmetric of the structuring element

outside the imageA.

[edit]Properties of the basic operators

Here are some properties of the basic binary

morphological operators (dilation, erosion, opening and

closing):
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They are translation invariant.

They are increasing, that is, if ,

then ,

and , etc.

The dilation iscommutative.

If the origin ofEbelongs to the structuring

element B,

then

.

The dilation isassociative,

i.e., .

Moreover, the erosion

satisfies

.

Erosion and dilation satisfy the

duality .

Opening and closing satisfy the

duality .

The dilation isdistributive overset union

The erosion isdistributive overset intersection

The dilation is apseudo-inverse of the erosion, and

vice-versa, in the following

sense: if and only

if .

Opening and closing are idempotent.

Opening is anti-extensive, i.e., ,

whereas the closing is extensive,

i.e., .[edit]Other operators and tools
http://en.wikipedia.org/wiki/Translational_invariancehttp://en.wikipedia.org/wiki/Increasinghttp://en.wikipedia.org/wiki/Increasinghttp://en.wikipedia.org/wiki/Commutativehttp://en.wikipedia.org/wiki/Commutativehttp://en.wikipedia.org/wiki/Associativehttp://en.wikipedia.org/wiki/Associativehttp://en.wikipedia.org/wiki/Associativehttp://en.wikipedia.org/wiki/Distributivehttp://en.wikipedia.org/wiki/Distributivehttp://en.wikipedia.org/wiki/Set_unionhttp://en.wikipedia.org/wiki/Set_unionhttp://en.wikipedia.org/wiki/Distributivehttp://en.wikipedia.org/wiki/Distributivehttp://en.wikipedia.org/wiki/Set_intersectionhttp://en.wikipedia.org/wiki/Set_intersectionhttp://en.wikipedia.org/wiki/Pseudo-inversehttp://en.wikipedia.org/wiki/Pseudo-inversehttp://en.wikipedia.org/wiki/Idempotenthttp://en.wikipedia.org/w/index.php?title=Anti-extensive&action=edit&redlink=1http://en.wikipedia.org/wiki/Extensivehttp://en.wikipedia.org/wiki/Extensivehttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=11http://en.wikipedia.org/wiki/Translational_invariancehttp://en.wikipedia.org/wiki/Increasinghttp://en.wikipedia.org/wiki/Commutativehttp://en.wikipedia.org/wiki/Associativehttp://en.wikipedia.org/wiki/Distributivehttp://en.wikipedia.org/wiki/Set_unionhttp://en.wikipedia.org/wiki/Distributivehttp://en.wikipedia.org/wiki/Set_intersectionhttp://en.wikipedia.org/wiki/Pseudo-inversehttp://en.wikipedia.org/wiki/Idempotenthttp://en.wikipedia.org/w/index.php?title=Anti-extensive&action=edit&redlink=1http://en.wikipedia.org/wiki/Extensivehttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=11


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Hit-or-miss transform

Morphological skeleton

Filtering by reconstruction

Ultimate erosions andconditional bisectors

Granulometry

Geodesic distance functions

[edit]Grayscale morphology

Cardiac MRI image

Watershed of the gradient of the cardiac image

Ingrayscalemorphology, images

arefunctionsmapping a Euclidean space or

grid Einto , where is the set
http://en.wikipedia.org/wiki/Hit-or-miss_transformhttp://en.wikipedia.org/wiki/Morphological_skeletonhttp://en.wikipedia.org/w/index.php?title=Filtering_by_reconstruction&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Ultimate_erosions&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Conditional_bisector&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Conditional_bisector&action=edit&redlink=1http://en.wikipedia.org/wiki/Granulometry_(morphology)http://en.wikipedia.org/w/index.php?title=Geodesic_distance_function&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=12http://en.wikipedia.org/wiki/Grayscalehttp://en.wikipedia.org/wiki/Grayscalehttp://en.wikipedia.org/wiki/Grayscalehttp://en.wikipedia.org/wiki/Grayscalehttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/File:Watershed_of_gradient_of_MRI_heart_image_.pnghttp://en.wikipedia.org/wiki/File:Watershed_of_gradient_of_MRI_heart_image_.pnghttp://en.wikipedia.org/wiki/File:Heart_MRI_image.pnghttp://en.wikipedia.org/wiki/File:Heart_MRI_image.pnghttp://en.wikipedia.org/wiki/Hit-or-miss_transformhttp://en.wikipedia.org/wiki/Morphological_skeletonhttp://en.wikipedia.org/w/index.php?title=Filtering_by_reconstruction&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Ultimate_erosions&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Conditional_bisector&action=edit&redlink=1http://en.wikipedia.org/wiki/Granulometry_(morphology)http://en.wikipedia.org/w/index.php?title=Geodesic_distance_function&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=12http://en.wikipedia.org/wiki/Grayscalehttp://en.wikipedia.org/wiki/Grayscalehttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Euclidean_space


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ofreals, is an element larger than any real number,

and is an element smaller than any real number.

Grayscale structuring elements are also functions of the

same format, called "structuring functions".

Denoting an image by f(x) and the structuring function

by b(x), the grayscale dilation offby b is given by

,

where "sup" denotes thesupremum.

Similarly, the erosion offby b is given by

,

where "inf" denotes theinfimum.

Just like in binary morphology, the opening

and closing are given respectively by

, and

.[edit]Flat structuring

functions

It is common to use flat

structuring elements in

morphological applications. Flat

structuring functions are

functions b(x) in the form

,

where .

In this case, the dilation

and erosion are greatly

simplified, and given

respectively by
http://en.wikipedia.org/wiki/Real_numbershttp://en.wikipedia.org/wiki/Real_numbershttp://en.wikipedia.org/wiki/Real_numbershttp://en.wikipedia.org/wiki/Supremumhttp://en.wikipedia.org/wiki/Supremumhttp://en.wikipedia.org/wiki/Supremumhttp://en.wikipedia.org/wiki/Infimumhttp://en.wikipedia.org/wiki/Infimumhttp://en.wikipedia.org/wiki/Infimumhttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=13http://en.wikipedia.org/wiki/Real_numbershttp://en.wikipedia.org/wiki/Supremumhttp://en.wikipedia.org/wiki/Infimumhttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=13


11/90


12/90

morphological

operators

depend only on

the relative

ordering

ofpixel values,

regardless their

numerical

values, and

therefore are

especially

suited to the

processing

ofbinary

images and gra

yscale

images whosel

ight transfer

functionis not

known.

[edit]Other

operators

and tools

Morphologi

cal

Gradients

Top-hat

transform

Watershed

algorithm

By combining

these operators

one can obtain

algorithms for

many image

processingtasks, such
http://en.wikipedia.org/wiki/Pixelhttp://en.wikipedia.org/wiki/Pixelhttp://en.wikipedia.org/wiki/Binary_imageshttp://en.wikipedia.org/wiki/Binary_imageshttp://en.wikipedia.org/wiki/Binary_imageshttp://en.wikipedia.org/wiki/Grayscale_imageshttp://en.wikipedia.org/wiki/Grayscale_imageshttp://en.wikipedia.org/wiki/Grayscale_imageshttp://en.wikipedia.org/w/index.php?title=Light_transfer_function&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Light_transfer_function&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Light_transfer_function&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Light_transfer_function&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Light_transfer_function&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=14http://en.wikipedia.org/wiki/Morphological_Gradienthttp://en.wikipedia.org/wiki/Morphological_Gradienthttp://en.wikipedia.org/wiki/Morphological_Gradienthttp://en.wikipedia.org/wiki/Morphological_Gradienthttp://en.wikipedia.org/wiki/Top-hat_transformhttp://en.wikipedia.org/wiki/Top-hat_transformhttp://en.wikipedia.org/wiki/Top-hat_transformhttp://en.wikipedia.org/wiki/Watershed_(algorithm)http://en.wikipedia.org/wiki/Watershed_(algorithm)http://en.wikipedia.org/wiki/Pixelhttp://en.wikipedia.org/wiki/Binary_imageshttp://en.wikipedia.org/wiki/Binary_imageshttp://en.wikipedia.org/wiki/Grayscale_imageshttp://en.wikipedia.org/wiki/Grayscale_imageshttp://en.wikipedia.org/wiki/Grayscale_imageshttp://en.wikipedia.org/w/index.php?title=Light_transfer_function&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Light_transfer_function&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Light_transfer_function&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=14http://en.wikipedia.org/wiki/Morphological_Gradienthttp://en.wikipedia.org/wiki/Morphological_Gradienthttp://en.wikipedia.org/wiki/Morphological_Gradienthttp://en.wikipedia.org/wiki/Top-hat_transformhttp://en.wikipedia.org/wiki/Top-hat_transformhttp://en.wikipedia.org/wiki/Watershed_(algorithm)http://en.wikipedia.org/wiki/Watershed_(algorithm)


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as feature

detection, imag

e

segmentation,i

mage

sharpening,im

age filtering,

andclassificatio

n.

[edit]Mathematicalmorphologyon complete lattices

Complete

lattices areparti

ally ordered

sets, where

every subsethas

aninfimumand

a supremum. In

particular, it

contains a least

element and

a greatest

element (also

denoted

"universe").

[edit]Adjunc

tions

(Dilation

and

Erosion)
http://en.wikipedia.org/wiki/Feature_extractionhttp://en.wikipedia.org/wiki/Feature_extractionhttp://en.wikipedia.org/wiki/Segmentation_(image_processing)http://en.wikipedia.org/wiki/Segmentation_(image_processing)http://en.wikipedia.org/wiki/Segmentation_(image_processing)http://en.wikipedia.org/wiki/Unsharp_maskinghttp://en.wikipedia.org/wiki/Unsharp_maskinghttp://en.wikipedia.org/wiki/Unsharp_maskinghttp://en.wikipedia.org/wiki/Unsharp_maskinghttp://en.wikipedia.org/wiki/Filter_(signal_processing)http://en.wikipedia.org/wiki/Filter_(signal_processing)http://en.wikipedia.org/wiki/Filter_(signal_processing)http://en.wikipedia.org/wiki/Statistical_classificationhttp://en.wikipedia.org/wiki/Statistical_classificationhttp://en.wikipedia.org/wiki/Statistical_classificationhttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=15http://en.wikipedia.org/wiki/Complete_latticehttp://en.wikipedia.org/wiki/Complete_latticehttp://en.wikipedia.org/wiki/Complete_latticehttp://en.wikipedia.org/wiki/Complete_latticehttp://en.wikipedia.org/wiki/Partially_ordered_sethttp://en.wikipedia.org/wiki/Partially_ordered_sethttp://en.wikipedia.org/wiki/Partially_ordered_sethttp://en.wikipedia.org/wiki/Partially_ordered_sethttp://en.wikipedia.org/wiki/Partially_ordered_sethttp://en.wikipedia.org/wiki/Infimumhttp://en.wikipedia.org/wiki/Infimumhttp://en.wikipedia.org/wiki/Infimumhttp://en.wikipedia.org/wiki/Supremumhttp://en.wikipedia.org/wiki/Least_elementhttp://en.wikipedia.org/wiki/Least_elementhttp://en.wikipedia.org/wiki/Greatest_elementhttp://en.wikipedia.org/wiki/Greatest_elementhttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=16http://en.wikipedia.org/wiki/Feature_extractionhttp://en.wikipedia.org/wiki/Feature_extractionhttp://en.wikipedia.org/wiki/Segmentation_(image_processing)http://en.wikipedia.org/wiki/Segmentation_(image_processing)http://en.wikipedia.org/wiki/Segmentation_(image_processing)http://en.wikipedia.org/wiki/Unsharp_maskinghttp://en.wikipedia.org/wiki/Unsharp_maskinghttp://en.wikipedia.org/wiki/Unsharp_maskinghttp://en.wikipedia.org/wiki/Filter_(signal_processing)http://en.wikipedia.org/wiki/Filter_(signal_processing)http://en.wikipedia.org/wiki/Statistical_classificationhttp://en.wikipedia.org/wiki/Statistical_classificationhttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=15http://en.wikipedia.org/wiki/Complete_latticehttp://en.wikipedia.org/wiki/Complete_latticehttp://en.wikipedia.org/wiki/Complete_latticehttp://en.wikipedia.org/wiki/Complete_latticehttp://en.wikipedia.org/wiki/Partially_ordered_sethttp://en.wikipedia.org/wiki/Partially_ordered_sethttp://en.wikipedia.org/wiki/Partially_ordered_sethttp://en.wikipedia.org/wiki/Infimumhttp://en.wikipedia.org/wiki/Supremumhttp://en.wikipedia.org/wiki/Least_elementhttp://en.wikipedia.org/wiki/Least_elementhttp://en.wikipedia.org/wiki/Greatest_elementhttp://en.wikipedia.org/wiki/Greatest_elementhttp://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=16


14/90

Let be

a complete

lattice, with

infimum andminimum

symbolized

by and ,

respectively. Its

universe and

least element

are symbolized

by Uand ,

respectively.

Moreover,

let {Xi} be a

collection of

elements

from L.

A dilation is any

operator

that distributes

over the

supremum, and

preserves the

least element.

I.e.:

,

.


15/90

An erosion is

any

operator

that distributes

over the

infimum, and

preserves the

universe. I.e.:

,

.

Dilations and

erosions

form Galois

connections.

That is, for all

dilation there

is one and only

one erosion

that satisfies

for

all

.

Similarly,

for all

erosion

there is

one and
http://en.wikipedia.org/wiki/Galois_connectionhttp://en.wikipedia.org/wiki/Galois_connectionhttp://en.wikipedia.org/wiki/Galois_connectionhttp://en.wikipedia.org/wiki/Galois_connection


16/90

only one

dilation

satisfying

the above

connectio

n.

Furtherm

ore, if two

operators

satisfy the

connectio

n,

then mu

st be a

dilation,

and an

erosion.

Pairs of

erosions

and

dilations

satisfying

the above

connectio

n are

called

"adjunctio

ns", and

the

erosion is

said to be

the

adjoint

erosion of

the

dilation,

and vice-

versa.


17/90

[edit]Op

ening

and

Closin

g

For all

adjunctio

n ,

the

morpholo

gical

opening

and

morpholo

gical

closing

are

defined

as

follows:

, and

.

T

he

m

o

r

p

h

o

l

o
http://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=17http://en.wikipedia.org/w/index.php?title=Mathematical_morphology&action=edit&section=17


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