07 - Morphological operators€¦ · What are morphological operators? Local pixel transformaon for processing region shapes Most oTen used on binary images Logical transforma:on

Morphologicaloperators

PartI

Elaborazionedisegnalieimmaginiperbioinforma6ca SimonaSchiavi

Morphology

Whatismorphology?▶ Commonlydenotesabranchofbiologythatdealswiththeformandstructureof

animalsandplants

Whatismathema:calmorphology?▶ Atoolforextrac:ngimagecomponentsthatareusefulinthe

- representa:onanddescrip:onofregionshapes(boundaries,skeletons,convexhull,…)- pre-orpost-processing(morphologicalfiltering,thinning,andpruning)

▶ Mathema:calmorphologydealswithsettheory

Whenisituseful?▶ pre-processing(noisefiltering,shapesimplifica:on,boundariessmoothing,…)

▶ enhancingobjectstructure(skeletoniza:on,convexhull,…)

▶ segmenta:on(watershed,…)

▶ quan:ta:vedescrip:on(area,perimeter,…)


Whatisaset?▶ Collec6onofdis:nctobjects(e.g.forimagescollec:onofpixels)

Basicsetopera:ons

Recallmathema6caltools

AB

SubsetA ✓ B

A B

UnionASB

A B

Intersec:onATB

A B

ComplementAc = {w | w /2 A}

AB

DisjointATB = ;

AB

DifferenceB \A = {w | w 2 B,w /2 A}


Recallmathema6caltools

Originalset

Transla:on

Reflec:on

B̂ = {w | w = �b, for b 2 B}

(B)z = {c | c = b+ z, for b 2 B}


Whataremorphologicaloperators?

Localpixeltransforma0onforprocessingregionshapes

MostoTenusedonbinaryimages

Logicaltransforma:onbasedoncomparisonofpixelneighbourhoodswithaspecificpaVern(structuringelement)▶ Erosion

▶ Dila:on

▶ Opening

▶ Closing

▶ Hit-or-misstransforma:on


Binary morphology


Structuringelement(SE)

Smallsetwithapredefinedshapetoprobetheimageunderstudy▶ simpleshape

▶ binaryimage

▶ welldefinedorigin(usuallythecenterofgravityorsymmetry)

Itisusefultoadapttheshapetoberectangular


HowweuseSE?

Wecomparethestructuringelementtotheneighbourhoodofeachpixel▶ Inparallelforeachpixelwe- checkifSEis“sa:sfied”- wemarkthepixelsthatsa:sfytheSEandthenweperformthedesiredmorphologicalopera:on(erosion,dila:on,…)

Example


FiDng,hiDngandmissing

GivenastructuringelementSandabinaryimageI

SfitsIatxif

ShitsIatxif

SmissesIatxif

{y | y = x+ s, s 2 S} ⇢ I

{y | y = x� s, s 2 S}T

I 6= ;

{y | y = x� s, s 2 S}T

I = ;


ErosionofasetAbyastructuringelementB▶ allthepointzinAsuchthatBisincludedinAwhentheoriginofBcoincideswithz

(i.e.allthepointsinAatwhichBfitsA)

Erosion

A B = {z | (B)z ✓ A} = {z | (B)z\

Ac = ;}

= {z | z + b 2 A for every b 2 B} N.B.Here“white”meansthatthevaluecanbe0or1


Erosionexample1


Erosionexample2


Erosionproper6es

itshrinksorthinsobjectsinabinaryimages▶ actslikea“filter”

itisnotcommuta:ve▶

Itisanincreasingtransforma:on▶

A B 6= B A

A ✓ C ) A B ✓ C B


Dila6on

Dila:onofasetAbyastructuringelementB▶ allthepointzinAsuchthatBflippedandtranslatedbyzhasnon-emptyintersec:on

withA(i.e.allthepointsinAatwhichflippedBhitsA)

�

A�B =nz | (B̂)z

\A 6= ;

o=

nz |

h(B̂)z

\Ai✓ A

o

= {z | z = a+ b, a 2 A and b 2 B}


Dila6onexample1


Dila6onexample2


Dila6onproper6es

itgrowsorthickensobjectsinabinaryimages▶ actslikea“spa:alconvolu:on”usingtheSEasconvolu:onmask

itiscommuta:ve▶

itisassocia:ve▶

Itisanincreasingtransforma:on▶

A�B = B �A

A� (B � C) = (A�B)� C

A ✓ C ) A�B ✓ C �B


ErosionvsDila6on

Erosion▶ RemovalofstructuresofcertainshapeandsizegivenbytheSE

Dila:on▶ FillingofholesoraddingbordersofcertainshapeandsizegivenbytheSE

Erosionanddila:onaredualopera:ons▶

▶

IfBissymmetricwithrespecttoitsorigin,wecanobtaintheerosionoftheimageAbyBsimplybydila:ngthebackgroundoftheimagewith

thesameBandcomplemen:ngtheresult.

(A B)c = Ac � B̂

(A�B)c = Ac B̂


Dualityexample

A

B

B̂

Ac

A B

Ac � B̂


Combiningerosionanddila6on

Why?▶ Toremovestructuresand/orfillholeswithoutaffec:ngtheremainingparts

How?▶ bycombiningerosionanddila:onusingthesamestructuringelement


Opening

OpeningofasetAbythestructuringelementB▶ erosionofAbyBfollowedbyadila:onoftheresultbyB

Proper:es▶ smoothesthecontour

▶ breaksnarrowisthmuses

▶ eliminatesprotrusions

�

A�B = (A B)�BA�B


Closing

ClosingofasetAbythestructuringelementB▶ dila:onofAbyBfollowedbyanerosionoftheresultbyB

Proper:es▶ smoothessec:onsofthecontour

▶ fusesnarrowbreaksandlongthingulfs

▶ eliminatessmallholes

▶ fillsgapsincontour

A•B = (A�B) B

•


Opening/Closingexample1


Opening/Closingexample2

Closing

Opening

Structuringelement


Proper6es

Opening▶ isasubset(subimage)ofA

▶ Itisanincreasingfunc:on

▶ Itisidempotent,i.e.

Closing▶ Aisasubset(subimage)of

▶ Itisanincreasingfunc:on

▶ Itisidempotent,i.e.

Openingandclosingaredualopera:ons▶

A•BC ✓ D ) C•B ✓ D•B

(A•B)•B = A•B

A�B

C ✓ D ) C �B ✓ D �B

(A�B)�B = A�B

(A•B)c = (Ac � B̂)


Morphologicalfiltering

Toremoveholesintheforegroundandislandsinthebackground▶ dobothopeningandclosing

ThesizeandshapeofthestructuringelementdetermineswhichfeaturewillsurviveaTerthefiltering

Intheabsenceofknowledgeabouttheshapeofthefeaturestoremove,useacircularSE


Morphologicalfilteringexample

Closeandthenopen

Openandthenclose

Structuringelement


Hit-or-MissTransforma6on

Searchesforanexactmatchofthestructuringelement

GivenastructuringelementB=(B1UB2),thematchofBinAis

▶ findsamatchin(hit)

▶ findsamatchin(miss)

~

A~B = (A B1)T(Ac B2)

B1

B2

A

Ac


Hit-or-MissTransforma6onexample1


Hit-or-MissTransforma6onexample2

B = (S1, S2)

B = (S2, S3)

B = (S4, S5)

Morphologicaloperators

PartII


Applications


Whydoweusemorphologicaloperators?

Representa:onanddescrip:onofashape▶ extrac:ngboundaries

▶ extrac:ngconnectedcomponents

▶ extrac:ngtheconvexhull

▶ skeletonizearegion

Pre/Post-processingsteps▶ regionfilling

▶ regionthinning

▶ regionthickening


Boundaryextrac6on

CombiningerosionbyastructuringelementBanddifferencebetweensets

�(A) = A \ (A B)

\


Boundaryextrac6onexample


Holesfilling

Combiningsetdila:on,complementa:onandintersec:on▶ Aholeisdefinedasabackgroundregionsurroundedbyaconnectedborderof

foregroundpixels

Algorithm▶ GivenapointPinsideaholeset

▶ whiledo end

▶

Wecallthiscondi0onaldila0on

X0 = P

Xk 6= Xk�1

Xk = (Xk�1 �B)T

Ac

XF = XkSA


Holesfillingexample


Extrac6onofconnectedcomponents

Combiningsetdila:onandintersec:on▶ AconnectedcomponentSisdefinedasallthepixelsinSforwhichthereexistsapath

thatconnectthemconsis:ngen:relyofpixelsinS

Algorithm▶ GivenapointPinsideAset

▶ whiledo end

Itisagainacondi0onaldila0on

Xk 6= Xk�1

Xk = (Xk�1 �B)T

Ac

X0 = P


Connectedcomponentsexample1


Combininghit-or-misstransforma:onandunion▶ AsetAissaidtobeconvexifthestraightlinesegmentjoininganytwopointsinAlies

en0relywithinA

▶ TheconvexhullC(A)ofanarbitrarysetAisthesmallestconvexsetcontainingA

Algorithm▶ Define4structuringelements

▶ Ini:alize

▶ whiledo end

▶

▶ TogetthesmallestconvexsetcontainingAwelimitgrowthtotheminandmaxcoordinatesofA

N.B.Here“x” meansthatthevaluecanbe0or1

ConvexHull

Xik = (Xi

k�1 ~Bi)SA

Bi i = 1, 2, 3, 4

Xi0 = A i = 1, 2, 3, 4

Xik 6= Xi

k�1

C(A) =4[

i=1

Xik


Thinning

Combininghit-or-misstransforma:onandintersec:on▶ removespixelsfromthesetAusingastructuringelementBun:lonlyanarrowset

remains

Algorithm▶ GivenasequenceofnSEs

▶ Letand

▶ whiledo end

⌦

A⌦B = AT(A~B)c

{B} = (B1, . . . , Bn)

Xn = A

Xn 6= Y

Y = X0 = 0N⇥M

Y = Xn

X0 = Xn

Xi = Xi�1

\(Xi�1 ~Bi)c i = 1, . . . , n


Thinningexample


Thickening �

Hit-or-misstransforma:onandunion▶ isthemorphologicaldualofthinningandisdefinedby

Algorithm▶ Compute

▶ PerformthinningalgorithmofusingthesequenceofnSEs

▶ TakeonlytheconnectedcomponentcontainingA

A�B = AS(A~B)

Ac

{B} = (B1, . . . , Bn)

A Ac

Thinning Complementofthinning

Thickeningresults


Skeletoniza6on

Itisexpressedintermsoferosionsandopenings▶ Maximumdisk:largestdiskincludedinA,touchingtheboundaryofAattwoormore

differentplaces

▶ Skeleton:setofcentersofthemaximumdisks

Algorithm▶ UsingestructuringelementB

▶ Thek-thskeletonsubsetis

▶ Theskeletonistheunionofallthesubsets

▶ Wecanobtaintheoriginalimageusingdila:on

S(A) =K[

k=0

Sk(A)

Sk(A) = (A kB) \ (A kB)�B

K = max{k | (A \ kB) 6= ;}

A =K[

k=0

(Sk(A)� kB)


Skeletonexample1


Skeletoniza6onexample2


Applica6ontomedicalimagingVeryusefulinmedicalimagingtocomparepa:entsandhealthycontrolsbrainstructures.

Tract-BasedSpa:alSta:s:csTBSS

Documents

07 - Morphological operators€¦ · What are morphological operators? Local pixel transformaon for processing region shapes Most oTen used on binary images Logical transforma:on