Morphological filtering catalyst

morphological filtering for image enhancement and detection morphological filtering algorithm and How to turn on Morphological filtering using ccc
Dr.AlanCena Profile Pic
Dr.AlanCena,United States,Researcher
Published Date:30-10-2017
Your Website URL(Optional)
Comment
DiscreteAppliedMathematics216(2017)307–320 ContentslistsavailableatScienceDirect DiscreteAppliedMathematics journalhomepage:www.elsevier.com/locate/dam Morphologicalfilteringonhypergraphs a,∗ b BinoSebastianVadakkenveettil ,AvittathurUnnikrishnan , a c KannanBalakrishnan ,RamkumarPadinjarePisharathBalakrishna a DepartmentofComputerApplications,CochinUniversityofScienceandTechnology,India b RajagiriSchoolofEngineeringandTechnology,Cochin,India c DepartmentofMathematics,AdiShankaraInstituteofEngineeringandTechnology,India a r t i c l e i n f o a b s t r a c t Articlehistory: Inthisworkwestudytheframeworkofmathematicalmorphologyonhypergraphspaces. Received15April2014 Hypergraphswereintroducedinthe60sasanaturalgeneralizationofgraphs,whereedges Receivedinrevisedform5February2015 becomehyperedgesandcancontainmorethantwovertices.Mathematicalmorphology Accepted9February2015 isoneofthemostpowerfulframeworksforimageprocessing,andisheavilyusedfor Availableonline7March2015 manyapplications.However,morphologicaloperatorsonhypergraphspacesisnota conceptfullydevelopedintheliterature.Weconsiderlatticestructuresonhypergraphs Keywords: onwhichwebuildmorphologicaloperators.Weproposeseveralnewopenings,closings, Hypergraph granulometriesandalternatesequentialfiltersacting(i)onthesubsetsofthevertex Mathematicalmorphology andhyperedgesetofahypergraphand(ii)onthesubhypergraphsofahypergraph.We Granulometry illustratewithapplicationsinimageprocessingforfilteringobjectsdefinedonhypergraph Alternatingsequentialfilter spaces. ©2015ElsevierB.V.Allrightsreserved. 1. Introductionandrelatedwork Hypergraphswerefirstintroducedinthe1960sasageneralizationofgraphs1,whereedgesbecomehyperedgesand cancontainmorethantwovertices,andhavethenbeenintensivelystudied.Theyarewidelyusedinvariousfieldssuchas computerscience,gametheory,databases,datamining,optimization,imageprocessingandsegmentation4,6,10. Hypergraphscanbeconsideredasanaturalgeneralizationofgraphsinthesensethatagraphisa2-uniformhypergraph. Asimplicialcomplexcanbeconsideredasaspecificinstanceofahypergraph,whichcontainallsubsetsofeveryhyperedge, andagraphisa1-complex.Inthepast,severalauthorsstudiedmorphologicaloperatorsongraphs9,24andsimplicial complexes11,20.However,veryfewstudiesexistaboutbasicmorphologicaloperatorsonhypergraphs3–5,23and nonedealwiththefilteringproblem.Theobjectiveofthisarticleistohelpbridgingthisgap.Wedevelopaframework forbuildingmorphologicaloperatorsonhypergraphspaces.Asmainexamplesofapplication,wepresentasetofoperators (erosionsdilations,openingsclosings,granulometriesantigranulometriesandalternatesequentialfilters)acting(i)onthe subsetsofthevertexandhyperedgesetofahypergraphand(ii)onthesubhypergraphsofahypergraph. Thispaperisanextensionofthearticle21,whereadjunctionsandbasicmorphologicaloperatorswereintroduced. Theseoperatorscanbeusedasbuildingblocksfornewoperatorsandfilters.Inthiswork,weexploretheseoperators, introducingcompositionpropertiesanddefiningmorphologicalfilters.Wealsorevisittherelatedwork,showingthatmost oftheoperatorsfromtheliteraturecanbeexpressedusingouroperatorsbythesuitablechoiceofhyperedges. ∗ Correspondingauthor.Tel.:+919446213379. E-mailaddresses:binosebastianvgmail.com(B.SebastianVadakkenveettil),unnikrishnan_alive.com(A.Unnikrishnan), mullayilkannangmail.com(K.Balakrishnan),rkpbmathsyahoo.co.in(R.PadinjarePisharathBalakrishna). http://dx.doi.org/10.1016/j.dam.2015.02.008 0166-218X/©2015ElsevierB.V.Allrightsreserved.308 B.SebastianVadakkenveettiletal./DiscreteAppliedMathematics216(2017)307–320 2. Preliminaries Thegoalofthisworkistoexploremathematicalmorphologyonhypergraphspaces.Tothisend,westartbyrecalling basicdefinitionsandresultsaboutmathematicalmorphologyandhypergraphs. 2.1.Mathematicalmorphology Letusbrieflyrecallsomealgebraictoolsthatarefundamentalinmathematicalmorphology9,13,14,18,21.Giventwo latticesL andL,anyoperatorδ :L →L thatdistributesoverthesupremumandpreservestheleastelementiscalled 1 2 1 2 adilation(i.e.∀ε ⊆ L ,δ( ∨ ε)=∨ δ(X)X ∈ε).Similarlyanoperatorthatdistributesovertheinfimumandpreserves 1 1 2 thegreatestelementiscalledanerosion. Twooperatorsϵ :L →L andδ :L →L formanadjunction (ϵ,δ) ,ifforanyX ∈L andanyY ∈L,wehave 1 2 2 1 1 2 δ(X)≤ Y ⇔X≤ ϵ( Y),where≤ and≤ denotetheorderrelationsinL andL respectively13.Giventwooperators 1 2 1 2 1 2 ϵ andδ ,ifthepair(ϵ,δ) isanadjunction,thenϵ isanerosionandδ isadilation.If≤ ,≤ and≤ arethreelatticesandif 1 2 3 ′ ′ ′ ′ δ :L →L,δ :L →L,ϵ :L →L andϵ :L →L arefouroperatorssuchthat(ϵ,δ) and(ϵ ,δ )areadjunctions, 1 2 2 3 2 1 3 2 ′ ′ thenthepair(ϵ ◦ ϵ ,δ ◦ δ) isalsoanadjunction. Giventwocomplementedlattices,L andL,twooperatorsα andβ aredualwithrespecttothecomplementofeach 1 2 other,ifforeachX ∈ L,wehaveβ( X) = α( X).Ifα andβ aredualofeachother,thenβ isanerosionwheneverα isa 1 dilation. 2.2.Morphologicaloperatorsonhypergraphs • × • × Wedefineahypergraph1,3,21asapairH =(H ,H )whereH isasetofpointscalledverticesandH iscomposedof • × × afamilyofsubsetsofH calledhyperedges.WedenoteH byH =(e) whereIisafinitesetofindices.Thesetofvertices i i∈I • × formingthehyperedgeeisdenotedbyv(e).AhypergraphX =(X ,X )iscalledasubhypergraphofH,denotedbyX ⊆ H, • • × × ifX ⊆ H andX ⊆ H . • × • × HereaftertheworkspaceisahypergraphH = (H ,H )andweconsiderthesetsH ,H andHofrespectivelyall • × subsetsofH ,allsubsetsofH andallsubhypergraphsofH. Theset H ofallsubhypergraphsofahypergraphH formacompletelattice23. H isnotcomplementedasthe • × complementofasubhypergraphofHneednotbeasubhypergraphofH.H andH arebooleanlattices. LetS,S bethesetsofrespectivelythesubhypergraphsmadeofasinglevertexandthesubhypergraphscomposedof 0 1 • × theverticesofasinglehyperedgetogetherwiththathyperedge.i.e.,S = (x,φ) x ∈H andS = (v(e),e)e ∈H . 0 1 LetS =S ∪S. 0 1 Property1.ThelatticeHissupremumgeneratedbythesetS =S ∪S. 0 1 Proof.LetX ∈HbeanysubhypergraphofHandletF = X ,...,X isthefamilyofallelementsinSthataresmaller 1 ℓ × • thanX.Thenwehave,X =(∪ X ,∪ X ).ThereforeHissupremumgeneratedbythesetS.  i∈1,ℓ i∈1,ℓ i i Wedefinemorphologicaloperatorsontheselatticesbydefiningavertex–hyperedgecorrespondence21.Composing • × thesemappingsproducesmorphologicaloperatorsonthelatticesH ,H andH. • • × × × • • × • × × Definition1(21).LetX ⊆ H andX ⊆ H ,whereX = (e),j ∈JsuchthatJ ⊆ I.Thenδ ,ϵ :H →H andϵ ,δ : j • × H →H aredefinedasfollows. • × 1. δ (X )= ∪v(e); j j∈J × • • 2. ϵ (X )=e,i∈Iv(e)⊆ X ; i i • × 3. ϵ (X )= ∩v(e); j j∉ J × • • 4. δ (X )=e,i∈Iv(e)∩X = ̸ φ . i i Remark1(RevisitingtheRelatedWorkonGraphs).Theseoperatorsarethestraightforwardgeneralizationoftheoperators presentedinCousty9forgraphs,consideringagraphasa2-uniformhypergraph.Thusalltheoperatorspresentedinthis paperholdgoodforlatticesdefinedongraphs. Property2(Dilation,Erosion,Adjunction,Duality21). × × • • 1.Operatorsϵ andδ (resp.ϵ andδ )aredualofeachother. × • • × 2.Both(ϵ ,δ )and(ϵ ,δ )areadjunctions. • × 3.Operatorsϵ andϵ areerosions. • × 4.Operatorsδ andδ aredilations.B.SebastianVadakkenveettiletal./DiscreteAppliedMathematics216(2017)307–320 309 • • × • × Definition2(VertexDilation,VertexErosion21).Wedefineδ andϵ thatactonH byδ =δ ◦ δ andϵ =ϵ ◦ ϵ . • • Property3(21).ForanyX ⊆ H . • • • 1. δ(X )=x∈H ∃e,i∈Isuchthatx∈v(e)andv(e)∩X ̸=φ . i i i • • • 2. ϵ( X )=x∈H ∀e,i∈Isuchthatx∈v(e),v(e)⊆ X . i i i × × • Definition3(Hyper-edgeDilation,Hyper-edgeErosion21).Wedefine △and εthatactonH by △ = δ ◦ δ and × • ε=ϵ ◦ ϵ . × × × Property4(21).ForanyX ⊆ H ,X =(e) . j j∈J × 1. △(X )=e,i∈I∃e,j∈Jsuchthatv(e)∩v(e)= ̸ φ . i j i j × 2. ε(X )=e,j∈Jv(e)∩v(e)=φ, ∀i∈I\J. j j i Remark2(RevisitingtheRelatedWorkonHypergraphs).Blochetal.5defineddilation δ on (P(E),⊆ )whereEisthe hyperedgesetofahypergraphasδ(A) = e ∈Ev(A)∩v(e) ̸= φ ,∀A ⊆ E.Weachievethisasthehyper-edgedilation × • • definedbythecomposition△ = δ ◦ δ inDefinition3.Thedilationsdefinedinexamples3and4of3areδ andthe • × • compositionδ ◦ δ ◦ δ respectively. Definition4(HypergraphDilation,HypergraphErosion21).Wedefinetheoperators δ, △and ϵ,ε byrespectively • × • × δ, △(X)=(δ(X ),△(X ))andϵ,ε (X)=(ϵ( X ),ε(X )),foranyX ∈H. Theorem1(21). Theoperatorsδ, △andϵ,ε arerespectivelyadilationandanerosionactingonthelattice(H,⊆ ). Theorem2(21). (ϵ,ε ,δ, △)isanadjunction. • × • × × Letα (X)=(H ,X )andβ (X)=(δ (X ),X ),forX ∈H. 1 1 Theorem3. (α ,β )isanadjunction. 1 1 Proof.LetX,Y ∈H.Then, • × • × × β (X)⊆ Y ⇔ δ (X )⊆ Y andX ⊆ Y 1 • • × × • • × • • ⇔X ⊆ H andX ⊆ Y (sinceX ⊆ δ (X )⊆ Y ⊆ H ) ⇔X ⊆ α (Y). 1 Thus(α ,β )isanadjunction.  1 1 • × • • Letα (X)=(X ,ϵ (X ))andβ (X)=(X ,φ) ,forX ∈H. 2 2 Theorem4. (α ,β )isanadjunction. 2 2 • • × × • • × • × • Proof.LetX,Y ∈H.Ifβ (X)⊆ Ythen,X ⊆ Y andφ ⊆ Y .Sinceϵ isincreasing,X ⊆ Y impliesϵ (X )⊆ ϵ (Y ). 2 × × • • • × × • AlsoX ⊆ ϵ (X ).ThusX ⊆ Y andX ⊆ ϵ (Y ).ThereforeX ⊆ α (Y). 2 • • × × • • • × Conversely,ifX ⊆ α (Y)thenX ⊆ Y andX ⊆ ϵ (Y ).HenceX ⊆ Y andφ ⊆ Y .Thereforeβ (X) ⊆ Y.Thus 2 2 (α ,β )isanadjunction.  2 2 • × × • × • × • × • Letα (X)=(ϵ (X ),ϵ ◦ ϵ (X ))andβ (X)=(δ ◦ δ (X ),δ (X )),forX ∈H. 3 3 Theorem5. (α ,β )isanadjunction. 3 3 • × • • × • × × • × Proof.LetX,Y ∈ Hbesuchthatβ (X) ⊆ Y.Thenδ ◦ δ (X ) ⊆ Y andδ (X ) ⊆ Y .δ (X ) ⊆ Y impliesthat 3 • • × • × • • × • • × • × X ⊆ ϵ (Y ),since(ϵ ,δ )isanadjunction.Bythedefinitionofδ wehave,δ (X ) ⊆ X .Thusδ (X ) ⊆ ϵ (Y ).Since × • × × • × bytheadjunctionpropertyof(ϵ ,δ ),wehaveX ⊆ ϵ ◦ ϵ (Y ).Thereforeβ (X)⊆ Y ⇒X ⊆ α (Y). 3 3 • • × × × • × • • × Conversely,supposethatX ⊆ α (Y).Thatis,X ⊆ ϵ (Y )andX ⊆ ϵ ◦ ϵ (Y ).NowX ⊆ ϵ (Y )impliesthat 3 × • × • × • • × • • × δ (X ) ⊆ Y bytheadjunctionpropertyof(ϵ ,δ ).Sinceδ isadilation,itisincreasingandsoδ ◦ δ (X ) ⊆ δ (Y ). • × • • • × • • Nowδ (Y )⊆ Y bythedefinitionofδ .Thereforeδ ◦ δ (X )⊆ Y andsoβ (X)⊆ Y.Hence(α ,β )isanadjunction.  3 3 3 3. Filters Inmathematicalmorphology,afilter9,17isanoperatorα actingonalatticeL,whichisincreasing(i.e.∀X,Y ∈ L, X ≤ Y =⇒ α( X) ≤ α( Y))andidempotent(i.e. ∀X ∈ L,α(α( X)) = α( X)).AfilteronLwhichisextensive(i.e. ∀X ∈ L,X ≤ α( X))iscalledaclosingonLandafilteronLwhichisanti-extensive(i.e.∀X ∈ L,α( X) ≤ X)iscalled anopening.If(α,β) isanadjunctionthenα isanerosion,β isadilation,β ◦ α isanopeningandα ◦ β isaclosingonL.310 B.SebastianVadakkenveettiletal./DiscreteAppliedMathematics216(2017)307–320 Definition5(Opening,Closing). • 1.Wedefineγ andφ ,thatactonH ,byγ =δ ◦ ϵ andφ =ϵ ◦ δ . 1 1 1 1 × 2.WedefineΓ andΦ ,thatactonH ,byΓ =∆ ◦ εandφ =ε◦ ∆ . 1 1 1 1 • × • 3.Wedefineγ, Γ andφ, Φ ,thatactonHbyrespectivelyγ, Γ (X)=(γ (X ),Γ (X ))andφ, Φ (X)=(φ (X ), 1 1 1 1 1 1 1 × Φ (X ))foranyX ∈H. 1 TheopeningsandclosingsareillustratedinFig.1(e)–(q).ThechoiceofHisinsuchawaythateveryhyperedgeofH (Fig.1(a))isincidentwithexactlyfourvertices,andthechoiceofX(Fig.1(b))ismadetopresentarepresentativesampleof thedifferentpossibleconfigurationsonsubhypergraphs21.See21foranillustrationofvariousdilationsanderosions. Proposition1.Thefollowingstatementsaretrue. 1. γ, Γ =δ, ∆ ◦ ϵ,ε 1 2. φ, Φ =ϵ,ε ◦ δ, ∆ . 1 Proof.LetXbeanyhypergraphinH.Then • × γ, Γ (X) = (γ (X ),Γ (X )) 1 1 1 • × = ((δ ◦ ϵ)( X ),(∆ ◦ ε)(X )) • × • × = (δ(X ),∆ (X ))◦ (ϵ( X ),ε(X )) = δ, ∆ ◦ ϵ,ε (X). Thisproves1.Asimilarlineofargumentswillprove2.  (ϵ,ε ,δ, ∆ )isanadjunctiononHimpliesthatδ, ∆ ◦ ϵ,ε isanopeningandϵ,ε ◦ δ, ∆ isaclosingonH. Definition6(Half-opening,Half-closing). • • × • × 1.Wedefineγ andφ ,thatactonH ,byγ =δ ◦ ϵ andφ =ϵ ◦ δ . 1/2 1/2 1/2 1/2 × × • × • 2.WedefineΓ andΦ ,thatactonH ,byΓ =δ ◦ ϵ andφ =ϵ ◦ δ . 1/2 1/2 1/2 1/2 • × 3.Wedefineγ, Γ andφ, Φ ,thatactonHbyrespectivelyγ, Γ (X)=(γ (X ),Γ (X ))andφ, Φ (X) 1/2 1/2 1/2 1/2 1/2 1/2 • × =(φ (X ),Φ (X ))foranyX ∈H. 1/2 1/2 • • × × Property5.LetX ⊆ H andX ⊆ H .Thefollowingpropertiesaretrue. • 1. γ (X )= ∪ v(e) 1/2 i • i∈I,v(e)⊆ X i • • • 2. φ (X )=x∈H ∀e,i∈Iwithx∈v(e)andv(e)∩X = ̸ φ . 1/2 i i i × × × 3. Γ (X )=e,i∈I∃x∈v(e)withe ∈H withx∈v(e)⊆ X . 1/2 i i i i × 4. φ (X )=e,i∈Iv(e)⊆ ∪ v(e). 1/2 i i j j∈J • • × • Proof. 1. γ (X )=δ ◦ ϵ (X ) 1/2 • • × = δ ◦ e,i∈Iv(e)⊆ X (Bypropertyofϵ ) i i • = ∪ v(e)(Bypropertyofδ ). i • i∈I,v(e)⊆ X i • • × • 2. φ (X )=ϵ ◦ δ (X ) 1/2 • • = ϵ ◦ e,i∈Iv(e)∩X = ̸ φ i i • • = ϵ ◦ e ,k∈K;whereK ⊆ Iissomeindexsetande issuchthatv(e )∩X ̸=φ k k k × • (Bypropertyofδ (X )) = ∩ v(e ). k k∉ K × × • × 3. Γ (X )=δ ◦ ϵ (X ) 1/2 • × × = e,i∈Iv(e)∩ϵ (X )= ̸ φ (Bypropertyofδ ) i i × × = setofalledgesinH whichdonotbelongtoanyedgeofX × × = e,i∈I∃x∈v(e)withe ∈H withx∈v(e)⊆ X . i i i i × × • × 4. φ (X )=ϵ ◦ δ (X ) 1/2 × = ϵ ◦∪ v(e) j j∈J = e,i∈Iv(e)⊆ ∪ v(e).  i i j j∈JB.SebastianVadakkenveettiletal./DiscreteAppliedMathematics216(2017)307–320 311 (a)H. (b)X. (c)δ, ∆ (X). (d)ϵ,ε (X). (e)γ 1. (f)φ 1. (g)Γ 1. (h)ϕ1. (i)γ, Γ 1. (j)φ,ϕ 1. (k)γ 1/2. (l)φ 1/2. (m)Γ 1/2. (n)ϕ1/2. (o)γ, Γ 1/2. (p)φ,ϕ 1/2. Fig.1. Illustrationofopeningsandclosings. Remark3.Thefollowingstatementsaretrueaboutγ . 1/2 • • • 1. γ (X )=x∈X ∃e,i∈Iwithx∈v(e)andv(e)⊆ X . 1/2 i i i • • • • 2. γ (X )=X \x∈X ∀e,i∈Iwithx∈v(e)andv(e)⊈ X . 1/2 i i i312 B.SebastianVadakkenveettiletal./DiscreteAppliedMathematics216(2017)307–320 Nowwewillenumeratesomepropertiesoftheseoperatorsontherespectivelatticesbasedonthepartialorderrelations definedonthem. • • × × Property6.LetX ⊆ H andX ⊆ H .Thefollowingpropertiesholdtrue. • • • • • 1. γ (X )⊆ γ (X )⊆ X ⊆ φ (X )⊆ φ (X ). 1 1/2 1/2 1 × × × × × 2. Γ (X )⊆ Γ (X )⊆ X ⊆ Φ (X )⊆ Φ (X ). 1 1/2 1/2 1 3. γ, Γ (X)⊆ γ, Γ (X)⊆ X ⊆ φ, Φ (X)⊆ φ, Φ (X). 1 1/2 1/2 1 Proof. 1. • • γ (X ) = δ ◦ ϵ( X ) 1 • • = x∈H ∃e,i∈Isuchthatx∈v(e)andv(e)∩ϵ( X )̸=φ (Bypropertyofδ) i i i • • ⊆ x∈H ∃e,i∈Isuchthatx∈v(e)andv(e)⊆ X i i i = ∪ v(e) i • v(e)⊆ X i • = γ (X ). 1/2 • • • • • • Nowγ (X )=∪ v(e)⊆ X .AlsoX ⊆ φ (X ),becauseφ isaclosingonH . 1/2 v(e)⊆ X i 1/2 1/2 i • • × • φ (X ) = ϵ ◦ δ (X ) 1/2 • × • × = ϵ ◦ I◦ δ (X ),(whereIistheidentityonH ) • × • × • × × × • × × • ⊆ ϵ ◦ (ϵ ◦ δ )◦ δ (X )(∵I(X )=X ⊆ ϵ ◦ δ (X )andϵ ◦ δ isaclosing) • × • × • = (ϵ ◦ ϵ )◦ (δ ◦ δ )(X ) • = ϵ ◦ δ(X ) • = φ (X ). 1 Thisproves1. Properties2and3canbeprovedinasimilarmanner.  Property7(HypergraphOpening,HypergraphClosing). • × 1.Theoperatorsγ andγ (resp.Γ andΓ )areopeningsonH (resp.H )andφ andφ (resp.Φ andΦ )are 1/2 1 1/2 1 1/2 1 1/2 1 • × closingsonH (resp.H ). 2.ThefamilyHisclosedunderγ, Γ ,φ, Φ ,γ, Γ andφ, Φ . 1/2 1/2 1 1 3. γ, Γ andγ, Γ areopeningsonHandφ, Φ andφ, Φ areclosingsonH. 1/2 1 1/2 1 Proof. 1.If(α,β) isanadjunctiononalatticeLthenβ ◦ α isanopeningandα ◦ β isaclosingonL.Theresultfollows • × × • fromthefactthat(ϵ ,δ ),(ϵ ,δ ),(ϵ,δ) and(ε,∆ )areadjunctions. • × 2.Wewillprovethatγ, Γ (X)∈H,wheneverX ∈H.Sinceγ, Γ (X)=(γ (X ),Γ (X )),itisenoughtoshow 1/2 1/2 1/2 1/2 × • thatife∈Γ (X ),thenv(e)⊆ γ (X ). 1/2 1/2 × × • × e∈Γ (X ) ⇒e∈δ ◦ ϵ (X ) 1/2 × × • × • × × ⇒e∈X (sinceδ ◦ ϵ isanopeningδ ◦ ϵ (X )⊆ X ) × • × × • ⇒e∈ϵ (X )(sinceX ⊆ ϵ (X )). × • × • × • • Sincee ∈X ⇒ v(e) ⊆ δ (X ),wehavev(e) ⊆ δ ◦ ϵ (X ).Thusv(e) ⊆ γ (X ).HenceifXisahypergraph,then j j 1/2 γ, Γ (X)isahypergraphandsoHisclosedunderγ, Γ . 1/2 1/2 Hisclosedunderγ, Γ andφ, Φ duetoProposition1. 1 1 NowwewillprovethatHisclosedunderφ, Φ .ItisenoughtoshowthatifX ∈H,thenφ, Φ (X)∈H.We 1/2 1/2 • × × • knowthatφ, Φ (X)=(φ (X ),Φ (X )).Ife∈Φ (X ),weneedtoprovethatv(e)⊆ φ (X ). 1/2 1/2 1/2 1/2 1/2 × × • × e∈Φ (X ) ⇒e∈ϵ ◦ δ (X ) 1/2 • × ⇒e∈e,i∈Iv(e)⊆ δ (X ) i i • × ⇒ v(e)⊆ δ (X ). • × • • × • • × • • × • × • Butδ (X )=∪ × v(e)⊆ X .Sinceϵ ◦ δ isaclosing,X ⊆ ϵ ◦ δ (X ).Thereforeδ (X )⊆ ϵ ◦ δ (X ).Thus e∈X × • × • e∈Φ (X ) ⇒ v(e)⊆ ϵ ◦ δ (X ) 1/2 • ⇒ v(e)⊆ φ (X ). 1/2 Thusφ, Φ (X)∈H.HenceHisclosedunderφ, Φ . 1/2 1/2B.SebastianVadakkenveettiletal./DiscreteAppliedMathematics216(2017)307–320 313 3.Inordertoprovethat γ, Γ isanopening,itisenoughtoshowthat γ, Γ isincreasingandidempotenton 1/2 1/2 • • • × × × H.LetX,Y ∈ HbesuchthatX ⊆ Y ⊆ H.ThusX ⊆ Y ⊆ H andX ⊆ Y ⊆ H .Wehave γ, Γ (X) 1/2 • × =(γ (X ),Γ (X )).Therefore 1/2 1/2 • × γ, Γ ◦ γ, Γ (X) = γ, Γ ◦ (γ (X ),Γ (X )) 1/2 1/2 1/2 1/2 1/2 • × = (γ ◦ γ (X ),Γ ◦ Γ (X )) 1/2 1/2 1/2 1/2 • × = (γ (X ),Γ (X ))(sinceγ andΓ areopenings) 1/2 1/2 1/2 1/2 = γ, Γ (X). 1/2 Thereforeγ, Γ isidempotent. 1/2 • • × × Sinceγ andΓ areopenings,γ (X ) ⊆ γ (Y )andΓ (X ) ⊆ Γ (Y ).Thusγ, Γ (X) ⊆ γ, Γ (Y)so 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 thatγ, Γ isincreasingonH.Henceγ, Γ isanopening.Similarly,wecanprovethatγ, Γ isanopening,and 1/2 1/2 1 thatφ, Φ andφ, Φ areclosings.  1/2 1 4. Granulometries Granulometries9,17dealwithfamiliesofopeningsandclosingsthatareparametrizedbyapositivenumber.Afamily γ λ ∈NofopeningsactingonalatticeL,isagranulometrywhen,λ ≥ µ ⇒ γ ≤ γ .Similarly,afamilyofclosings λ λ µ φ λ ≥ 0isananti-granulometryif,giventwopositiveintegersλ andµ ,wehaveλ ≥ µ ⇒ φ ≥ φ .Theseconditions λ λ µ arecalledMatheron’saxiomsforgranulometry. i j i Definition7.Letλ ∈N.Wedefineγ, Γ (resp.φ, Φ )asfollows.γ, Γ = δ, ∆ ◦ (γ, Γ ) ◦ ϵ,ε (resp. λ/ 2 λ/ 2 λ/ 2 1/2 i j i φ, Φ =δ, ∆ ◦ (φ, Φ )◦ ϵ,ε ),whereiandjarerespectivelythequotientandremainderwhenλ isdividedby2. λ/ 2 1/2 Theorem6.Thefamilyγ, Γ λ ∈N(resp.φ, Φ λ ∈N)isagranulometry(resp.anti-granulometry). λ/ 2 λ/ 2 i i Proof.Weknowthatif(α,β) isanadjunction,then(α ◦ α,β ◦ β) isanadjunctionandso(α ,β )isalsoanadjunction i i foreveryi∈N14.Now(ϵ,ε ,δ, ∆ )isanadjunctionforeveryi∈N,since(ϵ,ε ,δ, ∆ )isanadjunctiononH.This i i i i impliesδ, ∆ ◦ ϵ,ε isanopening.Butδ, ∆ ◦ ϵ,ε =γ, Γ .Thisimpliesγ, Γ isanopeningifλ iseven.Ifλ 2i/2 λ/ 2 i i isodd(i.e.ifλ = (2i+1)/2)thenγ, Γ = γ, Γ = δ, ∆ ◦ γ, Γ ◦ ϵ,ε .If(α,β) isanadjunctionand λ/ 2 (2i+1)/2 1/2 i i γ isanopeningonalatticeL,thenβ ◦ γ ◦ α isanopening13.Since(ϵ,ε ,δ, ∆ )isanadjunctionandγ, Γ isan 1/2 i i opening,wehaveδ, ∆ ◦ γ, Γ ◦ ϵ,ε isanopening. 1/2 Nowweneedtoprovethatγ, Γ (X) ⊆ γ, Γ (X),forλ ≤ µ,λ,µ ∈ NandX ∈ H.Wehaveγ, Γ (X) ⊆ µ/ 2 λ/ 2 1 i i γ, Γ (X) ⊆ X,foreveryX ∈H.(ByProperty6).ReplacingXwithϵ,ε (X),wehaveγ, Γ ◦ ϵ,ε (X) ⊆ γ, Γ 1/2 1 1/2 i i i i i i ◦ ϵ,ε (X)⊆ ϵ,ε (X).Nowδ, ∆ isadilation,because(ϵ,ε ,δ, ∆ )isanadjunction.Thisimpliesδ, ∆ ◦ γ, Γ ◦ 1 i i i i i i+1 i+1 i ϵ,ε (X) ⊆ δ, ∆ ◦ γ, Γ ◦ ϵ,ε (X) ⊆ δ, ∆ ◦ ϵ,ε (X).Thatisδ, ∆ ◦ ϵ,ε (X) ⊆ δ, ∆ ◦ γ, Γ ◦ 1/2 1/2 i i i ϵ,ε (X) ⊆ δ, ∆ ◦ ϵ,ε (X).Henceγ, Γ (X) ⊆ γ, Γ (X) ⊆ γ, Γ (X)foreveryi ∈N.Thisimplies (2i+2)/2 (2i+1)/2 2i/2 γ, Γ (X)⊆ γ, Γ (X)foreveryµ ≥ λ ,µ,λ ∈N.Thereforethefamilyγ, Γ λ ∈Nisagranulometry.Similar µ/ 2 λ/ 2 λ/ 2 lineofargumentscanbeusedtoprovethatφ, Φ λ ∈Nisananti-granulometry.  λ/ 2 Definition8.Letλ ∈NandX ∈H.WedefinetheoperatorASF by λ/ 2  X ifλ =0 ASF (X)= λ/ 2 γ, Γ ◦ φ, Φ ◦ ASF (X) ifλ = ̸ 0. λ/ 2 λ/ 2 (λ − 1)/2 ItispossibletodefineasecondfamilyofoperatorssimilartoASF =ASF λ ∈Nbyreplacingthesequenceγ, Γ ◦ λ/ 2 λ/ 2 φ, Φ bythesequenceφ, Φ ◦ γ, Γ .ThefollowingresultisadirectconsequenceofTheorem6. λ/ 2 λ/ 2 λ/ 2 Corollary1.ThefamilyASF λ ∈Nisafamilyofalternatesequentialfilters. λ/ 2 Remark4.GivenahypergraphX ∈ H,itmustbenoticedthatthevertexset(resp.hyperedgeset)ofγ, Γ (X)and λ/ 2 φ, Φ (X)dependsonlyonthevertexset(resp.hyperedgeset)ofX.ThusDefinition7alsoinducesgranulometriesand λ/ 2 • × • alternatingsequentialfiltersonH andH .Ifweconsiderthecasewhenλ iseven,observethatwhenH isasubsetof 2 × thegridpointsZ andH ischosenasinFig.1(a),thenthevertexpartsofγ, Γ andφ, Φ correspondtotheusual λ/ 2 λ/ 2 openingandclosingbya3× 3squarestructuringelement22ofsizeλ/ 2.Alsonotethattheframeworkpresentedinthis paperallowsvariouschoicesforstructuringelementsthroughtheselectionofthehyperedges.Hence,wecanseethat,in thecaseofasetofpoints,theproposedframeworkcompletesthegranulometriesandfilterswhichareclassicallyusedin applicationsbyconsideringoddvaluesofλ . 5. Illustrationonbinaryregularimages Inthissectionweconsidertheapplicationofouralternatingsequentialfiltersonbinaryimages.Forthisend,weneed tocreateahypergraphbasedontheimage.Severalmethodsincludingimageadaptiveneighborhoodhypergraphs6–8,10314 B.SebastianVadakkenveettiletal./DiscreteAppliedMathematics216(2017)307–320 (a)Hyperedgesforming3uniform (b)Hyperedgesforming4uniform (c)Hyperedgesforming5uniform hypergraphs. hypergraphs. hypergraphs. Fig.2. Illustrativeexamplesforthemethodusedtoconstructhypergraphsbasedonaregularimage. (a)Originalimage. (b)Noisyversion,MSE =19.56%. Fig.3. Originaltestimageanditsnoisyversion9,11,20. canbeusedandthechoiceishighlyapplicationdependent.Fortheillustrationsinimageprocessingshowninthispaper • therestrictionofthefilterstoH areconsidered. 5.1.Generatinghypergraphsfromimages Inthiswork,wechoosetocreateavertexforeachpixel.Hyperedgesareplacedamongacollectionofvertices.Avarietyof methodscanbeusedtoformthehyperedges.Forillustrationandcomparisonofouroperators,weformuniformhypergraphs asshowninFig.2. Fig.2(a),correspondtoahexagonalgridinwhicheachvertexbelongtoexactlysixhyperedges.Thisstructureresembles therelatedworkonsimplicialcomplexes20,byconsideringonlythe2-simplicesortriangles.Forclarityofvisualization, wedrawthehyperedgesastriangles,whereeachhyperedgeisincidentwithverticesatthecornerofthetriangles.Fig.2(b) isthehypergraphusedthroughoutthisworkforillustration.Hereeachvertexbelongtoexactlyfourhyperedges.InFig.2 (c)weforma5-uniformhypergraphbyselectingthe4-connectedneighborhoodofeachpixel.Eachvertexbelongtoexactly fivehyperedgesinthisrepresentation.Forclarityofvisualization,wedrawonlyonehyperedgeinFig.2(c). WeusethehypergraphsgiveninFig.2fortheillustrationofimagefiltering.Notethatallthesehypergraphsareuniform (r-uniformforr =3,4and5)hypergraphs. • Remark5(RestrictingtheFilterstoH ).Wegenerateahypergraphfromanimageasillustratedabove.FromRemark4we • canseethatDefinition7inducesgranulometriesandalternatingsequentialfiltersonH .Thiscompletesgranulometries andalternatingsequentialfiltersonimages. • • • Remark6(FiltersonH ).IfHisanyoneofther-uniformhypergraphsillustratedinFig.2(a)to(c)andletX ⊆ H .Then ′ rc rc • werepresentASF (resp.ASF )bythealternatingsequentialfilterdefinedonthelatticeH ,consideringaclosing(resp. λ/ 2 λ/ 2 opening)operatorfirst.ThusbyDefinition8,wehave  • X ifλ =0 rc • ASF (X )= rc • λ/ 2 φ ◦ γ ◦ ASF (X ) ifλ = ̸ 0 λ/ 2 λ/ 2 (λ − 1)/2B.SebastianVadakkenveettiletal./DiscreteAppliedMathematics216(2017)307–320 315 ′ 3c 3c (a)ASF ,MSE =1.86%. (b)ASF ,MSE =2.68%. 5/2 7/2 ′ 4c 4c (c)ASF ,MSE =2.31%. (d)ASF ,MSE =4.27%. 5/2 7/2 ′ 5c 5c (e)ASF ,MSE =2.59%. (f)ASF ,MSE =5.45%. 3/2 5/2 Fig.4. IllustrationofthebestresultsobtainedwiththeoperatorsbasedonhypergraphASF. and  • ′ X ifλ =0 rc • ′ ASF (X )= rc • λ/ 2 γ ◦ φ ◦ ASF (X ) ifλ ̸=0 λ/ 2 λ/ 2 (λ − 1)/2 i j i i j i where,γ =δ ◦ (γ ) ◦ ϵ (resp.φ =δ ◦ (φ ) ◦ ϵ ).Notethatiandjarerespectivelythequotientandremainder λ/ 2 1/2 λ/ 2 1/2 whenλ isdividedby2. 5.2.Illustrationonregularimages Inthissectionweconsidertheapplicationofouralternatingsequentialfiltersonregularimages. Tobeabletocompareourresultswiththeliterature,weconsiderthesameimagesusedbyCoustyetal.9andDias etal.11,20shownonFig.3.Weusethemeansquareerrorastheerrormeasure.Forbinaryimages,thisvalueisequivalent tothenumberofwrongpixelswithrespecttotheoriginalimageandwechoosetoexpressitinpercentage.Thenoisyimage shownonFig.3(b)hasMSEequalto19.56%.316 B.SebastianVadakkenveettiletal./DiscreteAppliedMathematics216(2017)307–320 (a)ClassicalASFwith9iterationsandtripleresolution (b)GraphASF 20,MSE =3.27%. 6/2 20,MSE =2.54%. c (c)SimplicialcomplexASF ,20MSE =1.91%. (d)Medianfilter,MSE =2.32%. 3 Fig.5. Comparisonwithsomeoftheliteratureresults. FirstweformahypergraphbasedontheimageusinganyoneoftheconfigurationsrepresentedinFig.2.Thenoisyimage shownonFig.3(b)wasthenprocessed,usingthealternatingsequentialfiltersuptofiltersofsize7,meaningseveniterations ′ rc rc fortheoperatorsASF andASF . λ/ 2 λ/ 2 Fig.4showstheresultingimagesfortheoperatorsontherespectivehypergraphswithbestresultsthatarebasedon ′ rc rc ASF andASF .Onalltheimages,wecanseethepresenceofsmallartifactsonthebackgroundandontheobject. InFig.5wecompareourresultswithsomeoftheresultsfromtheliteraturepresentedbyCoustyetal.9,11andDias etal.11,20theoriginalsourcesoftheimages.TheresultsareshowninFig.5(a),(b)and(c).Wealsopresenttheresultof Medianfiltering12inFig.5(d).Thebestresultformedianfilterisobtainedbytakinganeighborhoodofsize11with2.32% asMSE. Figs.6–8showthegraphoferrorversussizeofthefilterfortheconsideredhypergraphs.Thepointscorrespondingto rc theoperatorsthatapplyfirstaclosingoperator(ie.ASF )areconnectedwithablueline,andthepointscorrespondingto λ/ 2 ′ rc operatorsthatapplyfirstanopeningoperator(ie.ASF )areconnectedwithagreenline.Theoperatorsthatapplyfirsta λ/ 2 closingoperatorobtainedbetterresults,withsimilarresultsforallvariants. Fromtheresultspresentedinthissection,weconcludethatouroperatorsare,onthisexample,onacompetitivelevel withtheoperatorspresentedintheliterature. 6. Extensiontoweightedhypergraphs InthissectionweextendtheframeworkproposedinSections2–4tovertexandorhyperedgeweightedhypergraphs.This inturnallowustoworkwithgrayscaleimages.Givenagrayscaleimage,weformavertexweightedhypergraphbasedon theimage.Forillustration,weusethe4-uniformhypergraphgiveninFig.9(a).Theextensionoftheframeworktoweighted hypergraphsgeneratenewdilations,erosionsgranulometriesandAlternatingSequentialFilters. LetndenoteanypositiveintegerandK =0,...,c,n.LetEbeanyset.LetFun(E)denotethesetofallmapsfromEto K.Letk ∈KandletF ∈Fun(E).Theksection(k-threshold)ofFisthesubsetχ (F)ofEwhereχ (F) = x ∈EF(x) ≥ k. k k ThenFun(E)withtheorderrelation≤ inferredbythresholddecompositionfromtherelation⊆ onEisacompletelattice. Itssupremumandinfimumarecharacterizedbythresholddecomposition9. • Bythresholddecomposition,thelatticeHofallsubhypergraphsofHgiveninSection2.2inducesalatticeFun(H )⊗ × Fun(H )ofpairsoffunctionsweightingrespectivelytheverticesandthehyperedgesofHsuchthatthesimultaneous thresholdofthesetwofunctionsatanygivenlevelyieldsasubhypergraphofH.B.SebastianVadakkenveettiletal./DiscreteAppliedMathematics216(2017)307–320 317 ′ 3c 3c Fig. 6. MSEversussizeofthefilterfortheoperatorsASF andASF .(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderis referredtothewebversionofthisarticle.) ′ 4c 4c Fig. 7. MSEversussizeofthefilterfortheoperatorsASF andASF .(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderis referredtothewebversionofthisarticle.) • × TheoperatorsactingonthelatticesH ,H ,orH,presentedinSections2–4inducestackoperators2,15,16,19,25acting • × • × onthelatticesFun(H ),Fun(H ),andFun(H )⊗ Fun(H ).ThisimpliesthatanypropertypresentedinSections2–4,forop- • × • × • × eratorsonthelatticesH ,H ,orHalsoholdgoodforoperatorsonthelatticesFun(H ),Fun(H ),andFun(H )⊗ Fun(H ). Definition1presentedinSection2actastheelementarybuildingblockforalltheoperatorsthatfollows.Thefollowing • × • × definitionisthestackanaloguestoDefinition1,whichlocallycharacterizesδ ,ϵ ,ϵ ,δ onweightedhypergraphs. • • × × Definition9.LetF ∈Fun(H )andletF ∈Fun(H ). • × × × • δ (F )(x)= maxF (e)e ∈ H ∀x∈H i i x∈v(e) i × • • × ϵ (F )(e)=minF (x)x∈v(e)∀e ∈H i i i • × × × • ϵ (F )(x)= minF (e)e ∈H ∀x∈H i i x∈v(e) i × • • × δ (F )(e)=maxF (x)x∈v(e)∀e ∈H . i i i Thereforebyflatmathematicalmorphology,theopeningclosingandgranulometricpropertiesextendfromsetsto weightfunctionsbasedonDefinition9.Thisallowustoworkwithgrayscaledilationerosion,openingclosing,granu- lometriesandAlternatingSequentialFilters.318 B.SebastianVadakkenveettiletal./DiscreteAppliedMathematics216(2017)307–320 ′ 5c 5c Fig. 8. MSEversussizeofthefilterfortheoperatorsASF andASF .(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderis referredtothewebversionofthisarticle.) × • × (a)X. (b)δ . (c)δ =δ ◦ δ . • × × • × (d)φ =ϵ ◦ δ . (e)ϵ . (f)ϵ =ϵ ◦ ϵ . 1/2 Fig.9. Illustrationofpropagationofvertexweightsalonghyperedges. Weconsiderthevertexweightedhypergraphcorrespondingtoagrayscaleimagebyformingahypergraph.Asanillus- trationweformthehyperedgesasinFig.9(a)where,thenumberassociatedwithavertexrepresentstheintensityvalue ofthecorrespondingpixel.WepropagatevertexweightsalongthehyperedgesaccordingtoDefinition9toformadilation andanerosionrespectively,asillustratedinFig.9(c)and(f).HerewereassignthevertexweightsaccordingtoDefinition9 toformvariousoperators,asillustratedinFig.9(b)to(f). ′ rc rc NowweconsiderthestackextensionofthefamiliesASF andASF andapplyittoweightedhypergraphs.For λ/ 2 λ/ 2 thisend,weconsideragrayscaleimageshowninFig.10(a),whichisobtainedbyconvertinga944 × 847colorimage intoagrayscaleimage.Fig.10(b)isthegrayscaleimageobtainedbyaddingrandomimpulsenoisetothegrayscaleimageB.SebastianVadakkenveettiletal./DiscreteAppliedMathematics216(2017)307–320 319 (a)Originalimage. (b)Noisyimage. 3c (c)GraphASF ,MSE =33.19. (d)HypergraphASF ,MSE =17.91. 5/2 5/2 Fig.10. Illustrationoffilteringongrayscaleimages. 3c inFig.10(a).Fig.10(c)and(d)resp.showstheresultsgivenbytheGraphASF andHypergraphASF .VisuallyHyper- 5/2 5/2 3c graphASF removesmorenoisethanGraphASF .Moreprecisely,comparedtotheoriginalimageofFig.10(a),themean 5/2 5/2 squareerror(MSE)achievedbytheHypergraphfilterequals17.91,whereasitequals33.19fortheGraphASF .Weexpect 5/2 thefilteringresultstobeevenbetterwhenconsideringmorecomplexhyperedges. 7. Conclusionandfutureworks Thisarticleproposesaframeworkthatallowstobuildmorphologicaloperatorsforanalyzingandfilteringobjectsdefined onhypergraphspaces.Inparticular,weproposeasetofoperatorsthatareshowntobeusefulforimagefiltering.Weextend theproposedoperatorstoweightedhypergraphs,therebygeneratinganewsetofoperatorstoprocessgrayscaleimages. Theoperatorspresentedinthisworkareonlyasmallsampleofwhatcanbedoneusinghypergraphs.Futureworkincludes thedefinitionofmoreoperators,alongwithotherclassicalusesofmathematicalmorphology. Acknowledgments Wethanktheanonymousrefereesfortheiradvicetoimprovethequalityofthepaper.Theworkofthefirstauthor (onleavefromMarAthanasiusCollege,Kothamangalam)wassupportedbytheUniversityGrantsCommission(OrderNo. F.Fip/11thPlan/KLMG038TF05dated08-03-2011),Govt.ofIndia,undertheFIPscheme. References 1 ClaudeBerge,Hypergraphs:Combinatoricsoffinitesets,1989. 2 GillesBertrand,Ontopologicalwatersheds,J.Math.ImagingVision22(2–3)(2005)217–230.320 B.SebastianVadakkenveettiletal./DiscreteAppliedMathematics216(2017)307–320 3 IsabelleBloch,AlainBretto,Mathematicalmorphologyonhypergraphs:Preliminarydefinitionsandresults,in:DiscreteGeometryforComputer Imagery,Springer,2011,pp.429–440. 4 IsabelleBloch,AlainBretto,Mathematicalmorphologyonhypergraphs,applicationtosimilarityandpositivekernel,Comput.Vis.ImageUnderst.117 (4)(2013)342–354. 5 IsabelleBloch,AlainBretto,AurélieLeborgne,Similaritybetweenhypergraphsbasedonmathematicalmorphology,in:MathematicalMorphology andItsApplicationstoSignalandImageProcessing,Springer,2013,pp.1–12. 6 AlainBretto,J.Azema,HocineCherifi,BernardLaget,Combinatoricsandimageprocessing,Graph.ModelsImageProcess.59(5)(1997)265–277. 7 AlainBretto,HocineCherifi,DrissAboutajdine,Hypergraphimaging:anoverview,PatternRecognit.35(3)(2002)651–658. 8 AlainBretto,LucGillibert,Hypergraph-basedimagerepresentation,in:Graph-basedRepresentationsinPatternRecognition,Springer,2005,pp.1–11. 9 JeanCousty,LaurentNajman,FabioDias,JeanSerra,Morphologicalfilteringongraphs,Comput.Vis.ImageUnderst.117(4)(2013)370–385. 10 R.Dharmarajan,K.Kannan,Ahypergraph-basedalgorithmforimagerestorationfromsaltandpeppernoise,AEU-Int.J.Electron.Commun.64(12) (2010)1114–1122. 11 FabioDias,JeanCousty,LaurentNajman,Dimensionaloperatorsformathematicalmorphologyonsimplicialcomplexes,PatternRecognit.Lett.(2014). 12 RafaelC.Gonzalez,ERichard,Woods,DigitalImageProcessing,PrenticeHallPress,ISBN:0-201-18075-8,2002. 13 HenkJ.A.M.Heijmans,Composingmorphologicalfilters,IEEETrans.ImageProcess.6(5)(1997)713–723. 14 HenkJ.A.M.Heijmans,ChristianRonse,Thealgebraicbasisofmathematicalmorphologyi.dilationsanderosions,Comput.Vis.Graph.ImageProcess. 50(3)(1990)245–295. 15 RomainLerallut,ÉtienneDecencière,FernandMeyer,Imagefilteringusingmorphologicalamoebas,ImageVis.Comput.25(4)(2007)395–404. 16 PetrosMaragos,RonaldW.Schafer,Morphologicalfilters–parti:Theirset-theoreticanalysisandrelationstolinearshift-invariantfilters,IEEETrans. Acoust.SpeechSignalProcess.35(8)(1987)1153–1169. 17 LaurentNajman,HuguesTalbot,MathematicalMorphology,JohnWiley&Sons,2013. 18 ChristianRonse,Whymathematicalmorphologyneedscompletelattices,Signalprocess.21(2)(1990)129–154. 19 ChristianRonse,Flatmorphologyonpowerlattices,J.Math.ImagingVision26(1–2)(2006)185–216. 20 FabioAugustoSalveDias,Astudyofsomemorphologicaloperatorsinsimplicialcomplexspaces,(Ph.D.thesis)UniversitéParis-Est,2012. 21 Bino Sebastian, A. Unnikrishnan, Kannan Balakrishnan, P.B. Ramkumar, Mathematical morphology on hypergraphs using vertex-hyperedge correspondence,ISRNDiscreteMath.(2014). 22 FrankY.Shih,ImageProcessingandMathematicalMorphology:FundamentalsandApplications,CRCpress,2010. 23 JohnG.Stell,Relationsonhypergraphs,in:RelationalandAlgebraicMethodsinComputerScience,Springer,2012,pp.326–341. 24 LucVincent,Graphsandmathematicalmorphology,SignalProcess.16(4)(1989)365–388. 25 P.Wendt,EdwardJ.Coyle,NealC.GallagherJr.,Stackfilters,IEEETrans.Acoust.SpeechSignalProcess.34(4)(1986)898–911.

Advise: Why You Wasting Money in Costly SEO Tools, Use World's Best Free SEO Tool Ubersuggest.