Table Of ContentDimensional operators for mathematical morphology on simplicial complexes
F. Diasa,b,∗, J. Coustya, L. Najmana
aUniversit´e Paris-Est, Laboratoire d’Informatique Gaspard-Monge, Equipe A3SI, ESIEE, Paris, France
bCollege of Physical Education, State University of Campinas, Brazil
Abstract
In this work we study the framework of mathematical morphology on simplicial complex spaces.
Simplicial complexes are widely used to represent multidimensional data, such as meshes, that are two dimensional
4
complexes, or graphs, that can be interpreted as one dimensional complexes.
1
Mathematical morphology is one of the most powerful frameworks for image processing, including the processing of
0
2digitalstructures,andisheavilyusedformanyapplications. However,mathematicalmorphologyoperatorsonsimplicial
complex spaces is not a concept fully developed in the literature.
n
a Specifically, we explore properties of the dimensional operators,small, versatile operators that can be used to define
Jnewoperatorsonsimplicialcomplexes,whilemaintainingpropertiesfrommathematicalmorphology. Theseoperatorscan
2alsobeusedtorecovermanymorphologicaloperatorsfromtheliterature. Matlabcodeandadditionalmaterial,including
2theproofsoftheoriginalproperties,arefreelyavailableathttps://code.google.com/p/math-morpho-simplicial-complexes.
]Keywords: Mathematical morphology,simplicial complexes, granulometries, alternating sequential filters, image
M
filtering.
D
.
s1. Introduction and related work tant frameworks for non-linear image processing, provid-
c
[ ing tools for many applications. It was later extended
Simplicialcomplexeswerefirstintroducedby Poincar´e
byHeijmansandRonse(Heijmans and Ronse,1990)using
1in 1895 (Poincar´e, 1895) to study the topology of spaces
complete lattices, allowing the use of more complex digi-
v
of arbitrary dimension, and are basic tools for algebraic
2 tal structures, such as graphs (Cousty et al., 2009b, 2013;
topology(Maunder,1996),imageanalysis(Bertrand,2007;
0 Ta et al., 2008; L´ezorayand Grady, 2012; Vincent, 1989),
6Couprie and Bertrand,2009;Kong,1997)anddiscretesur- hypergraphs (Bloch and Bretto, 2011; Bloch et al., 2013;
5faces(Evako,1996;Evako et al.,1996;Daragon et al.,2005),
Stell, 2010) and simplicial complexes (Dias et al., 2011;
.among many other domains.
1 Dias, 2012; Lom´enie and Stamon, 2008).
0 In the form of meshes they are widely used in many The use ofa digitalstructure as supportto imagepro-
4contexts to express tridimensional data. Some graphs can
cessingisnotnew. InVincent(1989),Vincentusesthelat-
1be represented as a form of simplicial complexes, and we
tice approachto mathematicalmorphologyto define mor-
:
vcan build simplicial complexes based on regular, matri-
phological operators on neighborhood graphs, where the
icial, images. This versatility is the reasonwe chose to use
X graph structure is used to define neighborhood relation-
simplicial complexes as the operating space.
ships between unorganized data, expressed as vertices.
r
a Considering operators on simplicial complex spaces, it By allowing the propagation of values from vertices
is fairly common to change the complexity of the mesh
to the edges, therefore using the graph structure to ex-
structure (Chiang et al., 2011; De Floriani et al., 1999).
press more than just neighborhood relation, Cousty et
Evenwhenadditionaldataisassociatedwiththeelements
al. (Cousty et al., 2013) obtained different morphologi-
of the complex, they are mostly used to guide the change
caloperators,includingopenings,closingsandalternating
in the structure, the values themselves are not changed.
sequential filters. Those operators are capable of deal-
Here,wepursuitadifferentoption,ourobjectiveistofilter
ing with smaller noise structures, acting in a smaller size
values associated to the elements of the complex, without
than the classical operators. Similarly, Meyer and Staw-
changing its structure, using the frameworkof mathemat-
iaski(Meyer and Stawiaski,2009)andMeyerandAngulo(Meyer and
ical morphology.
2007) obtain a new approach to image segmentation and
Mathematical morphology was introduced by Math-
levellings, respectively.
eron and Serra in 1964 and it is one of the most impor-
Recently, Bloch and Bretto (Bloch and Bretto, 2011;
Bloch et al., 2013) introduced mathematical morphology
∗Correspondingauthor onhypergraphs,defininglatticesandoperators. Theirlat-
Email addresses: [email protected] (F.Dias), tices and operatorsare similar to the ones presented here,
[email protected] (J.Cousty),[email protected] (L.Najman)
Preprint submittedto Pattern Recognition Letters January 23, 2014
taking into account the differences between hypergraphs
and complexes.
This work is focused on mathematical morphology on
simplicial complexes, specifically to process values asso-
ciated to elements of the complex in an unified manner,
withoutalteringthestructureitself. InLom´enie and Stamon (a) (b) (c) (d)
(2008), Lom´enie and Stamon explore mathematical mor-
phologyoperatorsonmeshspacesfrompointspaces. How- Figure 1: Graphical representation of (a) a 0-simplex, (b) a 1-
ever, the complex only provides structural information, simplex,(c)a2-simplex,and(d)asmall2-complex.
while the information itself is associated only to triangles
or edges of the mesh.
Importantnotations. Inthiswork,thesymbolCdenotes
Our approach for mathematical morphology on sim-
anon-emptyn-complex,withn∈N. Thesetofallsubsets
plicial complexes has been studied before and this arti-
of C is denoted by P(C). Any subset of C that is also
cle is an extension of the conference article (Dias et al.,
a complex is called a subcomplex (of C). We denote by C
2011), where interesting new operators were introduced.
the set of all subcomplexes of C.
Theseoperators,calleddimensional operators canbeused
If X is a subset ofC, we denote by X the complement
as building blocks for new operators. In this work we ex-
ofX (inC):X =C\X. Thecomplementofasubcomplex
plore these operators, introducing composition properties
of C is usually not a subcomplex. Any subset X of C
anddefiningnew morphologicaloperators. Theproofsare
whosecomplementX is asubcomplex iscalledastar. We
omitted here, but they are available in Dias (2012). We
denotebySthesetofallstarsinC. TheintersectionC∩S
also revisit the related work, showing that most of the
is non-empty since it always contains at least ∅ and C.
operators from the literature can be expressed by the di-
In the domain of simplicial complexes, some opera-
mensional operators.
tors are well known, such as the closure and star (J¨anich,
1984). We define the closurexˆ andthe starxˇ ofasimplex
2. Basic theoretical concepts x as:
Theobjectiveofthisworkistoexplorethedimensional
operatorsformathematicalmorphologyonsimplicialcom- ∀x∈C, xˆ={y |y ⊆x,y 6=∅} (1)
plex spaces. To this end, we start by reminding useful ∀x∈C, xˇ={y ∈C |x⊆y} (2)
definitions about simplicial complexes and mathematical
morphology. Inotherwords,the closureoperatorgivesasresultthe
set of all simplices that are subsets of the simplex x, and
2.1. Simplicial complexes thestargivesasresultthesetofallsimplicesofC thatcon-
Oneofthemostknownformsofcomplexistheconcept tainthesimplexx. Theseoperatorscanbeeasilyextended
of mesh, often used to express tridimensionaldata on var- to sets of simplices. The operators Cl : P(C) → P(C)
ious domains, such as computer aided design, animation and St:P(C)→P(C) are defined by:
and computer graphics in general. However, in this work
weprefertoapproachcomplexesbythecombinatorialdef-
∀X ∈P(C), Cl= {xˆ|x∈X} (3)
inition of an abstract complex (J¨anich, 1984).
[
The basic element of a complex is a simplex. In this ∀X ∈P(C), St= {xˇ|x∈X} (4)
work, a simplex is a finite, nonempty set. The dimension
[
of a simplex x, denoted by dim(x), is the number of its
2.2. Mathematical morphology
elements minus one. A simplex of dimension n is also
We approach mathematical morphology through the
called an n-simplex. We call simplicial complex, or simply
framework of lattices (Ronse, 1990). We start with the
complex, any set X of simplices such that, for any x ∈
concept of partially ordered set. It is composed by a set
X, any non-empty subset of x also belongs to X. The
and a binary relation. The binary relation is defined only
dimension ofa complexis equaltothe greatestdimension
between certain pairs of elements of the set, represent-
of its simplices and, by convention, we set the dimension
ing precedence, and must be reflexive, antisymmetric and
of the empty set to −1. In the following, a complex of
transitive.
dimension n is also called an n-complex.
A lattice is a partially ordered set with a least up-
Figure 1(a) (resp. b, and c) graphically represents a
per bound, called supremum, and a greatest lower bound,
simplex x = {a} (resp. y = {a,b} and z = {a,b,c})
calledinfimum. Forinstance,thesetP(S)={{a,b,c},{a,b},{a,c},
of dimension 0 (resp. 1, 2). Figure 1(d) shows a set of
that is the power set of the set S = {a,b,c}, ordered by
simplices composed of one 2-simplex ({a,b,c}), three 1-
the inclusion relation, is a lattice. The supremum of two
simplices ({a,b},{b,c} and {a,c}) and three 0-simplices
elements of this lattice is given by the union operatorand
({a},{b} and {c}).
2
the infimum by the intersectionoperator. This lattice can 3. Dimensional operators
be denoted by hP(S), , ,⊆i.
InmathematicalmSorpThology(see,e.g.,Ronse and Serra InDias et al.(2011), weintroducedfournew basicop-
(2010)), any operator that associates elements of a lattice erators that act on simplices of given dimensions. These
L [1] to elements of a lattice L [2] is called a dilation if it operators can be composed into new operators which be-
commuteswiththesupremum. Similarly,anoperatorthat havior can be finely controlled. We proceed with a brief
commutes with the infimum is called an erosion. reminder of their definition andexplore some new proper-
Let L [1] and L [2] be two lattices whose order relations ties.
andsupremaaredenotedby≤ ,≤ , ∨ ,and∨ . Twoop- We start by introducing a new notation that allows
1 2 1 2
erators α : L →L and β : L →L form an adjunction onlysimplicesofagivendimensiontoberetrieved. LetX ⊆
2 1 1 2
(β,α) if α(a)≤1 b ⇔ a≤2 β(b) for every element a in L2 C and let i ∈ [0,n], we denote by Xi the set of all i-
(a2n0d10b))inthLa1t.,gIitveisnwtwelolokpneorwatnor(sseαe,ane.dg.β,,Rifotnhseepaanidr(Sβe,rαra) Csimi ipslitcheessoeftXo:fXalil =i-s{imxp∈licXes|odfimC(.x)W=eid}e.nIontepbayrtiPcu(Clair),
is an adjunction, then β is an erosion and α is a dilation. the set of all subsets of Ci.
Furthermore,ifαisadilation,thereisauniqueerosionβ, Leti∈Nsuchthati∈[0,n]. ThestructurehP(Ci), , ,⊆i
called the adjoint of α, such that (β,α) is an adjunction. is a lattice. S T
This erosion is characterizedby: Definition 1. Let i,j ∈ N such that 0 ≤ i < j ≤ n,
X ∈ P(C ) and Y ∈ P(C ). We define the operators δ+
i j i,j
andε+ actingfromP(C )intoP(C )andtheoperatorsδ−
∀a∈L [1], β(a)=∨2{b∈L [2]|α(b)≤1 a} (5) i,j i j j,i
and ε− acting from P(C ) into P(C ) by:
j,i j i
In certaincases,we will denote the adjoint operatorof
an operator α by αA, to explicit the relationship between
them. δi+,j(X)={x∈Cj |∃y ∈X,y ⊆x} (8)
In mathematical morphology, an operator α, acting ε+ (X)={x∈C |∀y ∈C ,y ⊆x⇒y ∈X} (9)
i,j j i
froma lattice L to L , thatis increasing(∀a,b∈L , a≤
1 2 1 δ−(Y)={x∈C |∃y ∈Y,x⊆y} (10)
b =⇒ α(a) ≤ α(b)) and idempotent (∀a ∈ L 1, α(a) = j,i i
α(α(a))) is a filter. If a filter is anti-extensive (∀a ∈ ε−(Y)={x∈C |∀y ∈C ,x⊆y ⇒y ∈Y} (11)
j,i i j
L , α(a) ≤ a) it is called an opening. Similarly, an ex-
1
tensive filter (∀a∈L , a≤α(a)) is called a closing. In other words, δ+(X) is the set of all j-simplices
1 i,j
One wayofobtainingopenings andclosingsis bycom- of C that include an i-simplex of X, δ−(X) is the set
j,i
biningdilationsanderosions(Ronse and Serra,2010). Let of all i-simplices of C that are included in a j-simplex
α : L → L be a dilation. Then the operator ζ = αAα is of X, ε+ (X) is the set of all j-simplices of C whose sub-
i,j
a closing and the operator ψ = ααA is an opening. Both setsofdimensioniallbelongtoX,andε−(X)isthesetof
operators act on L . alli-simplicesofC thatarenotcontainedj,iinanyj-simplex
A family of openings Ψ = {ψ | λ ∈ N} acting on
λ of X.
L , is a granulometry if, given two positive integers i and
The dimensionaloperatorscanalsobe recoveredusing
j, we have i ≥ j =⇒ ψ (a) ⊆ ψ (a), for any a ∈
i j the classical star and closure operators:
L (Ronse and Serra, 2010). Similarly, a family of closings
Z ={ζ |λ≥0},is aanti-granulometry if,giventwopos- Property 2. We have:
λ
itive integers i and j, we have i ≤ j =⇒ ζ (a) ⊆ ζ (a),
i j 1. ∀X ⊆C , δ+(X)=[St(X)] ;
for any a∈L . i i,j j
Afamilyoffilters{αλ,λ∈N}isafamilyofalternating 2. ∀X ⊆Cj, δj−,i(X)=[Cl(X)]i;
sequential filters if, giventwo positive integersi and j, we 3. ∀X ⊆C , ε+ (X)= St X ;
i i,j
have i>j =⇒ αiαj =αi. h (cid:0) (cid:1)ij
Let Ψ = {ψλ, λ ∈ N} be a granulometry and Z = 4. ∀X ⊆Cj, ε−j,i(X)= Cl X .
{ζλ, λ ∈ N} be an anti-granulometry. We can construct h (cid:0) (cid:1)ii
two alternating sequential filters by composing operators Thedimensionaloperatorscanbeusefulwhenthecon-
from both families. Let i∈N and a∈L : sidered data is associated only with simplices of a given
dimensionofthecomplex,whichisfairlycommon. Inthis
situation,theseoperatorscanbeusedtopropagatetheval-
νi(a)=(ψiζi)(ψi−1ζi−1)...(ψ1ζ1)(a) (6) ues to the other dimensions of the complex, or even filter
ν′(a)=(ζ ψ )(ζ ψ )...(ζ ψ )(a) (7) the values directly, depending on the application. How-
i i i i−1 i−1 1 1
ever, since the objective of this work is to find interesting
operatorsactingonsubcomplexes,wemostlyusetheseop-
eratorsasbuildingblockstodefinenewoperators. Thefol-
lowingadjunctionpropertycanbe provedbyconstructing
3
the adjoint erosion of the dilation operators and verifying 3.1. Extension to weighted complexes
that they correspond to the provided erosion, the proper- In this section, we extend the operators we defined to
ties regarding duality are trivial results from property 2. weighted simplicial complexes. Let k and k be two
min max
Property 3. Let i,j ∈N such that 0≤i<j ≤n. distinct, positive integers. We define the set K as the set
1. The pairs (ε+ ,δ−) and (ε−,δ+) are adjunctions; of the integers between these two numbers, K = {k ∈
i,j j,i j,i i,j N | k ≤ k ≤ k }. Now, let M be a map from C to
2. The operators δ+ and ε+ are dual of each other: min max
i,j i,j K,thatassociatesanelementofKto everyelementofthe
∀X ⊆Ci, ε+i,j(X)=Cj \δi+,j(Ci\X); simplicial complex C. Let x ∈ C, in this work, M(x) is
3. The operators δ− and ε− are dual of each other:
j,i j,i called the value of the simplex x.
∀X ⊆Cj, ε−j,i(X)=Ci\δj−,i(Cj \X). Wecanextendthenotionofsubcomplexesandstarsto
Wecanusethedimensionaloperatorsfromdefinition1 thedomainofweightedcomplexes. AmapM fromC inK
todefinenewoperators,leadingtonewdilations,erosions, isasimplicial stack(seeCousty et al.(2009a))ifthevalue
openings, closings and alternating sequential filters. Be- of each simplex is smaller than or equal to the value of
fore we start composing these operators, let us consider the simplices it includes, ∀x ∈ X, ∀y ⊆ x,M(x) ≥ M(y).
the following results, that can guide the exploration of Ontheotherhand,whenthecomparisonisreversed,when
new compositions. ∀x∈C, ∀y ⊆x,M(x)≤M(y),wesaythatM isastarred
stack. The dual M of a map M is defined using the value
Property 4. Let i,j,k∈N such that 0≤i<j <k ≤n. k : ∀x∈C,M(x)=k −M(x).
1. ∀X ⊆P(Ci), δj+,kδi+,j(X)=δi+,k(X); maLxet M be a map frommaanx arbitrary set E in K and let
2. ∀X ⊆P(Ci), ε+j,kε+i,j(X)=ε+i,k(X); k ∈ K. We denote by M[k] the set of elements in E with
3. ∀X ⊆P(C ), δ−δ− (X)=δ− (X); valuegreaterthanorequaltok, M[k]={x∈E |M(x)≥
k j,i k,j k,i
4. ∀X ⊆P(C ), ε−ε− (X)=ε− (X). k}. This set is called the k-threshold of M.
k j,i k,j k,i
The following lemma, which canbe easily provedfrom
Property4statesthatanycompositionofthesameop-
thedefinitions,clarifiesthelinksbetweenstars,complexes,
erator is equivalent to the operator acting from the initial
andthek-thresholdsofsimplicialstacksandstarredstacks.
to the final dimension. The proof of this property can be
done by contradiction, where if δ+ δ+(X) 6= δ+ (X) is Lemma 7. The following relations hold true:
j,k i,j i,k
true, our space is not a simplicial complex. 1. M is a simplicial stack ⇔ ∀k ∈K, M[k]∈C;
To explore the possible combinations of the operators 2. M is a starred stack ⇔∀k ∈K, M[k]∈S;
from definition 1, we start by considering only operators
3. M is a simplicial stack ⇔M is a starred stack.
actingonthesamedimension. Thefollowingpropertycan
be deduced from property 2: Weapproachtheproblemofextendingthedimensional
operators to weighted complexes using threshold decom-
Property 5. Let i,j,k∈N such that 0≤i<j <k ≤n.
position and stack reconstruction (see, e.g. Serra (1982)).
1. ∀X ⊆P(C ), δ−δ+(X)=δ− δ+ (X);
i j,i i,j k,i i,k The main idea of this method is that, if the considered
2. ∀X ⊆P(Ci), ε−j,iε+i,j(X)=ε−k,iε+i,k(X); operatoris increasing,we canapply itto eachk-threshold
3. ∀X ⊆P(C ), ε−δ+(X)=ε− δ+ (X); and then combine the results to obtain the final values.
i j,i i,j k,i i,k
4. ∀X ⊆P(Ci), δj−,iε+i,j(X)=δk−,iε+i,k(X). More precisely, let E1 and E2 be two sets and α an in-
creasingoperatorfromP(E )toP(E ),theextendedstack
Property 5 states that the result of the compositions 1 2
operator of α, also denoted by α, is:
of dilations and erosions that use a higher intermediary
dimensionisindependentofthe dimensionchosen. There-
fore, we can obtain only one dilation, one erosion, one ∀M :E →K,∀x∈E ,
1 2
opening and one closing using those compositions. How-
[α(M)](x)=max{k ∈K|x∈α(M[k])} (12)
ever,this is not entirely true when we considera lowerdi-
mension as intermediary dimension for the compositions.
As erosions and dilations, the dimensional operators
The following property can be deduced from property 2:
areincreasing. Thus,theycanbeextendedtomaps. Their
Property 6. Let i,j,k∈N such that 0≤i<j <k ≤n. extended stack operators are characterized by the follow-
1. ∀X ∈P(C ), δ+ δ− (X)⊇δ+ δ− (X); ing property.
k i,k k,i j,k k,j
2. ∀X ∈P(Ck), ε+i,kε−k,i(X)⊆ε+j,kε−k,j(X); Property 8. Let i,j ∈ K such that i ≤ j, let Mi :
3. ∀X ∈P(Ck), ε+i,kδk−,i(X)=ε+j,kδk−,j(X); P(Ci)→K and Mj :P(Cj)→K.
4. ∀X ∈P(Ck), δi+,kε−k,i(X)=δj+,kε−k,j(X). ∀x∈C ,
i
Property6statesthatcompositionsfromdilationsand 1. [δ−(M )](x)=max{M (y)|x⊆y};
erosions using a lower intermediary dimension are equal, j,i j y∈Cj j
independent of the chosen dimension and that composi- ∀x∈Ci,
tions of only dilations and erosions are related, but not 2. [ε−(M )](x)= min{M (y)|x⊆y});
always equivalent. j,i j y∈Cj j
4
∀x∈C , Stawiaski (Meyer and Stawiaski, 2009) obtained new op-
j
3. [δ+(M )](x)=max{M (y)|y ⊆x}); erators.
i,j i y∈Ci i Cousty et al. (Cousty et al., 2009b, 2013) considered
∀x∈C , a graph G = (G•,G×). For any X× ⊆ G× and Y• ⊆ G•,
j
4. [ε+ (M )](x)= min{M (y)|y ⊆x}). the operators ε×, δ×, ε• and δ• are defined by:
i,j i y∈Ci i
3.2. Revisiting the related work ε×(Y•)= ex,y ∈G× |x∈Y• and y ∈Y• (17)
In section 3 we defined operators acting between spe- δ×(Y•)=(cid:8)ex,y ∈G× |x∈Y• or y ∈Y• (cid:9) (18)
cific dimensions of the complex. Here, we use these op- ε•(X×)=(cid:8)x∈G• |∀e ∈G×,e ∈X(cid:9)× (19)
x,y x,y
erators, considering in particular property 8, to express
δ•(X×)=(cid:8)x∈G• |∃e ∈X× (cid:9) (20)
operators from the literature. x,y
(cid:8) (cid:9)
We start by the classical star and closure operators. If the considered space C is the 1-complex C = G•∪
Let X ⊆C. G×, using the dimensional operators,we have:
St(X)= δ+(X )|i,j ∈N,i≤j (13)
i,j i ε×(Y•)=ε+ (Y•) (21)
[(cid:8) (cid:9) 0,1
Cl(X)= δj−,i(Xj)|i,j ∈N,i≤j (14) δ×(Y•)=δ0+,1(Y•) (22)
[(cid:8) (cid:9)
ε•(X×)=ε− (X×) (23)
Vincent (1989) defined operators acting on a vertex 1,0
weighted graph (V,E,f), where V is a finite set (of ver- δ•(X×)=δ− (X×) (24)
1,0
tices), E is a set of unordered pairs of V, called edges,
and f is a map from V in K. By abuse of terminology, f Later, in Cousty et al. (2013), Cousty et al. extended
is called a weighted graph. these operators to weighted graphs, but the relations pre-
Letv ∈V,thesetofneighborsofavertexv isgivenby sentedherearestilltrue. Meyer,AnguloandStawiaski(Meyer and
N (v) = {v′ ∈V |{v,v′}∈E}. The dilated graph Γ(f) 2007; Meyer and Stawiaski, 2009) also defined operators
E
and the eroded graph Γ0(f) of the graph f are given, for capable of dealing with weighted graphs. They consider
any vertex v, by: thespaceasagraphG=(N,E),whereN = n1,n2,...,n|N|
isthesetofverticesandE ={eij |i,j ∈N+,0(cid:8)<i<j ≤|N|}(cid:9)
1. [Γ(f)](v)=max{f(v′)|v′ ∈N (v)∪{v}};
E is the set of edges. For two functions n and e weighting
2. [Γ0(f)](v)=min{f(v′)|v′ ∈NE(v)∪{v}}. the vertices and edges of G, they consider the following
operators:
In other words, these operators replace the value of
each vertex with the maximum (or minimum) value of its
neighbors, as morphological operators often do. To be
[ε n] =n ∧n (25)
able to draw a parallel between these operators and the en ij i j
dimensional operators presented in this work, let us con- [δ e] = {e } (26)
ne i ik
sider the 1-complex C defined as the union of the vertex _kneighborsofi
and edge sets of the graph G : C =V ∪E. Observe then [εnee]i = {eik} (27)
that C = V and C = E and that f is a map weighting ^kneighborsofi
0 1 [δ n] =n ∨n (28)
the 0-simplicesofC. Usingthe dimensionaloperators,we en ij i j
can recover the operators from Vincent (1989): IftheconsideredspaceC isthe1-complexC ={1,...,|N|}∪
{{i,j} | e ∈ E}, using the dimensional operators from
ij
definition 1, we have:
Γ(f)=δ− δ+ (f) (15)
1,0 0,1
Γ0(f)=ε− ε+ (f) (16)
1,0 0,1
[ε n] =[ε+ (n)]({i,j}) (29)
en ij 0,1
FromthesebasicdilationsanderosionsVincent(1989)
[δ n] =[δ+ (n)]({i,j}) (30)
derives severalinteresting operators,which, thanks to the en ij 0,1
previousrelations,canbe recoveredusingtheoperatorsof [ε e] =[ε− (e)]({i}) (31)
ne i 1,0
this article. [δ e] =[δ− (e)]({i}). (32)
So far the graphs were used only to provide structural ne i 1,0
information about the considered space. By considering Meyer,Angulo,andStawiaski(Meyer and Angulo,2007;
the edges and vertices in an uniform way, allowing the Meyer and Stawiaski,2009)definedseveraloperatorsbased
propagation of the values also to the edges of the graph, onthefourpresentedabove,allofthemarerecoverableby
both Cousty et al. (Cousty et al., 2009b) and Meyer and the dimensional operators. They also defined operators
5
that rely onthe particularstructure ofthe hexagonalgrid
and cannot be easily expressed using our operators.
3.3. Morphological operators on C using a higher interme-
diary dimension
In this section, we define new operators, acting on
subcomplexes, whose result is a complex of the same di-
mension of its argument, using an higher intermediary
dimension, exploring the effects of property 5. For in-
stance, if we consider a complex X of dimension i, with
i ∈ N, 0 < i ≤ n, we would like the dilation of X to be
also an i-complex. To that end, the operators proposed
nextactindependently oneachdimensionofthe complex:
Definition 9. We define:
∀X ∈C, δþ(X)=
δ− δ+ (X )
(cid:26) i∈[0...(n−1)] i+1,i i,i+1 i (cid:27)
[ [
δ+ δ− (X ) (33)
n−1,n n,n−1 n
∀X(cid:8)∈C, εþ(X)= (cid:9) (a) Y
ClA ε− ε+ (X )
(cid:18)(cid:26) i∈[0...(n−1)] i+1,i i,i+1 i (cid:27)
[ [
ε+ ε− (X ) (34)
n−1,n n,n−1 n (cid:19)
(cid:8) (cid:9)
As expected, the set(δþ(X)) , made ofthe i-simplices
i
of δþ(X), depends only on the set X , made of the i-
i
simplices of X. Intuitively, for i < n, the set (δþ(X))
i
contains all i-simplices of C that either belong to X or
i
arecontainedina(i+1)-simplexthatincludesani-simplex (b) δ(Y) (c) δþ(Y)
of X . For i = n, the operator will return all n-simplices
i
that contains an (n−1)-simplex of X.
Someresultsoftheoperatorsδþandεþ,alongwiththe
results of the operators δ and ε introduced by Dias et al.
(2011), are depicted as gray simplices in the figure 2. As
expected,theseoperatorsresultinasubcomplexmoresim-
ilar to the argument. The dilation included less simplices
intotheset,whiletheerosionremovedlesssimplicesofthe
set.
Itcaneasilybeproventhattheoperatorsεþandδþact
(d) ε(Y) (e) εþ(Y)
on C and form an adjunction. Therefore, we can compose
them to define new operators.
Figure2: Illustrationoftheoperatorsδþ andεþ [seetext].
Definition 10. Let i∈N. We define:
γþ = δþ i εþ i (35)
i
φþ =(cid:0)εþ(cid:1)i(cid:0)δþ(cid:1)i (36)
i
(cid:0) (cid:1) (cid:0) (cid:1)
SimilarlytotheoperatorsdefinedinCousty et al.(2013)
andDias et al.(2011),the parametericontrolshowmuch
of the complex will be affected by the operator. Figure 3
illustrates the operatorsγþ and φþ ontwo subcomplexes,
i i
depicted in gray. Since the dilation and erosion used to
6
different operators, using the variation of the temporary
dimension as parameter. However, we chose to explore
only the operators that affects the smallest possible num-
ber of simplices, because such operators usually lead to
more controlled filters. Additionally, one would need a
spaceofhigherdimensionalityinordertoproperlyexploit
these families.
Definition 12. Let X ∈ C. We define the operators δß
and εß by:
(a) Y (b) Z
δß(X)= δ+ δ− (X)
(cid:26) i∈[1...n] i−1,i i,i−1 (cid:27)
[ [
δ− δ+ (X) (39)
1,0 0,1
(cid:8) (cid:9)
εß(X)=ClA ε+ ε− (X)
(cid:18)(cid:26) i∈[1...n] i−1,i i,i−1 (cid:27)
[
ε− ε+ (X) (40)
1,0 0,1 (cid:19)
[(cid:8) (cid:9)
(c) φþ1(Y) (d) γ1þ(Z) However,the following property states that the opera-
tors from definition 12 are the same operators from defi-
Figure3: Illustrationofoperators γþ andφþ. nition 9.
i i
Property 13. Let i∈N such that 1≤i≤(n−1).
composetheseoperatorsaffectlesselementsthantheclas- 1. ∀X ∈P(Ci), δi+−1,iδi−,i−1(X)=δi−+1,iδi+,i+1(X);
sical operators, we can expect the same behavior from 2. ∀X ∈P(C ), ε+ ε− (X)=ε− ε+ (X).
i i−1,i i,i−1 i+1,i i,i+1
them as well.
Thispropertycanbeprovedbyanalysingtheelements
It can be easily proven that the operators from defini-
tion 10actoncomplexes,thatγþ is anopeningandφþ is of the space that are included or removed by each opera-
tor. Following this property,the operatorsobtained using
a closing. The families of operators formed by these two
a lower intermediary dimension are identical to the ones
operators,indexedbytheintegeri,areagranulometryand
obtained using a higher intermediary dimension. For this
ananti-granulometry,respectively. Therefore,theycanbe
reason,wewillonlyillustratethe resultsofoneofthemin
used to define new alternating sequential filters acting on
C. the next section.
Definition 11. Let i∈N. We define:
4. Illustrations of some operators
∀X ∈C, ASFþ(X)=
i We definedvariousoperatorsandfilters actingonsub-
γþφþ γþ φþ ... γþφþ (X) (37) complexes. In this section we illustrate these operators,
(cid:16) i i (cid:17)(cid:16) (i−1) (i−1)(cid:17) (cid:16) 1 1(cid:17) acting on values associated with elements of a mesh and
∀X ∈C, ASFþ′(X)= on subcomplexes created from regular images. More re-
i
sults and quantitative comparisons are available in Dias
φþγþ φþ γþ ... φþγþ (X) (38)
i i (i−1) (i−1) 1 1 (2012).
(cid:16) (cid:17)(cid:16) (cid:17) (cid:16) (cid:17)
Inthissectionweexploredoperatorsactingonsubcom- 4.1. Illustration on a tridimensional mesh
plexes composed by dimensional operators using a higher
As illustration, we processed the curvature values as-
intermediary dimension. We defined an adjunction, fam-
sociated with a 3D mesh, courtesy of the French Museum
ilies of openings and closings. We composed these gran-
Center for Research. We computed the curvature for the
ulometry and anti-granulometry into two alternating se-
vertices and propagated these values to the edges and tri-
quential filters.
angles,followingtheproceduredescribedinAlcoverro et al.
(2008), resulting in values between 0 and 1. These values
3.4. Morphological operators on C using a lower interme-
were then processed using our filters. For visualization
diary dimension
purposesonly,wethresholdedthevaluesat0.51,asshown
We just explored compositions of dimensional opera- inblack onfigure4(a)thatdepicts the thresholdedsetfor
tors using a higher intermediary dimension. We will ex- the original curvature data. The renderings presented in
plore compositions that use a lower intermediary dimen- this section consider only the values associated with the
sion. As theorem 6 suggests, we can define a family of vertices of the mesh, and no interpolation was used.
7
(a) X (inblack). (b) ASFþ3(X). (c) ASFþ3′(X).
Figure4: Renderingofthemeshconsidered. Thesetsarerepresented
inblack. [seetext]
4.2. Illustration on binary regular images
Inthis sectionweconsiderthe applicationofouralter-
(a) Originalimage. (b) Noisyversion.
nating sequential filters on regular images. For this end,
weneedtocreateasimplicialcomplexbasedontheimage.
Severalmethods canbe usedand the choice is application
dependent. Here, we create a vertex for each pixel, with
edgesbetweenthevertices,sixforeachvertex,correspond-
ing to an hexagonal grid. Triangles are placed between
three vertices, so each vertex belongs to six triangles. We
thenconsiderthegreatestcomplexthatcanbemadeusing
the value of the vertices. For visualization purposes, the (c) ASFþ6. (d) ASFþ′3.
images presented depict only the values associated with
the vertices.
We compare our results with the literature consider-
ing the same image used by Cousty et al. (Cousty et al.,
2009b), shown on figure 5(a). The noisy image shown on
figure 5(b) was processed. Figure 5 shows some results of
the operators ASFþ and ASFþ′, along with some results
fromCousty et al.(2013)andDias(2012),forvisualcom-
parison. TheoperatorASFþ removedmostofthefeatures (e) GraphASF6/2. (f) ASFc3.
of the zebra and left some noise on the background. The
operator ASFþ′ removed most of the background noise, Figure 5: Illustration of some results obtained with the operators
ASFþ and related literature results. Images (a), (b) and (e) are
while preserving some of the gaps between the stripes.
fromCoustyetal.(2009b)andimage(f)isfromDias(2012).
Howeverit alsoremovedthe smallerfeaturesofthe object
and left small holes. From these results, we may conclude
thatouroperatorsare,forthistypeofimage,onacompet-
itive level with the operators presented in the literature.
Additionally, Mennillo et al. (Mennillo et al., 2012)
used the dimensional operators for document processing
as a pre-processingstage to boost OCRperformance with
encouraging results.
Whiletheresultsforregularimageprocessingaregood,
they do not fully exploit the structure of the simplicial
complex, nor the flexibility of the operators. We expect
the filtering results to be even better when considering
more complex scenarios.
8
the FUI6 project ”Demat-Factory”. F´abio Dias received
additionalfundingbyFAPESP2012/15811−1. JeanCousty
and Laurent Najman received funding from the Agence
NationaledelaRecherche,contractANR−2010−BLAN−
0205−03.
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