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%% | PAPER TITLE |
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\title
	[Adjacency Stable Connected Oper. and Set Levelings] % short title (maximum of 50 characters)
	{Adjacency Stable Connected Operators and Set Levelings} % full title

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%% | AUTHORS |
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\begin{Authors}
	\Author
   		{Jose Crespo}
		\Affil{GIB - LIA, 
Facultad de Informática,
Universidad Politécnica de Madrid,
Campus de Montegancedo,
28660 Boadilla del Monte (Madrid), Spain \\*
		\email{jcrespo@fi.upm.es}}
\end{Authors}


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%% Abstract and keywords are written inside two environments,
%% the "Abstract" amd the "Keywords" environments, respectively.
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\begin{Abstract}
This paper studies morphological connected operators. 
Particularly, it focuses on an adjacency 
constraint, as well as on the so-called set levelings. 
Two important findings are reported in this work. 
First, the relationships between the so-called 
adjacency stable operators and set levelings 
%
are investigated, 
and an equivalence is established.      
This is an important result about how 
these concepts have been chronologically introduced, 
and it permits to apply some properties 
to the related operator class. 
%
Second, the implications and limits of a property about expressing 
certain connected operators as 
a 
sequential composition 
of an opening and a closing 
(and vice-versa) 
based on markers 
are discussed.
%
Then, a commutative property for 
attribute alternated filters is presented. 
\end{Abstract}



\begin{Keywords}
	image processing,
	image analysis,
	mathematical morphology,
	\Index{connected operator},
	\Index{adjacency stable operator},
	\Index{leveling}.
\end{Keywords}




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%% | SECTIONS |
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\vspace*{-0.3377cm}


%
\section{Introduction}
\label{\fullpaperid:sec:Introduction}


This paper investigates some aspects of morphological 
connected operators, which preserve well shapes and 
do not introduce discontinuities. 
%


It happens that usual morphological connected operators 
(such as those composed by openings and closings) 
impose certain adjacency constraints between the 
input and the output. 
This was previously researched, and the so-called 
\emph{adjacency stable connected} operator concept 
\cite{Crespo:crespo93} \cite{Crespo:crespo93c} \cite{Crespo:crespo94e} 
was established in the set or binary framework. 

In addition, the so-called \emph{levelings} 
\cite{Crespo:Meyer1998CoOpLe} \cite{Crespo:Meyer1998Le} \cite{Crespo:MeyeMara:00}
\cite{Crespo:Meyer2004}
were defined in the non-binary framework 
in such a way that constrain 
the output variations depending on 
the input variations. 
%
Those levelings within the 
set or binary framework are called set or binary levelings 
%
\cite{Crespo:Meyer2004}. 



A question that naturally arises is 
whether there are relationships between 
adjacency stable connected operators 
and set levelings. 
As will be shown later, this is one of the results 
that will be established in this work. 
%
In scientific research, it is key being able 
to indicate how and when research concepts have been 
developed and introduced. 
%
Besides, in the case where such relationships exist, 
properties that are valid for one type of operators 
could perhaps apply to the other related operator class. 


A second aspect researched in this paper is 
the possibility of expressing certain 
connected operators as 
the 
sequential composition 
of an opening and a closing (and vice-versa). 
Some clarifications about the implications, and limits, 
of a previously presented result are provided. 
Furthermore, 
a commutative property for attribute alternated filters 
will be presented. 


Section~\ref{\fullpaperid:sec:Background} 
provides some general background. 
The adjacency stability 
and leveling concepts are recalled in, respectively, 
Sections~\ref{\fullpaperid:sec:AdjacencyStability} 
and~\ref{\fullpaperid:sec:SetLevelings}. 
The relationships between 
adjacency stable connected operators and set levelings 
are established in Section~\ref{\fullpaperid:sec:RelationshipsAS_SL}. 
%
The expression of certain connected operators as 
sequential compositions of an opening and a closing is treated 
in Section~\ref{\fullpaperid:sec:OpeningClosingProduct}. 
%
Finally, a conclusion section ends the paper. 


\vspace*{-0.1177cm}
\vspace*{-0.1177cm}


\section{Background}
\label{\fullpaperid:sec:Background}

\vspace*{-0.1177cm}
\vspace*{-0.1177cm}

\subsection{General Definitions}

\vspace*{-0.1177cm}

Mathematical morphology 
%
deals with 
the application of set theory
concepts to image processing and analysis, 
and it considers that 
images are composed of geometrical shapes with 
intensity or multi-band profiles \cite{Crespo:ronse:guest}. 
Some general references are
\cite{Crespo:matheron75} \cite{Crespo:serra82}
\cite{Crespo:serra88}
\cite{Crespo:giardina88} 
\cite{Crespo:SchmittMattioli93} 
\cite{Crespo:heijmans94} 
\cite{Crespo:Banon:2000:FoInDi}
\cite{Crespo:soille2003sv} 
\cite{Crespo:DougLotu:03}%
.



A basic set of notions on morphological filtering can be
the following:

\vspace*{-0.1177cm}

\begin {itemize} 
\item {
Mathematical morphology focuses on increasing
mappings defined on a complete lattice
\cite{Crespo:serra88}.
In a complete lattice there exists an ordering relation,
and two basic operations called infimum and supremum
(denoted by $\bigwedge$ and $\bigvee$, respectively). 
} 

\vspace*{-0.1177cm}

\item { 
A transformation $\psi$ 
is increasing if
and only if it preserves ordering.  }
\vspace*{-0.1177cm}
\item { 
A transformation $\psi$ is idempotent if and only if $\psi \psi =
\psi$.  }

\vspace*{-0.1177cm}

\item { 
A transformation $\psi$ is a morphological filter if and only if it is
increasing and idempotent.  }

\vspace*{-0.1177cm}

\item { 
An opening (denoted by $\gamma$) is an anti-extensive
morphological filter. 
Since $\gamma$ is anti-extensive, we can say that 
$\gamma \leq \CRESPOID$, 
where $\CRESPOID$ symbolizes the identity operator that 
leaves the input unchanged. 
}

\vspace*{-0.1177cm}

\item { 
A closing (denoted by $\varphi$) is an extensive morphological
filter. 
Since $\varphi$ is extensive, we can say that 
$\varphi \geq \CRESPOID$. 
}
\end {itemize} 

\vspace*{-0.1177cm}

In the theoretical expressions in this paper, 
we will be working on the lattice ${\mathcal {P}} (E)$, 
where $E$ is a given set of points (the space) 
and 
${\mathcal {P}} (E)$ denotes 
the set of all subsets of~$E$ 
(i.e., ${\mathcal {P}} (E) = \{ A : A \subseteq E \}$). 
In other words, inputs and outputs are supposed 
to be sets or, equivalently, binary functions. 
In this lattice, the sup $\bigvee$ and the inf $\bigwedge$ 
operations are, respectively, 
the set union $\bigcup$ and set intersection $\bigcap$ operations, 
while the order relation is the set inclusion relation $\subseteq$. 
%
Even though we will work on the lattice ${\mathcal {P}} (E)$, 
results are extendable to gray-level 
functions by means of the so called flat operators 
\cite{Crespo:serra82} 
\cite{Crespo:soille2003sv}. 
%


Two morphological operators $\psi _{1}$ and $\psi _{2}$ 
are dual if 
$\psi _{1} = \complement \psi _{2} \complement$, 
%
where $\complement$ symbolizes the complement operator. 
%
%



\subsection{Some Background on Connectivity and Connected Operators}



Some references to the topic of connectivity and connected filtering 
are: 
\cite{Crespo:serra93} 
\cite{Crespo:crespo93} 
\cite{Crespo:vincent93b} 
\cite{Crespo:salembier95} 
\cite{Crespo:crespo94} 
%
\cite{Crespo:bb2233} 
%
\cite{Crespo:crespo94e} 
%
\cite{Crespo:crespo95f}  
\cite{Crespo:GarrSaleGarc1998} 
%
\cite{Crespo:Ronse1998} 
%
\cite{Crespo:Jone:99} 
%
\cite{Crespo:heijmans99} 
\cite{Crespo:SalembierGarrido2000}
\cite{Crespo:CresMaojSanaBillMuno2002} 
\cite{Crespo:serra98} 
\cite{Crespo:Meyer1998CoOpLe} 
\cite{Crespo:Meyer1998Le} 
\cite{Crespo:Serra2000} 
\cite{Crespo:MeyeMara:00} 
\cite{Crespo:RonseSerra2001} 
\cite{Crespo:TeroVarg2002} 
\cite{Crespo:soille2003sv}
\cite{Crespo:VargCresMaoj2003} 
\cite{Crespo:braga-goutsias2003CVIU} 
\cite{Crespo:braga-goutsias2003JMIV} 
\cite{Crespo:Meyer2004} 
\cite{Crespo:braga-neto:multiscale}%
.



Connectivity is established 
%
in 
%
%
\cite[(pp.~51-57)]{Crespo:serra88}     
by means
of the {\it connected class} concept.  
%
A connected class ${\mathcal {C}}$
in ${\mathcal {P}} (E)$ is a subset of ${\mathcal {P}} (E)$ such that (a)
$\CRESPOEMPTYSET \in {\mathcal {C}}$ and for all $x \in E$, 
$\{ x \} \in {\mathcal {C}}$; 
and (b) for each family $C _{i}$ in ${\mathcal {C}}$, $\bigwedge
_{i} C _{i} \neq \CRESPOEMPTYSET$ implies 
$\bigvee _{i} C _{i} \in {\mathcal {C}}$.
The subclass ${\mathcal {C}} _{x}$ that has all members of ${\mathcal {C}}$
that contain $x$ 
(i.e., ${\mathcal {C}} _{x} = \{ C \in {\mathcal {C}} : x \in C \}$) 
leads to the definition of an opening $\CRESPOOPENINGX{\CRESPOXXS}$  
called {\CRESPOEMPH {point opening}} 
\cite{Crespo:serra88}.  
For all $x \in E$, $A \in {P} (E)$, 

\vspace*{-0.1177cm}
\vspace*{-0.1177cm}
%
\begin {equation} \label{\fullpaperid:eq:gammax} 
\CRESPOOPENINGX{\CRESPOXXS} (\CRESPOSETA) = \bigvee \{ C: C \in {\mathcal {C}} _{x}, C \leq A \}. 
\end {equation} 
%

\vspace*{-0.1177cm}
\vspace*{-0.1177cm}


The dual operation of $\CRESPOOPENINGX{\CRESPOXXS}$ is the closing $\varphi _{x}$ that
is equal to $\complement \CRESPOOPENINGX{\CRESPOXXS} \complement$. 
%
%
%
If a point $x$ does not belong to a set $\CRESPOSETA$, i.e., it 
belongs to a pore of $\CRESPOSETA$, then we can obtain such a pore 
with $\CRESPOOPENINGX{\CRESPOXXS} \complement$ 
(or, equivalently, with $\complement \varphi _{x}$). 


In sets 
(or, equivalently, binary images), 
the {\CRESPOEMPH {flat zone}} of a point (or pixel) $\CRESPOXXS$ 
is the 
%
grain 
or the 
%
pore 
(whichever is not empty) 
which $\CRESPOXXS$ belongs to. 
I.e., the flat zone of $\CRESPOXXS$ in a set $\CRESPOSETA$ is 
equal to: 
%
%
$
\CRESPOFLATZONEX{\CRESPOXXS} (\CRESPOSETA) \CRESPOEQSSL 
%
\CRESPOOPENINGX{\CRESPOXXS} (\CRESPOSETA) \CRESPOBIGVEESS 
\CRESPOOPENINGX{\CRESPOXXS} \CRESPOCOMPLEMENT (\CRESPOSETA). 
$
%
%
%
(Note that either 
the grain 
%
or 
%
the pore of a point $\CRESPOXXS$ 
is empty.) 
%
In the non-binary case, 
the flat zones of a function 
are its piecewise-constant regions, i.e., 
the set of 
%
connected sets with 
the same function value.




%
An operator $\psi$ is \CRESPOEMPH{connected} 
\cite{Crespo:serra93} \cite{Crespo:salembier95} 
if, 
for all $A \in {P} (E)$, 
each flat zone (grain or pore) of $A$ is 
included in a flat zone (grain or pore) of $\psi (A)$. 
%



\vspace*{-0.1177cm}
\vspace*{-0.1177cm}


\subsection {Connectivity Requirement} 
\label{\fullpaperid:unusual_connectivities} 


A general requirement for the space connectivity 
is assumed in this work. 
%
Particularly, the space connectivity is supposed 
to be a \CRESPOEMPH{strong connectivity} 
(see 
%
\cite{Crespo:Ronse1998} \cite{Crespo:heijmans99}). 
%
%
The usual four- and eight-connectivities in 
connected subsets of $\CRESPOZZTWO$ 
are cases of strong connectivities. 

\vspace*{-0.1177cm}
\vspace*{-0.1177cm}


\section {Adjacency Stability} 
\label{\fullpaperid:sec:AdjacencyStability} 




\vspace*{-0.1177cm}
\vspace*{-0.1177cm}


This section discusses and summarizes the 
\Index{adjacency stability} concept 
for connected operators, 
which 
appeared first in 
\cite{Crespo:crespo93}, 
and 
was further studied in 
\cite{Crespo:crespo93c} \cite{Crespo:crespo94e}. 
%
A closely related concept was later discussed in 
\cite{Crespo:heijmans99}. 





Let us define the concept of {\CRESPOEMPH {adjacency between flat zones}}, 
which formalizes 
the intuitive 
notion of contiguity. 

\begin{definition}
Two (disjoint) flat zones $\CRESPOFLATZONEX{\CRESPOXXS} (\CRESPOSETA)$ 
and $\CRESPOFLATZONEX{\CRESPOYYS} (\CRESPOSETA)$ in 
a space $\CRESPOEE$ (endowed  with $\CRESPOOPENINGX{\CRESPOXXS}$, $\CRESPOXXS \in \CRESPOEE$) 
are said to be {\CRESPOEMPHDEFI {adjacent}} 
if $\CRESPOFLATZONEX{\CRESPOXXS} (\CRESPOSETA) \bigvee \CRESPOFLATZONEX{\CRESPOYYS} (\CRESPOSETA)$ 
%
is a connected set, 
i.e., 
if $\CRESPOFLATZONEX{\CRESPOXXS} (\CRESPOSETA) \bigvee \CRESPOFLATZONEX{\CRESPOYYS} (\CRESPOSETA) =$ 
$\CRESPOOPENINGX{\CRESPOXXS} (\CRESPOFLATZONEX{\CRESPOXXS} (\CRESPOSETA) \bigvee \CRESPOFLATZONEX{\CRESPOYYS} (\CRESPOSETA))$. 
%
\label{\fullpaperid:defi:adjacency_flatzones} 
%
%
\end{definition}

\noindent 
(Note that $\CRESPOFLATZONEX{\CRESPOXXS} (\CRESPOSETA) = \CRESPOOPENINGX{\CRESPOXXS} (\CRESPOSETA) \bigvee \CRESPOOPENINGX{\CRESPOXXS} \CRESPOCOMPLEMENT (\CRESPOSETA)$.) 
%
%
The relationship of Definition~\ref{\fullpaperid:defi:adjacency_flatzones} 
is symmetric (and not reflexive). 
%
%
Definition~\ref{\fullpaperid:defi:adjacency_flatzones} 
can be extended to sets: 
two sets $\CRESPOSETA$ and $\CRESPOSETB$ are adjacent if some flat zone of 
$\CRESPOSETA$ is adjacent to some flat zone of $\CRESPOSETB$. 



\begin{figure} [h] 
\begin {center} 
\begin{tabular}{cccc}
\includegraphics[width=1.811cm]{\fullpaperpdirectory/images/stable51}
& 
\includegraphics[width=1.811cm]{\fullpaperpdirectory/images/stable52b}
& 
\includegraphics[width=1.811cm]{\fullpaperpdirectory/images/stable53}
& 
\includegraphics[width=1.811cm]{\fullpaperpdirectory/images/stable54}
\\
{\CRESPOmyfigletterssize (a) Input set (in dark)} 
& {\CRESPOmyfigletterssize (b) AU Output} 
& {\CRESPOmyfigletterssize (c) AS Output}
& {\CRESPOmyfigletterssize (d) AS Output} \\
\end {tabular} 
\end {center} 
\caption [Adjacency stability example]{ 
Adjacency stability example. 
Part~(a) shows an input set $\CRESPOSETA$, 
and 
parts~(b), (c) and~(d) 
display three possible outputs of connected operators. 
Part~(b) would be adjacency unstable: 
note that a pore of the input set $\CRESPOSETA$ is in a grain 
in~(b) but is not surrounded by grains of $\CRESPOSETA$. 
This situation does not happen in cases (c) and (d). 
(Note: 
``AU'' refers to adjacency unstable, and 
``AS'' denotes adjacency stable.) 
(Note: the space 
is endowed with a usual four- or eight-connectivity.) 
}
\label{\fullpaperid:fig-stable}
\end{figure}







%
The {\CRESPOEMPH {adjacent flat zones}} of a point 
$\CRESPOXXS$ in an input set $\CRESPOSETA$, 
symbolized by ${{\CRESPOADJACENT}} _{\CRESPOXXS} (\CRESPOSETA)$, 
are the pores 
(if $\CRESPOXXS \in \CRESPOSETA$) 
or the grains (if $\CRESPOXXS \not \in \CRESPOSETA$) that are 
adjacent 
to 
$\CRESPOFLATZONEX{\CRESPOXXS} (\CRESPOSETA)$, 
i.e., 

\vspace*{-0.1177cm}
\vspace*{-0.1177cm}

\begin{center} 
${{\CRESPOADJACENT}} _{\CRESPOXXS} (\CRESPOSETA) = \bigvee _{\CRESPOYYS} \{ \CRESPOFLATZONEX{\CRESPOYYS} (\CRESPOSETA): 
\CRESPOYYS \in \CRESPOEE, 
\CRESPOFLATZONEX{\CRESPOYYS} (\CRESPOSETA) \bigvee \CRESPOFLATZONEX{\CRESPOXXS} (\CRESPOSETA) 
= \CRESPOOPENINGX{\CRESPOXXS} (\CRESPOFLATZONEX{\CRESPOYYS} (\CRESPOSETA) \bigvee \CRESPOFLATZONEX{\CRESPOXXS} (\CRESPOSETA) \} 
$. 
\end{center} 



\vspace*{-0.1177cm}

The important concept of {\CRESPOEMPH {adjacency stability}} 
\cite{Crespo:crespo93} \cite{Crespo:crespo93c} \cite{Crespo:crespo94e}
is established next. 
This requirement 
concerns how adjacent grains and pores are treated by 
an operation. 


\vspace*{-0.1177cm}

%
\begin{definition}
Let $\CRESPOEE$ be a space 
%
endowed 
with $\CRESPOOPENINGX{\CRESPOXXS}$, $\CRESPOXXS \in \CRESPOEE$. 
An operator 
$\psi : {\mathcal {P}} (E) \longrightarrow {\mathcal {P}} (E)$ 
is {\bf adjacency stable} if, for all $x \in E$: 

\vspace*{-0.1177cm}
\vspace*{-0.1177cm}

%
\begin {equation} \label{\fullpaperid:eq:stable} 
\CRESPOOPENINGX{\CRESPOXXS} (\CRESPOID \bigvee \psi) = 
\CRESPOOPENINGX{\CRESPOXXS} \bigvee \CRESPOOPENINGX{\CRESPOXXS} \psi. 
\end {equation} 

\label{\fullpaperid:defi:stable} 
%
%
\end{definition}

\vspace*{-0.3377cm}
\noindent 
Note that 
$\CRESPOOPENINGX{\CRESPOXXS}$ 
commutes under the inf 
($\CRESPOOPENINGX{\CRESPOXXS} (\bigwedge _{i} \psi _{i}) = 
\bigwedge _{i} \CRESPOOPENINGX{\CRESPOXXS} \psi _{i}$%
)
but not in general under the sup. 


The adjacency stability equation~(\ref{\fullpaperid:eq:stable}) treats grains and
pores symmetrically.  
%
The
reason is that what matters is the switch from grain to pore and
vice-versa.  
%
%
We can state as well 
that the dual of an adjacency stable operator 
is adjacency stable. 


The grain-pore relationship is illustrated in Figure~\ref{\fullpaperid:fig-stable}. 
%
%
The consequences of adjacency stability 
on the relationships between the grains 
of an input set $\CRESPOSETA$ and the output $\psi (\CRESPOSETA)$ 
are the following: 
the grains of $\psi (\CRESPOSETA)$ are a union of 
%
(a) grains of $\CRESPOSETA$, and (b) pores of $\CRESPOSETA$ 
surrounded by grains in (a). 
%
%
For the particular case in Figure~\ref{\fullpaperid:fig-stable}(b), 
Figure~\ref{\fullpaperid:fig-stable-test} shows 
that the 
adjacency stability equation does not hold for 
the point marked as $x$ (this point is not the only one). 
%
%
The adjacency stability equation must hold for all $A \in {\mathcal {P}} (E)$ 
and for all $x \in E$. 







Lemma~\ref{\fullpaperid:lem:stable_adjacent_0} 
is useful to relate the input and the output. 

%
\begin {lemma} \label{\fullpaperid:lem:stable_adjacent_0} 
Let $\CRESPOEE$ be a space endowed with $\CRESPOOPENINGX{\CRESPOXXS}$, $\CRESPOXXS \in \CRESPOEE$. 
A connected operator 
$\psi : {\mathcal {P}} (E) \longrightarrow {\mathcal {P}} (E)$ 
is adjacency stable if and only if, 
for all $A \in {\mathcal {P}} (E)$, 
$\psi (\CRESPOSETA)$ and $A \setminus \psi (\CRESPOSETA)$ 
are not connected to each other 
%
(i.e., are not adjacent). 
\end {lemma} 


\vspace*{-0.1177cm}
\vspace*{-0.1177cm}
\vspace*{-0.1177cm}

\begin{figure} [h] 
\begin {center} 
\begin{tabular}{cc}
\includegraphics[width=1.811cm]{\fullpaperpdirectory/images/stable_5x1}
& 
\includegraphics[width=1.811cm]{\fullpaperpdirectory/images/stable_5x2}
\\ 
{\CRESPOmyfigletterssize (a) Input set $\CRESPOSETA$ (in dark)} 
& 
{\CRESPOmyfigletterssize (b) $\psi (\CRESPOSETA)$ ($\psi$ is adjacency unstable)} \\ 
\includegraphics[width=1.811cm]{\fullpaperpdirectory/images/stable_5x3}
& 
\includegraphics[width=1.811cm]{\fullpaperpdirectory/images/stable_5x4}
\\
{\CRESPOmyfigletterssize (c) $\CRESPOOPENINGX{\CRESPOXXS} (\CRESPOSETA) \bigvee \CRESPOOPENINGX{\CRESPOXXS} \psi (\CRESPOSETA)$}
& {\CRESPOmyfigletterssize (d) $\CRESPOOPENINGX{\CRESPOXXS} (\CRESPOID \bigvee \CRESPOOPENINGX{\CRESPOXXS} \psi) (\CRESPOSETA)$} \\
\end {tabular} 
\end {center} 
\caption [Adjacency stability equation]{ 
Adjacency stability equation. 
Parts (a) and (b) display, respectively, an input set $\CRESPOSETA$ 
and the output $\psi (\CRESPOSETA)$, where $\psi$ is 
an adjacency unstable connected operator. 
The adjacency stability equation does not hold: 
for the point $\CRESPOXXS$ signaled (among others), 
$\CRESPOOPENINGX{\CRESPOXXS} (\CRESPOSETA) \bigvee \CRESPOOPENINGX{\CRESPOXXS} \psi (\CRESPOSETA)$ 
(part~(c)) is not equal to 
$\CRESPOOPENINGX{\CRESPOXXS} (\CRESPOID \bigvee \CRESPOOPENINGX{\CRESPOXXS} \psi) (\CRESPOSETA)$ (part~(d)). 
}
\label{\fullpaperid:fig-stable-test}
\end{figure}


\vspace*{-0.1177cm}
\vspace*{-0.1177cm}
\vspace*{-0.1177cm}
\vspace*{-0.1177cm}


\begin{property} \label{\fullpaperid:prop:stable_extensive_antiextensive} 
Extensive and anti-extensive mappings 
are adjacency stable. 
\label{\fullpaperid:prop:ext_antiext} 
\end{property} 

\vspace*{-0.1177cm}
\vspace*{-0.1177cm}
\vspace*{-0.1177cm}

\begin{property} \label{\fullpaperid:prop:stable_stable} 
The class 
of adjacency stable connected operators 
is closed under the sup, the inf and 
the sequential composition 
operations. 

\end{property} 



\vspace*{-0.1177cm}


If $\psi$ is an adjacency stable connected operator, 
%
then, for all $A \in {\mathcal {P}} (E)$: 
%
%
%
(a) 
%
$\CRESPOXXS \not \in \CRESPOSETA$, $\CRESPOXXS \in \psi (\CRESPOSETA)$ 
$\Longrightarrow$ 
${{\CRESPOADJACENT}} _{x} (\CRESPOSETA) \leq \psi (\CRESPOSETA)$
;
%
%
%
~(b) 
%
$\CRESPOXXS \in \CRESPOSETA$, $\CRESPOXXS \not \in \psi (\CRESPOSETA)$ 
$\Longrightarrow$ 
${{\CRESPOADJACENT}} _{x} (\CRESPOSETA) \leq \CRESPOCOMPLEMENT \psi (\CRESPOSETA)$. 


\vspace*{-0.1177cm}
\vspace*{-0.1177cm}


\section{Levelings and Set Levelings} 
\label{\fullpaperid:sec:Levelings}
\label{\fullpaperid:sec:SetLevelings}

\vspace*{-0.1177cm}


A \emph{\Index{leveling}} 
\cite{Crespo:Meyer1998CoOpLe} \cite{Crespo:Meyer1998Le} \cite{Crespo:MeyeMara:00}
\cite{Crespo:Meyer2004} 
is defined next. 



%
%
\begin{definition}
An image $g$ is a leveling of an input image $f$ 
if and only if:

\vspace*{-0.3377cm}
\vspace*{-0.1177cm}


%
\begin{equation}\label{\fullpaperid:eq:Leveling01}
\forall ~(p,q) 
%
{\rm ~neighboring~pixels}: 
g_p > g_q ~\Rightarrow~ f_p \geq g_p {\rm ~and~} g_q \geq f_q 
\end{equation} 

\label{\fullpaperid:defi:Leveling01}
%
%
\end{definition}


The previous definition of leveling is that in 
\cite[Definition~4 (p.~193)]{Crespo:Meyer1998CoOpLe} 
%
\cite[Definition~2.2 (p.~4)]{Crespo:MeyeMara:00}. 
A more general and complex definition is 
introduced in \cite[Definition~10 (p.~62)]{Crespo:Meyer2004}, 
but a leveling as established by expression~(\ref{\fullpaperid:eq:Leveling01}) 
is also a leveling as defined in 
\cite[Definition~10 (p.~62)]{Crespo:Meyer2004}. 
Moreover, we focus on set operators. 


\emph{\Index{Set levelings}} are those defined in the 
set or binary framework. 
%
%
Expression~(\ref{\fullpaperid:eq:Leveling01}) 
will be particularized for the set or binary case 
in Section~\ref{\fullpaperid:sec:SetLevelings_InputOutputRestriction}. 



\vspace*{-0.1177cm}
\vspace*{-0.1177cm}

\section{Relationships between Adjacency Stable Connected Operators and Set Levelings} 
\label{\fullpaperid:sec:RelationshipsAS_SL} 


Both adjacency stable connected operators and levelings impose 
certain constraints on the input and output variations, 
which will analyzed next. 
Then, the relationships between these types of operators 
will be discussed. 

\vspace*{-0.1177cm}

\subsection{Adjacency Stable Connected Operators: Input/Output Variations Restriction} 
\label{\fullpaperid:sec:AdjacencyStableConnectedOperators_InputOutputRestriction} 



First, we will formalize 
the neighborhood relationship 
%
between two pixels 
using $\CRESPOOPENINGX{\CRESPOXXS}$. 
%
As in Section~\ref{\fullpaperid:sec:SetLevelings} 
%
(Definition~\ref{\fullpaperid:defi:Leveling01}), 
we will use the symbols $p$ and $q$ 
to refer to a pair of pixels that are neighbors. 
%
%
Two pixels $\CRESPOXXSN$ and $\CRESPOYYSN$ 
are {\CRESPOEMPH {neighbors}} to each other if and only if 
they satisfy 

\vspace*{-0.1177cm}

%
\begin{equation} \label{\fullpaperid:eq:neighbors_gammax} 
\CRESPOOPENINGX{\CRESPOXXSN} (\{ \CRESPOXXSN, \CRESPOYYSN \}) = 
\CRESPOOPENINGX{\CRESPOYYSN} (\{ \CRESPOXXSN, \CRESPOYYSN \}) = 
\{ \CRESPOXXSN, \CRESPOYYSN \}. 
\end{equation} 



The next proposition, 
which states the restrictions imposed 
on the values of neighboring pixels 
between an input $\CRESPOII$ and the output $\CRESPOII'$ 
of an adjacency stable connected operator, 
follows directly from 
Definition~\ref{\fullpaperid:defi:stable}. 


\vspace*{-0.1177cm}

\begin{property} \label{\fullpaperid:prope:stable_set} 
%
Let $\CRESPOSETA$ be an input set. 
Let $\psi: \CRESPOPOWERSETE \longrightarrow \CRESPOPOWERSETE$ 
be an adjacency stable connected 
%
operator. 
%
If $\CRESPOXXSN$ and $\CRESPOYYSN$ are neighbors to each other 
(i.e., $\CRESPOXXSN$ and $\CRESPOYYSN$ satisfy equation~(\ref{\fullpaperid:eq:neighbors_gammax})), 
or if they belong to adjacent flat zones, 
the adjacency stability of $\psi$ 
%
implies the following restrictions between 
the input $\CRESPOSETA$ and output $\psi (\CRESPOSETA)$ sets: 

\vspace*{-0.1177cm}
\vspace*{-0.1177cm}


%
\begin{equation}\label{\fullpaperid:eq:stabilityswitch_set} 
\begin{tabular}{llll} 
(a) 
& 
$
\CRESPOOPENINGX{\CRESPOXXSN} (\CRESPOSETA) = \CRESPOOPENINGX{\CRESPOYYSN} (\CRESPOSETA) 
$
& 
$
\Longrightarrow 
$
& 
$
\CRESPOOPENINGX{\CRESPOXXSN} \psi (\CRESPOSETA) = \CRESPOOPENINGX{\CRESPOYYSN} \psi (\CRESPOSETA) 
$ 
\\ 
(b) 
& 
$
\CRESPOXXSN \in \CRESPOSETA, \CRESPOYYSN \not \in \CRESPOSETA 
$
& 
$
\Longrightarrow
$ 
& 
$
\left\{ 
     \begin{array}{c} 
       \CRESPOXXSN \in \psi (\CRESPOSETA), \CRESPOYYSN \not \in \psi (\CRESPOSETA) \\ 
       
       \mbox{{\rm or}}  \\ 
       \CRESPOOPENINGX{\CRESPOXXSN} \psi (\CRESPOSETA) = \CRESPOOPENINGX{\CRESPOYYSN} \psi (\CRESPOSETA) 
     \end{array} 
\right. 
$ 
\\ 
\end{tabular} 
\end{equation}

%
%
\end{property}




\noindent 
There is a case symmetric to~(b) above 
(interchanging $\CRESPOXXSN$ and $\CRESPOYYSN$ in~(b)) not shown. 


The cases 
not covered in 
%
expression~(\ref{\fullpaperid:eq:stabilityswitch_set}) 
are: 
(i)~$\CRESPOXXSN \in \CRESPOSETA, \CRESPOYYSN \not \in \CRESPOSETA 
$
and 
$
\CRESPOXXSN \not \in \psi (\CRESPOSETA), \CRESPOYYSN \in \psi (\CRESPOSETA)) 
$; 
and 
(ii)~$\CRESPOXXSN \not \in \CRESPOSETA, \CRESPOYYSN \in \CRESPOSETA 
$
and 
$
\CRESPOXXSN \in \psi (\CRESPOSETA), \CRESPOYYSN \not \in \psi (\CRESPOSETA)
$.
(Cases~(i) and~(ii) 
are symmetric to each other, interchanging $\CRESPOXXSN$ and $\CRESPOYYSN$.) 
%
Clearly, 
these cases 
do not 
satisfy 
equation~(\ref{\fullpaperid:eq:stable}) of Definition~\ref{\fullpaperid:defi:stable}. 
Let us prove that for one case. 




%
\begin {quote} 

{\CRESPOmyproofstyle 
{Proof that the case 
$
\CRESPOXXSN \not \in \CRESPOSETA, \CRESPOYYSN \in \CRESPOSETA 
$
and 
$
\CRESPOXXSN \in \psi (\CRESPOSETA), \CRESPOYYSN \not \in \psi (\CRESPOSETA)) 
$
does not 
satisfy 
equation~(\ref{\fullpaperid:eq:stable})}. 
%
We have that: 


%
\begin{itemize} 
\item{ 
%
$
\CRESPOOPENINGX{\CRESPOXXSN} (\CRESPOID \bigvee \psi) (\CRESPOSETA) =$ 
$\CRESPOOPENINGX{\CRESPOXXSN} (\CRESPOSETA \bigvee \psi (\CRESPOSETA)) =$ 
$\CRESPOOPENINGX{\CRESPOXXSN} \psi (\CRESPOSETA) \bigvee \CRESPOOPENINGX{\CRESPOYYSN} (\CRESPOSETA). 
$ 
} 
\item{ 
%
$
\CRESPOOPENINGX{\CRESPOXXSN} (\CRESPOSETA) \bigvee \CRESPOOPENINGX{\CRESPOXXSN} \psi (\CRESPOSETA) =$
$\CRESPOOPENINGX{\CRESPOXXSN} \psi (\CRESPOSETA). 
$
} 
\end{itemize} 

And 
$\CRESPOOPENINGX{\CRESPOXXSN} \psi (\CRESPOSETA) \bigvee \CRESPOOPENINGX{\CRESPOYYSN} (\CRESPOSETA) 
$ 
$>$ 
$\CRESPOOPENINGX{\CRESPOXXSN} \psi (\CRESPOSETA) 
$%
, 
since $\CRESPOYYSN \in \CRESPOSETA$, 
%
$\CRESPOOPENINGX{\CRESPOYYSN} (\CRESPOSETA) \neq \emptyset$, 
and 
$\CRESPOOPENINGX{\CRESPOYYSN} (\CRESPOSETA) \not \leq \CRESPOOPENINGX{\CRESPOXXSN} \psi (\CRESPOSETA)$. 

} 
\end {quote} 




We can straightforwardly extend 
%
Property~\ref{\fullpaperid:prope:stable_set} 
to binary functions (which can be considered as equivalent to sets). 
%
Let us assume that the binary values are 0 and 1. 
%
Let $\CRESPOII$ and $\CRESPOII'$ be, respectively, 
an input image and the output of 
an adjacency stable connected operator $\psi$ 
(i.e., $\CRESPOII' =$ $\psi (\CRESPOII)$). 
%



%
\begin{itemize} 
\item[] { 
{\underline {Binary functions case}}: 
If $\CRESPOXXSN$ and $\CRESPOYYSN$ are neighbors to each other 
%
(or if they belong to adjacent flat zones), 
then: 



%
\begin{equation}\label{\fullpaperid:eq:stabilityswitch_binary} 
\begin{tabular}{llll} 
(a) 
& 
$
\CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOXXSN} = \CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOYYSN}
$
& 
$
\Longrightarrow 
$
& 
$
\CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOXXSN} = \CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOYYSN}. 
$ 
\\ 
(b) 
& 
$
\CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOXXSN} = 1, \CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOYYSN} = 0 
$
& 
$
\Longrightarrow
$ 
& 
$
\left\{ 
     \begin{array}{c} 
       \CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOXXSN} = 1, \CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOYYSN} = 0 \\ 
       \mbox{or}  \\ 
       \CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOXXSN} = \CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOYYSN} 
     \end{array} 
\right. 
$ 
\\ 
\end{tabular} 
\end{equation}


%
\noindent 
Note: the case symmetric to~(b) 
(interchanging $\CRESPOXXSN$ and $\CRESPOYYSN$ in (b)) 
is not shown. 

The cases ruled out are: 
(i)~$\CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOXXSN} = 1, \CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOYYSN} = 0$, 
and 
$\CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOXXSN} = 0, \CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOYYSN} = 1$; 
and 
(ii)~$\CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOXXSN} = 0, \CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOYYSN} = 1$, 
and 
$\CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOXXSN} = 1, \CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOYYSN} = 0$%
. 
(Cases~(i) and~(ii) 
are symmetric to each other, interchanging $\CRESPOXXSN$ and $\CRESPOYYSN$.) 


} 

\end{itemize} 





\subsection{Set Levelings: Input/Output Variation Restriction} 
\label{\fullpaperid:sec:SetLevelings_InputOutputRestriction} 


%
Definition~\ref{\fullpaperid:defi:Leveling01} 
can of course 
be applied to binary images (or, equivalently, sets). 
If we particularize for binary images 
then 
we have the following, concerning 
expression~(\ref{\fullpaperid:eq:Leveling01}). 
%
%
%
Similarly as in 
Section~\ref{\fullpaperid:sec:AdjacencyStableConnectedOperators_InputOutputRestriction}, 
$\CRESPOII$ and $\CRESPOII'$ denote the input and output images 
(i.e., $\CRESPOII$ and $\CRESPOII'$ have been substituted for 
$\CRESPOFFLA$ and $\CRESPOGGLA$, respectively, in expression~(\ref{\fullpaperid:eq:Leveling01})). 
%
%
An inequality such as 
$\CRESPOVALUEIMPIXELAA{\CRESPOGGLB}{\CRESPOXXSN} > \CRESPOVALUEIMPIXELAA{\CRESPOGGLB}{\CRESPOYYSN}$ 
can only occur 
when there is a discontinuity where 
$\CRESPOVALUEIMPIXELAA{\CRESPOGGLB}{\CRESPOXXSN}$ and $\CRESPOVALUEIMPIXELAA{\CRESPOGGLB}{\CRESPOYYSN}$ are 1 
and 0, respectively. 
Then, expression~(\ref{\fullpaperid:eq:Leveling01}) 
can only be: 


%
\begin{equation}\label{\fullpaperid:eq:Leveling01_partbinary}
%
1 > 0 ~\Rightarrow~ 1 \geq 1 {\rm ~and~} 0 \geq 0 
\end{equation}

\noindent 
I.e., 
$\CRESPOVALUEIMPIXELAA{\CRESPOFFLB}{\CRESPOXXSN}$ has to be 1, 
and $\CRESPOVALUEIMPIXELAA{\CRESPOFFLB}{\CRESPOYYSN}$ must be 0. 


Therefore, 
if $\CRESPOVALUEIMPIXELAA{\CRESPOGGLB}{\CRESPOXXSN} = 1$ 
and $\CRESPOVALUEIMPIXELAA{\CRESPOGGLB}{\CRESPOYYSN} = 0$, 
the case 
where 
$\CRESPOVALUEIMPIXELAA{\CRESPOFFLB}{\CRESPOXXSN}$ = 0 
and $\CRESPOVALUEIMPIXELAA{\CRESPOFFLB}{\CRESPOYYSN}$ = 1 
is 
%
excluded. 



\vspace*{-0.1177cm}

\subsection{Discussion}
%

\vspace*{-0.1177cm}

We can now precisely state the relationships between 
stable connected operators and set levelings: 
%
(a) 
Both adjacent stable connected operators 
and set levelings impose restrictions on the 
input/output variations. 
(b) 
The imposed restrictions are identical in both cases: 
if $\CRESPOVALUEIMPIXELAA{\CRESPOGGLB}{\CRESPOXXSN} = 1$ 
and $\CRESPOVALUEIMPIXELAA{\CRESPOGGLB}{\CRESPOYYSN} = 0$, 
then 
$\CRESPOVALUEIMPIXELAA{\CRESPOFFLB}{\CRESPOXXSN}$ 
and $\CRESPOVALUEIMPIXELAA{\CRESPOFFLB}{\CRESPOYYSN}$ 
must be 1 and 0, respectively. 



Thus, 
\textbf
{the set leveling concept 
and 
the adjacency stable connected operator concept 
are equivalent}. 
%
The adjacency stable connected operator notion 
\cite{Crespo:crespo93} \cite{Crespo:crespo93c} \cite{Crespo:crespo94e}
is prior in time to levelings 
\cite{Crespo:Meyer1998CoOpLe} \cite{Crespo:Meyer1998Le} \cite{Crespo:MeyeMara:00}
\cite{Crespo:Meyer2004}. 


A final question can arise: 
can the adjacency stable connected operator concept 
be applied to a gray-level framework? 
The answer is affirmative, 
%
%
we can directly extend 
such input and output variation restrictions 
to 
flat gray-level 
%
connected operators 
that commute with thresholding, 
%
where the operator is level-by-level connected 
and where equation~(\ref{\fullpaperid:eq:stable}) 
must hold for all sections of the 
input and output gray-level functions. 
%
%
Let $\CRESPOII$ and $\CRESPOII'$ be, respectively, 
an input image and the output of 
a flat gray-level
adjacency stable connected operator $\psi$ 
(i.e., $\CRESPOII' =$ $\psi (\CRESPOII)$). 


\vspace*{-0.1177cm}

\begin{itemize} 
\item[] 
{\underline {Gray-level (non-binary) functions case}}: 
If $\CRESPOXXSN$ and $\CRESPOYYSN$ are neighbors to each other 
%
(or if they belong to adjacent flat zones), 
then: 


\vspace*{-0.1177cm}
\vspace*{-0.222cm} 
%

%
\begin{equation}\label{\fullpaperid:eq:stabilityswitch_nonbinary} 
\begin{tabular}{llll} 
(a) 
& 
$
\CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOXXSN} = \CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOYYSN}
$
& 
$
\Longrightarrow 
$
& 
$
\CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOXXSN} = \CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOYYSN}. 
$ 
\\ 
(b) 
& 
$
\CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOXXSN} > \CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOYYSN} 
$
& 
$
\Longrightarrow
$ 
& 
$
\left\{ 
     \begin{array}{c} 
       \CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOXXSN} > \CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOYYSN} \\ 
       \mbox{or}  \\ 
       \CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOXXSN} = \CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOYYSN} 
     \end{array} 
\right. 
$ 
\\ 
\end{tabular} 
\end{equation}
%

\vspace*{-0.222cm} 

\vspace*{-0.1177cm}

%
\noindent 
Note: the case symmetric to~(b) is not shown. 

\vspace*{-0.1177cm}

The case ruled out is: 
$\CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOXXSN} < \CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOYYSN}$, 
and 
$\CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOXXSN} > \CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOYYSN}$ 
(as well as the symmetric one: 
$\CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOXXSN} > \CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOYYSN}$, 
and 
$\CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOXXSN} < \CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOYYSN}$%
). 
%
This case is also excluded by 
expression~(\ref{\fullpaperid:eq:Leveling01}) of levelings. 
Moreover, 
disregarding trivial cases
%
{(such as those where, processing level by level, 
$\psi (\emptyset) = \CRESPOEE$ or 
$\psi (\CRESPOEE) = \emptyset$)}%
,
in flat morphological (increasing) 
connected operators 
the variation range of the output 
%
is 
equal or smaller than that of the input. 
Otherwise, the flat zone inclusion relationship 
of connected operators 
would not necessarily hold between sections 
at same level 
of the input and output images. 
For example, in the case where 
$\CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOXXSN} > \CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOYYSN}$ 
and 
$\CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOXXSN} > \CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOYYSN}$, 
the next gradation would exist
%
:
$\CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOXXSN} \geq 
\CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOXXSN} > 
\CRESPOVALUEIMPIXELAA{\CRESPOII'}{\CRESPOYYSN} \geq 
\CRESPOVALUEIMPIXELAA{\CRESPOII}{\CRESPOYYSN}$. 



\end{itemize} 


\vspace*{-0.1177cm}

Now that the equivalence of 
adjacency stable connected operators 
and set levelings has been established, 
researchers know that 
the adjacency stable connected operator properties 
also hold for (flat) levelings. 
Let us point out that the properties included 
in Section~\ref{\fullpaperid:sec:AdjacencyStability} 
(Lemma~\ref{\fullpaperid:lem:stable_adjacent_0} 
and Properties~\ref{\fullpaperid:prop:stable_extensive_antiextensive} 
and~\ref{\fullpaperid:prop:stable_stable}), 
altogether with 
the adjacency stability equation~(\ref{\fullpaperid:eq:stable}), 
are useful 
for manipulating expressions composed of 
connected operators, 
especially when they are also 
connected-component local 
(i.e., they treat grains or pores independently from the rest) 
\cite{Crespo:crespo93} \cite{Crespo:crespo93c} \cite{Crespo:crespo94e}.
%
(In fact, they will be employed in a proof 
in the following section.) 
%





%
\section{On Commutative Properties of Open-Close and Close-Open Filters for Connected Operators}
\label{\fullpaperid:sec:OpeningClosingProduct}

\vspace*{-0.1177cm}

In this section we will first comment on an interesting commutative property 
that has been presented elsewhere, 
and on some of its implications and limits. 
Afterwards, we will also present another commutative property 
that is satisfied by certain alternated connected filters, 
particularly by attribute alternated filters. 



\vspace*{-0.1177cm}
\vspace*{-0.1177cm}
\vspace*{-0.1177cm}

\vspace*{-0.1177cm}

%
%
\subsection{Marker-Based Operators} 
\label{\fullpaperid:sec:CommutativeProperty_StrongProperty_Clarification}


Let us define the 
%
~``\CRESPORELATIONAUXAA''~ 
relationship 
%
presented in \cite[Definition~7.3]{Crespo:Serra2000}: 

\vspace*{-0.1177cm}

\begin{definition} 
Let $A$ and $B$ be, respectively, a grain (a connected set) 
and a set. 
We say that $\CRESPOSETA ~\CRESPORELATIONAUXAA~ \CRESPOSETB$ if 
$\CRESPOSETA$ and $\CRESPOSETB$ have a non-empty intersection 
or are adjacent. 
\end{definition}


\vspace*{-0.1177cm}

Now, we are going to define two operators, 
$\CRESPOOPENINGOVER$ and $\CRESPOCLOSINGUNDER$, 
based on makers that are presented in \cite[p.~176]{Crespo:Serra2000}. 
Besides one normal input set, those operators use a second one, 
which is a marker set. 
%

\vspace*{-0.1177cm}

\begin{definition} \label{\fullpaperid:defi:openingover_markerbased} 
%
Let $\CRESPOSETA$ and $\CRESPOSETMARKER$ be two sets. 
The 
connected operator 
$\CRESPOOPENINGOVER$ 
%
of a set $\CRESPOSETA$ based on marker $\CRESPOSETMARKER$, 
symbolized by 
$\CRESPOOPENINGMARKERB{\CRESPOSETA}{\CRESPOSETMARKER}$, 
is defined as: 

\vspace*{-0.11111cm} 
\vspace*{-0.1177cm}


%
\begin{equation}\label{\fullpaperid:eq:openingover_markerbased} 
\CRESPOOPENINGMARKERB{\CRESPOSETA}{\CRESPOSETMARKER} = \bigcup
%
\{ \CRESPOOPENINGX{\CRESPOXXS} (\CRESPOSETA) : \CRESPOOPENINGX{\CRESPOXXS} (\CRESPOSETA) ~\CRESPORELATIONAUXAA~ \CRESPOSETMARKER \} 
\end{equation}


\end{definition}

\vspace*{-0.11111cm} 

\noindent 
Note that 
Definition~\ref{\fullpaperid:defi:openingover_markerbased} does not define 
$\CRESPOOPENINGOVER (\CRESPOSETA)$ 
but 
$\CRESPOOPENINGMARKERB{\CRESPOSETA}{\CRESPOSETMARKER}$. 

\vspace*{-0.1177cm}

\begin{definition} \label{\fullpaperid:defi:closingover_markerbased} 
%
Let $\CRESPOSETA$ and $\CRESPOSETMARKER$ be two sets. 
The 
connected operator 
%
$\CRESPOCLOSINGUNDER$ 
of a set $\CRESPOSETA$ based on marker $\CRESPOSETMARKERB$, 
symbolized by 
$\CRESPOCLOSINGMARKERB{\CRESPOSETA}{\CRESPOSETMARKERB}$, 
is defined as: 

\vspace*{-0.1177cm}


%
\begin{equation}\label{\fullpaperid:eq:closingunder_markerbased_Serra2000} 
\CRESPOCOMPLEMENT [\CRESPOCLOSINGMARKERB {\CRESPOSETA}{\CRESPOCOMPLEMENT \CRESPOSETMARKERB}] = 
\bigcup \{ 
\CRESPOOPENINGX{\CRESPOXXS}(\CRESPOCOMPLEMENT \CRESPOSETA), \CRESPOXXS \in \CRESPOEE: 
\CRESPOOPENINGX{\CRESPOXXS}(\CRESPOCOMPLEMENT \CRESPOSETA) ~\CRESPORELATIONAUXAA~ \CRESPOSETMARKERB 
        \} 
\end{equation}

\end{definition}


\vspace*{-0.11111cm} 

In \cite{Crespo:Serra2000}, it is established that, 
under some connectivity considerations, 
there exists a commutative property for 
$\CRESPOOPENINGOVER$ and $\CRESPOCLOSINGUNDER$ 
%
(\cite[Theorem~7.3]{Crespo:Serra2000}): 

\vspace*{-0.222cm} 
\vspace*{-0.1177cm}


%
\begin{equation} \label{\fullpaperid:eq:openingmarkerclosingmarkercommutativity} 
\CRESPOOPENINGMARKERB
{\CRESPOCLOSINGMARKERB{\CRESPOSETA}{\CRESPOCOMPLEMENT \CRESPOSETMARKERB}}
{\CRESPOSETMARKER} 
=
\CRESPOCLOSINGMARKERB
{\CRESPOOPENINGMARKERB{\CRESPOSETA}{\CRESPOSETMARKER}}
{\CRESPOCOMPLEMENT \CRESPOSETMARKERB} 
\end{equation}



It is indicated in \cite{Crespo:Serra2000} 
that 
$\CRESPOOPENINGMARKERB
{\CRESPOCLOSINGMARKERB{\CRESPOSETA}{\CRESPOCOMPLEMENT \CRESPOSETMARKERB}}
{\CRESPOSETMARKER}$ 
(or 
$\CRESPOCLOSINGMARKERB
{\CRESPOOPENINGMARKERB{\CRESPOSETA}{\CRESPOSETMARKER}}
{\CRESPOCOMPLEMENT \CRESPOSETMARKERB}$%
)
is a leveling and that is a \Index{strong filter}. 
%
%
Moreover, 
in 
\cite{Crespo:MeyeMara:00} \cite{Crespo:Meyer2004}, 
it is mentioned that levelings are 
strong filters
(see \cite[Section~3.4 (p.~9)]{Crespo:MeyeMara:00}
and \cite[Section~4.4.1 (p.~67)]{Crespo:Meyer2004})%
, and the discussion 
refers to a commutative expression 
similar to the aforementioned one. 
%
This should be clarified, because it seems 
there could be some confusion 
%
%
about levelings 
(Definition~\ref{\fullpaperid:defi:Leveling01}), 
particularly 
about whether all levelings 
%
%
can be formulated as 
%
sequential compositions of an opening and a closing, and vice-versa. 
%




There is an important remark, 
which concerns the main cause 
%
of possible misconceptions, 
to be made 
about 
equation~(\ref{\fullpaperid:eq:openingmarkerclosingmarkercommutativity}): 
the computation of the markers is \emph{not} considered. 
%
Thus, 
\textbf{saying that an operation can be expressed as 
in equation~(\ref{\fullpaperid:eq:openingmarkerclosingmarkercommutativity}) 
does not necessarily imply that such an operation can be expressed 
as a sequential composition of an opening and a closing, 
and vice-versa, \emph{when} 
%
the marker computation is considered}. 
%
%
For example, 
in 
$\CRESPOOPENINGMARKERB
{\CRESPOCLOSINGMARKERB{\CRESPOSETA}{\CRESPOCOMPLEMENT \CRESPOSETMARKERB}}
{\CRESPOSETMARKER}$, 
the marker $\CRESPOSETMARKER$ of 
$\CRESPOOPENINGOVER$ is not affected by the result 
of the previously applied $\CRESPOCLOSINGUNDER$; 
that is, the $\CRESPOSETMARKER$ marker can be considered as an input. 
%
%
This is not how we usually think about openings and closings. 
%
In fact, when the marker computation is adequately considered, 
not all levelings 
(Definition~\ref{\fullpaperid:defi:Leveling01}) 
can be expressed as a sequential composition 
of an opening and a closing, 
and vice-versa; 
and it would not be the case that all levelings 
are strong filters. 
%
(See 
\cite{Crespo:CresMaojSanaBillMuno2002} 
\cite{Crespo:CresMaoj200702:Sub} 
for further discussion about the strong property of 
connected alternated filters%
.) 




Next section will present a commutative property for 
certain connected filters that does not raise that objection 
but where the underlying marker 
is, as usual, not considered fixed. 
%


\vspace*{-0.1177cm}
\vspace*{-0.1177cm}
\vspace*{-0.1177cm}
\vspace*{-0.1177cm}

\subsection{A Commutative Property} 
%
\label{\fullpaperid:sec:CommutativeProperty_StrongProperty_AttributeAlternatedFilter}


\vspace*{-0.1177cm}


We present a commutative property for 
a certain type of connected alternated filters, 
particularly for 
\Index{alternated attribute filters}, 
which are strong filters 
composed of an \emph{attribute opening} 
\cite{Crespo:serra88} \cite{Crespo:serra93} 
%
\cite{Crespo:vincent93b} 
\cite{Crespo:soille2003sv} 
and an \emph{attribute closing}. 
%
Area openings and closings are examples of attribute 
openings and closings, respectively. 


Let $\CRESPOOPENINGATTRIB$ and $\CRESPOCLOSINGATTRIB$ 
denote, respectively an 
attribute 
%
opening and closing. 
%
In the following, 
unlike in equation~(\ref{\fullpaperid:eq:openingmarkerclosingmarkercommutativity}), 
when we write 
$\CRESPOCLOSINGATTRIB \CRESPOOPENINGATTRIB$ 
it is clear that 
the criterion 
(and associated marker) 
of $\CRESPOCLOSINGATTRIB$ 
is applied to the output computed by the previous $\CRESPOOPENINGATTRIB$. 
%
We have the following property: 



\begin{property} \label{\fullpaperid:prope:alternatedattributefilter_commutativity} 
An attribute alternated 
%
filter $\CRESPOCLOSINGATTRIB \CRESPOOPENINGATTRIB$ 
can be expressed as a commutative sequential composition 
of an opening and a closing as follows: 
%
\vspace*{-0.1177cm}
\vspace*{-0.1177cm}

%
\begin{equation} \label{\fullpaperid:eq:alternatedattributefilter_commutativity}
\CRESPOCLOSINGATTRIB \CRESPOOPENINGATTRIB 
= 
\CRESPOOPENINGATTRIB\, (\CRESPOID \bigvee \CRESPOCLOSINGATTRIB \CRESPOOPENINGATTRIB) 
= 
(\CRESPOID \bigvee \CRESPOCLOSINGATTRIB \CRESPOOPENINGATTRIB)\, \CRESPOOPENINGATTRIB 
\end{equation}
\end{property}

{

%


{\CRESPOmyproofstyle 
{Proof of Property~\ref{\fullpaperid:prope:alternatedattributefilter_commutativity}:} 
There are two equalities to consider. 

\vspace*{-0.1177cm}

\begin{itemize} 

\vspace*{-0.1177cm}


%
\item[(a)] 
%
%
Let $\CRESPOSETA$ be a set. 
Since $\tilde {\gamma}$ is 
%
connected-component~local 
%
%
we have 
%
$\tilde {\gamma} =$ 
$\bigvee _{x} \CRESPOOPENINGX{\CRESPOXXS} \tilde {\gamma} =$ 
$\bigvee _{x} \tilde {\gamma} \CRESPOOPENINGX{\CRESPOXXS}$%
. 
%
Thus, 
$\tilde {\gamma} (\CRESPOID \bigvee \tilde {\varphi} \tilde {\gamma}) (\CRESPOSETA) 
=  
\bigvee _{x} \CRESPOOPENINGX{\CRESPOXXS} \tilde {\gamma} (\CRESPOID \bigvee \tilde {\varphi} \tilde {\gamma}) (\CRESPOSETA) 
=$
$\bigvee _{x} \tilde {\gamma} \CRESPOOPENINGX{\CRESPOXXS}(\CRESPOID \bigvee \tilde {\varphi} \tilde {\gamma}) (\CRESPOSETA)$. 
%
%
From 
\CRESPOPROPERTY~\ref{\fullpaperid:prop:ext_antiext} and 
\CRESPOPROPERTY~\ref{\fullpaperid:prop:stable_stable}, 
$\tilde {\varphi} \tilde {\gamma}$ is adjacency stable, 
and, from Lemma~\ref{\fullpaperid:lem:stable_adjacent_0}, 
$\tilde {\varphi} \tilde {\gamma} (\CRESPOSETA)$ and 
$A \setminus \tilde {\varphi} \tilde {\gamma} (\CRESPOSETA)$ 
are not adjacent 
\cite{Crespo:crespo93c} \cite{Crespo:crespo94e}. 
Then, 

\vspace*{-0.1177cm}
\vspace*{-0.222cm} 
%
\begin {equation} \label{\fullpaperid:eq:proof_th} 
\tilde {\gamma} \CRESPOOPENINGX{\CRESPOXXS} (\CRESPOID \bigvee \tilde {\varphi} \tilde {\gamma}) (\CRESPOSETA) =  
\left \{ 
\begin  {array} {ll} 
\tilde {\gamma} \CRESPOOPENINGX{\CRESPOXXS} (\CRESPOSETA) = \CRESPOEMPTYSET, 
& x \in A \setminus \tilde {\varphi} \tilde {\gamma} (\CRESPOSETA). \\ 
\tilde {\gamma} \CRESPOOPENINGX{\CRESPOXXS} \tilde {\varphi} \tilde {\gamma} (\CRESPOSETA), 
& x \in \tilde {\varphi} \tilde {\gamma} (\CRESPOSETA). 
\end {array} 
\right. 
\end {equation} 

\vspace*{-0.1177cm}

%
Thus, 
$\bigvee _{x} \tilde {\gamma} \CRESPOOPENINGX{\CRESPOXXS} \tilde {\varphi} \tilde {\gamma} =$ 
$\bigvee _{x} \CRESPOOPENINGX{\CRESPOXXS} \tilde {\gamma} \tilde {\varphi} \tilde {\gamma} =$ 
$\tilde {\gamma} \tilde {\varphi} \tilde {\gamma}$. 
Finally, $\tilde {\gamma} \tilde {\varphi} \tilde {\gamma} =$ 
$\tilde {\varphi} \tilde {\gamma}$ 
(since 
$\tilde {\varphi} \tilde {\gamma} \leq$ 
$\tilde {\gamma} \tilde {\varphi}$ 
and 
$\tilde {\gamma} \tilde {\varphi} \tilde {\gamma} =$ 
$\tilde {\varphi} \tilde {\gamma}$ 
\cite{Crespo:serra93} \cite{Crespo:salembier95}). 



\item[(b)] 
%

$(\CRESPOID \bigvee \CRESPOCLOSINGATTRIB \CRESPOOPENINGATTRIB)\, \CRESPOOPENINGATTRIB$ 
=
$\CRESPOOPENINGATTRIB \bigvee \CRESPOCLOSINGATTRIB \CRESPOOPENINGATTRIB \CRESPOOPENINGATTRIB$ 
=
$\CRESPOOPENINGATTRIB \bigvee \CRESPOCLOSINGATTRIB \CRESPOOPENINGATTRIB$ 
=
$\CRESPOCLOSINGATTRIB \CRESPOOPENINGATTRIB$. 



\end{itemize} 

}


%


} 



\noindent 
Notes: (a)~$(\CRESPOID \bigvee \CRESPOCLOSINGATTRIB \CRESPOOPENINGATTRIB)$ 
is a closing 
%
(and different from 
$\CRESPOCLOSINGATTRIB$). 
%
(b)~Property~\ref{\fullpaperid:prope:alternatedattributefilter_commutativity} 
is different from 
%
%
\cite[Proposition~10.2]{Crespo:heijmans99}. 
%
%
(c)~This proof also provides an example of 
using adjacent stable connected operators properties 
to manipulate 
expressions. 
%
(d)~Concerning filter expressions and 
decompositions, see also~\cite{Crespo:crespo95f}. 


\vspace*{-0.1177cm}
\vspace*{-0.1177cm}

\section{Conclusion} 
\label{\fullpaperid:sec:Conclusion} 


\vspace*{-0.111111cm}

This paper has focused on connected morphological operators. 
%
First, the relationship between 
adjacency stable connected operators 
and set levelings 
has been investigated, 
and a close relationship, an equivalence, has been identified. 
This is important to establish how and when 
this concept has been introduced, 
and to clarify 
its origin. 
%
In addition, 
properties satisfied by one class of operators 
can be applied to the other equivalent one. 

Second, 
this work has analyzed 
a commutative property previously presented, 
%
as well as some of its implications and limits. 
%
%
The paper has presented also a commutative property 
for attribute alternated filters, in which, 
as is usually the case and unlike in the 
other commutative property analyzed, 
the underlying marker 
%
computation is taken into account. 





\vspace*{-0.1177cm}

\section*{Acknowledgement} 
\label{\fullpaperid:sec:Acknowledgement} 


The author wants to thank 
the comments of Dr. Ronse about strong connectivities, 
as well as the remarks of an anonymous reviewer. 



%
% BIBLIOGRAPHY
% for details, see ftp://ftp.ams.org/pub/tex/amsrefs/amsrdoc.pdf
%
\begin{bibsection}
\begin{biblist}

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\bib{Crespo:crespo94}{article}{
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}

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}

\bib{Crespo:Jone:99}{article}{
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}

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}

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     address={Dordrecht},
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}

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}

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}

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}

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}

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}

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}

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\end{biblist}

\end{bibsection}