\relax 
\ifx\hyper@anchor\@undefined
\global \let \oldcontentsline\contentsline
\gdef \contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global \let \oldnewlabel\newlabel
\gdef \newlabel#1#2{\newlabelxx{#1}#2}
\gdef \newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\let \contentsline\oldcontentsline
\let \newlabel\oldnewlabel}
\else
\global \let \hyper@last\relax 
\fi

\bibstyle{amsrn}
\citation{Crespo:crespo93c}
\citation{Crespo:crespo94e}
\citation{Crespo:crespo93}
\citation{Crespo:Meyer1998CoOpLe}
\citation{Crespo:Meyer1998Le}
\citation{Crespo:Meyer2004}
\citation{Crespo:MeyeMara:00}
\citation{Crespo:Meyer2004}
\select@language{english}
\@writefile{toc}{\select@language{english}}
\@writefile{lof}{\select@language{english}}
\@writefile{lot}{\select@language{english}}
\@writefile{toc}{\contentsline {chapter}{\numberline {1}Adjacency stable connected operators and set levelings}{215}{chapter.1}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{215}{section.1.1}}
\newlabel{xxx:sec:Introduction}{{1}{215}{Introduction\relax }{section.1.1}{}}
\citation{Crespo:ronse:guest}
\citation{Crespo:Banon:2000:FoInDi}
\citation{Crespo:DougLotu:03}
\citation{Crespo:giardina88}
\citation{Crespo:heijmans94}
\citation{Crespo:matheron75}
\citation{Crespo:SchmittMattioli93}
\citation{Crespo:serra82}
\citation{Crespo:serra88}
\citation{Crespo:soille2003sv}
\citation{Crespo:serra88}
\citation{Crespo:serra82}
\citation{Crespo:soille2003sv}
\citation{Crespo:braga-neto:multiscale}
\citation{Crespo:braga-goutsias2003CVIU}
\citation{Crespo:braga-goutsias2003JMIV}
\citation{Crespo:bb2233}
\citation{Crespo:crespo95f}
\citation{Crespo:CresMaojSanaBillMuno2002}
\citation{Crespo:crespo94e}
\citation{Crespo:crespo93}
\citation{Crespo:crespo94}
\citation{Crespo:GarrSaleGarc1998}
\citation{Crespo:heijmans99}
\citation{Crespo:Jone:99}
\citation{Crespo:Meyer1998CoOpLe}
\citation{Crespo:Meyer1998Le}
\citation{Crespo:Meyer2004}
\citation{Crespo:MeyeMara:00}
\citation{Crespo:Ronse1998}
\citation{Crespo:RonseSerra2001}
\citation{Crespo:SalembierGarrido2000}
\citation{Crespo:salembier95}
\citation{Crespo:serra98}
\citation{Crespo:Serra2000}
\citation{Crespo:serra93}
\citation{Crespo:soille2003sv}
\citation{Crespo:TeroVarg2002}
\citation{Crespo:VargCresMaoj2003}
\citation{Crespo:vincent93b}
\@writefile{toc}{\contentsline {section}{\numberline {2}Background}{216}{section.1.2}}
\newlabel{xxx:sec:Background}{{2}{216}{Background\relax }{section.1.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}General definitions}{216}{subsection.1.2.1}}
\citation{Crespo:serra88}
\citation{Crespo:serra88}
\citation{Crespo:salembier95}
\citation{Crespo:serra93}
\citation{Crespo:heijmans99}
\citation{Crespo:Ronse1998}
\citation{Crespo:crespo93}
\citation{Crespo:crespo93c}
\citation{Crespo:crespo94e}
\citation{Crespo:heijmans99}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Some background on connectivity and connected operators}{217}{subsection.1.2.2}}
\newlabel{xxx:eq:gammax}{{1}{217}{Some background on connectivity and connected operators\relax }{equation.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Connectivity requirement}{217}{subsection.1.2.3}}
\newlabel{xxx:unusual_connectivities}{{2.3}{217}{Connectivity requirement\relax }{subsection.1.2.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Adjacency stability}{217}{section.1.3}}
\newlabel{xxx:sec:AdjacencyStability}{{3}{217}{Adjacency stability\relax }{section.1.3}{}}
\newlabel{xxx:defi:adjacency_flatzones}{{1}{217}{Adjacency stability\relax }{definition.1}{}}
\citation{Crespo:crespo93c}
\citation{Crespo:crespo94e}
\citation{Crespo:crespo93}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Adjacency stability example}}{218}{figure.1.1}}
\newlabel{xxx:fig-stable}{{1}{218}{Adjacency stability\relax }{figure.1.1}{}}
\newlabel{xxx:eq:stable}{{2}{218}{Adjacency stability\relax }{equation.2}{}}
\newlabel{xxx:defi:stable}{{2}{218}{Adjacency stability\relax }{equation.2}{}}
\newlabel{xxx:lem:stable_adjacent_0}{{1}{218}{Adjacency stability\relax }{lemma.1}{}}
\citation{Crespo:Meyer1998CoOpLe}
\citation{Crespo:Meyer1998Le}
\citation{Crespo:Meyer2004}
\citation{Crespo:MeyeMara:00}
\citation{Crespo:Meyer1998CoOpLe}
\citation{Crespo:MeyeMara:00}
\citation{Crespo:Meyer2004}
\citation{Crespo:Meyer2004}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Adjacency stability equation}}{219}{figure.1.2}}
\newlabel{xxx:fig-stable-test}{{2}{219}{Adjacency stability\relax }{figure.1.2}{}}
\newlabel{xxx:prop:stable_extensive_antiextensive}{{1}{219}{Adjacency stability\relax }{property.1}{}}
\newlabel{xxx:prop:ext_antiext}{{1}{219}{Adjacency stability\relax }{property.1}{}}
\newlabel{xxx:prop:stable_stable}{{2}{219}{Adjacency stability\relax }{property.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4}Levelings and set levelings}{219}{section.1.4}}
\newlabel{xxx:sec:Levelings}{{4}{219}{Levelings and set levelings\relax }{section.1.4}{}}
\newlabel{xxx:sec:SetLevelings}{{4}{219}{Levelings and set levelings\relax }{section.1.4}{}}
\newlabel{xxx:eq:Leveling01}{{3}{219}{Levelings and set levelings\relax }{equation.3}{}}
\newlabel{xxx:defi:Leveling01}{{3}{219}{Levelings and set levelings\relax }{equation.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5}Relationships between adjacency stable connected operators and set levelings}{220}{section.1.5}}
\newlabel{xxx:sec:RelationshipsAS_SL}{{5}{220}{Relationships between adjacency stable connected operators and set levelings\relax }{section.1.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Adjacency stable connected operators: input/out\discretionary {-}{}{}put variations restriction}{220}{subsection.1.5.1}}
\newlabel{xxx:sec:AdjacencyStableConnectedOperators_InputOutputRestriction}{{5.1}{220}{Adjacency stable connected operators: input/out\-put variations restriction\relax }{subsection.1.5.1}{}}
\newlabel{xxx:eq:neighbors_gammax}{{4}{220}{Adjacency stable connected operators: input/out\-put variations restriction\relax }{equation.4}{}}
\newlabel{xxx:prope:stable_set}{{3}{220}{Adjacency stable connected operators: input/out\-put variations restriction\relax }{property.3}{}}
\newlabel{xxx:eq:stabilityswitch_set}{{5}{220}{Adjacency stable connected operators: input/out\-put variations restriction\relax }{equation.5}{}}
\citation{Crespo:crespo93c}
\citation{Crespo:crespo94e}
\citation{Crespo:crespo93}
\citation{Crespo:Meyer1998CoOpLe}
\citation{Crespo:Meyer1998Le}
\citation{Crespo:MeyeMara:00}
\citation{Crespo:Meyer2004}
\newlabel{xxx:eq:stabilityswitch_binary}{{6}{221}{Adjacency stable connected operators: input/out\-put variations restriction\relax }{equation.6}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Set levelings: input/output variation restriction}{221}{subsection.1.5.2}}
\newlabel{xxx:sec:SetLevelings_InputOutputRestriction}{{5.2}{221}{Set levelings: input/output variation restriction\relax }{subsection.1.5.2}{}}
\newlabel{xxx:eq:Leveling01_partbinary}{{7}{221}{Set levelings: input/output variation restriction\relax }{equation.7}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3}Discussion}{221}{subsection.1.5.3}}
\citation{Crespo:crespo93c}
\citation{Crespo:crespo94e}
\citation{Crespo:crespo93}
\citation{Crespo:Serra2000}
\newlabel{xxx:eq:stabilityswitch_nonbinary}{{8}{222}{Discussion\relax }{equation.8}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6}On commutative properties of open-close and close-open filters for connected operators}{222}{section.1.6}}
\newlabel{xxx:sec:OpeningClosingProduct}{{6}{222}{On commutative properties of open-close and close-open filters for connected operators\relax }{section.1.6}{}}
\citation{Crespo:Serra2000}
\citation{Crespo:Serra2000}
\citation{Crespo:Serra2000}
\citation{Crespo:Serra2000}
\citation{Crespo:Meyer2004}
\citation{Crespo:MeyeMara:00}
\citation{Crespo:MeyeMara:00}
\citation{Crespo:Meyer2004}
\citation{Crespo:CresMaojSanaBillMuno2002}
\citation{Crespo:CresMaoj200702:Sub}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.1}Marker-based operators}{223}{subsection.1.6.1}}
\newlabel{xxx:sec:CommutativeProperty_StrongProperty_Clarification}{{6.1}{223}{Marker-based operators\relax }{subsection.1.6.1}{}}
\newlabel{xxx:defi:openingover_markerbased}{{5}{223}{Marker-based operators\relax }{definition.5}{}}
\newlabel{xxx:eq:openingover_markerbased}{{9}{223}{Marker-based operators\relax }{equation.9}{}}
\newlabel{xxx:defi:closingover_markerbased}{{6}{223}{Marker-based operators\relax }{definition.6}{}}
\newlabel{xxx:eq:closingunder_markerbased_Serra2000}{{10}{223}{Marker-based operators\relax }{equation.10}{}}
\newlabel{xxx:eq:openingmarkerclosingmarkercommutativity}{{11}{223}{Marker-based operators\relax }{equation.11}{}}
\citation{Crespo:serra88}
\citation{Crespo:serra93}
\citation{Crespo:vincent93b}
\citation{Crespo:soille2003sv}
\citation{Crespo:crespo93c}
\citation{Crespo:crespo94e}
\citation{Crespo:salembier95}
\citation{Crespo:serra93}
\citation{Crespo:heijmans99}
\citation{Crespo:crespo95f}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.2}A commutative property}{224}{subsection.1.6.2}}
\newlabel{xxx:sec:CommutativeProperty_StrongProperty_AttributeAlternatedFilter}{{6.2}{224}{A commutative property\relax }{subsection.1.6.2}{}}
\newlabel{xxx:prope:alternatedattributefilter_commutativity}{{4}{224}{A commutative property\relax }{property.4}{}}
\newlabel{xxx:eq:alternatedattributefilter_commutativity}{{12}{224}{A commutative property\relax }{equation.12}{}}
\newlabel{xxx:eq:proof_th}{{13}{224}{A commutative property\relax }{equation.13}{}}
\@writefile{toc}{\contentsline {section}{\numberline {7}Conclusion}{224}{section.1.7}}
\newlabel{xxx:sec:Conclusion}{{7}{224}{Conclusion\relax }{section.1.7}{}}
\bibcite{Crespo:Banon:2000:FoInDi}{{1}{}}
\bibcite{Crespo:braga-neto:multiscale}{{2}{}}
\bibcite{Crespo:braga-goutsias2003CVIU}{{3}{}}
\bibcite{Crespo:braga-goutsias2003JMIV}{{4}{}}
\bibcite{Crespo:bb2233}{{5}{}}
\bibcite{Crespo:crespo93c}{{6}{}}
\bibcite{Crespo:crespo95f}{{7}{}}
\bibcite{Crespo:CresMaoj200702:Sub}{{8}{}}
\bibcite{Crespo:CresMaojSanaBillMuno2002}{{9}{}}
\bibcite{Crespo:crespo94e}{{10}{}}
\bibcite{Crespo:crespo93}{{11}{}}
\bibcite{Crespo:crespo94}{{12}{}}
\bibcite{Crespo:DougLotu:03}{{13}{}}
\bibcite{Crespo:GarrSaleGarc1998}{{14}{}}
\bibcite{Crespo:giardina88}{{15}{}}
\bibcite{Crespo:heijmans94}{{16}{}}
\newlabel{xxx:sec:Acknowledgement}{{7}{225}{\relax }{section.1.7}{}}
\bibcite{Crespo:heijmans99}{{17}{}}
\bibcite{Crespo:Jone:99}{{18}{}}
\bibcite{Crespo:matheron75}{{19}{}}
\bibcite{Crespo:Meyer1998CoOpLe}{{20}{}}
\bibcite{Crespo:Meyer1998Le}{{21}{}}
\bibcite{Crespo:Meyer2004}{{22}{}}
\bibcite{Crespo:MeyeMara:00}{{23}{}}
\bibcite{Crespo:Ronse1998}{{24}{}}
\bibcite{Crespo:ronse:guest}{{25}{}}
\bibcite{Crespo:RonseSerra2001}{{26}{}}
\bibcite{Crespo:SalembierGarrido2000}{{27}{}}
\bibcite{Crespo:salembier95}{{28}{}}
\bibcite{Crespo:SchmittMattioli93}{{29}{}}
\bibcite{Crespo:serra82}{{30}{}}
\bibcite{Crespo:serra88}{{31}{}}
\bibcite{Crespo:serra98}{{32}{}}
\bibcite{Crespo:Serra2000}{{33}{}}
\bibcite{Crespo:serra93}{{34}{}}
\bibcite{Crespo:soille2003sv}{{35}{}}
\bibcite{Crespo:TeroVarg2002}{{36}{}}
\bibcite{Crespo:VargCresMaoj2003}{{37}{}}
\bibcite{Crespo:vincent93b}{{38}{}}
\newlabel{[bibenv:1]}{13.21579pt}
