Ajout de la section sur la granulometrie.
authorGreg Burri <greg.burri@gmail.com>
Thu, 28 Jan 2016 22:40:40 +0000 (23:40 +0100)
committerGreg Burri <greg.burri@gmail.com>
Thu, 28 Jan 2016 22:40:40 +0000 (23:40 +0100)
rapport/img/RBC_noyau_cytoplasme.eps [new file with mode: 0644]
rapport/img/granulometrie-ellipse-cercle-closing-area.eps [new file with mode: 0644]
rapport/img/granulometrie-ellipse-cercle-closing-morpho.eps [new file with mode: 0644]
rapport/img/histogram_probleme_teinte.eps [new file with mode: 0644]
rapport/img/histogram_segmentation_grayFlat.eps [new file with mode: 0644]
rapport/img/pattern_spectrum_granulometry.eps [new file with mode: 0644]
rapport/main.tex

diff --git a/rapport/img/RBC_noyau_cytoplasme.eps b/rapport/img/RBC_noyau_cytoplasme.eps
new file mode 100644 (file)
index 0000000..d920f1e
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+[ 0.24 0 0 0.24 107 59.039368 ] concat
+/DeviceRGB setcolorspace
+8 dict dup begin
+  /ImageType 1 def
+  /Width 226 def
+  /Height 25 def
+  /Interpolate false def
+  /BitsPerComponent 8 def
+  /Decode [ 0 1 0 1 0 1 ] def
+  /DataSource currentfile /ASCII85Decode filter /FlateDecode filter def
+  /ImageMatrix [ 1 0 0 -1 0 25 ] def
+end
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+  /Decode [ 0 1 0 1 0 1 ] def
+  /DataSource currentfile /ASCII85Decode filter /FlateDecode filter def
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new file mode 100644 (file)
index 0000000..37b8d66
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new file mode 100644 (file)
index 0000000..e73bf13
--- /dev/null
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+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: cairo 1.14.1 (http://cairographics.org)
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+%%Pages: 1
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 3
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+%%EndComments
+%%BeginProlog
+save
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+/h { closepath } bind def
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+/S { stroke } bind def
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new file mode 100644 (file)
index 0000000..835e5d6
--- /dev/null
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+%%Pages: 1
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+cp } bd
+/_ctm matrix def
+/_tm matrix def
+/BT { _ctm currentmatrix pop matrix _tm copy pop 0 0 moveto } bd
+/ET { _ctm setmatrix } bd
+/iTm { _ctm setmatrix _tm concat } bd
+/Tm { _tm astore pop iTm 0 0 moveto } bd
+/ux 0.0 def
+/uy 0.0 def
+/F {
+  /Tp exch def
+  /Tf exch def
+  Tf findfont Tp scalefont setfont
+  /cf Tf def  /cs Tp def
+} bd
+/ULS {currentpoint /uy exch def /ux exch def} bd
+/ULE {
+  /Tcx currentpoint pop def
+  gsave
+  newpath
+  cf findfont cs scalefont dup
+  /FontMatrix get 0 get /Ts exch def /FontInfo get dup
+  /UnderlinePosition get Ts mul /To exch def
+  /UnderlineThickness get Ts mul /Tt exch def
+  ux uy To add moveto  Tcx uy To add lineto
+  Tt setlinewidth stroke
+  grestore
+} bd
+/OLE {
+  /Tcx currentpoint pop def
+  gsave
+  newpath
+  cf findfont cs scalefont dup
+  /FontMatrix get 0 get /Ts exch def /FontInfo get dup
+  /UnderlinePosition get Ts mul /To exch def
+  /UnderlineThickness get Ts mul /Tt exch def
+  ux uy To add cs add moveto Tcx uy To add cs add lineto
+  Tt setlinewidth stroke
+  grestore
+} bd
+/SOE {
+  /Tcx currentpoint pop def
+  gsave
+  newpath
+  cf findfont cs scalefont dup
+  /FontMatrix get 0 get /Ts exch def /FontInfo get dup
+  /UnderlinePosition get Ts mul /To exch def
+  /UnderlineThickness get Ts mul /Tt exch def
+  ux uy To add cs 10 mul 26 idiv add moveto Tcx uy To add cs 10 mul 26 idiv add lineto
+  Tt setlinewidth stroke
+  grestore
+} bd
+/QT {
+/Y22 exch store
+/X22 exch store
+/Y21 exch store
+/X21 exch store
+currentpoint
+/Y21 load 2 mul add 3 div exch
+/X21 load 2 mul add 3 div exch
+/X21 load 2 mul /X22 load add 3 div
+/Y21 load 2 mul /Y22 load add 3 div
+/X22 load /Y22 load curveto
+} bd
+/SSPD {
+dup length /d exch dict def
+{
+/v exch def
+/k exch def
+currentpagedevice k known {
+/cpdv currentpagedevice k get def
+v cpdv ne {
+/upd false def
+/nullv v type /nulltype eq def
+/nullcpdv cpdv type /nulltype eq def
+nullv nullcpdv or
+{
+/upd true def
+} {
+/sametype v type cpdv type eq def
+sametype {
+v type /arraytype eq {
+/vlen v length def
+/cpdvlen cpdv length def
+vlen cpdvlen eq {
+0 1 vlen 1 sub {
+/i exch def
+/obj v i get def
+/cpdobj cpdv i get def
+obj cpdobj ne {
+/upd true def
+exit
+} if
+} for
+} {
+/upd true def
+} ifelse
+} {
+v type /dicttype eq {
+v {
+/dv exch def
+/dk exch def
+/cpddv cpdv dk get def
+dv cpddv ne {
+/upd true def
+exit
+} if
+} forall
+} {
+/upd true def
+} ifelse
+} ifelse
+} if
+} ifelse
+upd true eq {
+d k v put
+} if
+} if
+} if
+} forall
+d length 0 gt {
+d setpagedevice
+} if
+} bd
+%%EndResource
+%%BeginResource: procset (Apache XML Graphics EPS ProcSet) 1.0 0
+%%Version: 1.0 0
+%%Copyright: (Copyright 2002-2003 The Apache Software Foundation. License terms: http://www.apache.org/licenses/LICENSE-2.0)
+/BeginEPSF { %def
+/b4_Inc_state save def         % Save state for cleanup
+/dict_count countdictstack def % Count objects on dict stack
+/op_count count 1 sub def      % Count objects on operand stack
+userdict begin                 % Push userdict on dict stack
+/showpage { } def              % Redefine showpage, { } = null proc
+0 setgray 0 setlinecap         % Prepare graphics state
+1 setlinewidth 0 setlinejoin
+10 setmiterlimit [ ] 0 setdash newpath
+/languagelevel where           % If level not equal to 1 then
+{pop languagelevel             % set strokeadjust and
+1 ne                           % overprint to their defaults.
+{false setstrokeadjust false setoverprint
+} if
+} if
+} bd
+/EndEPSF { %def
+count op_count sub {pop} repeat            % Clean up stacks
+countdictstack dict_count sub {end} repeat
+b4_Inc_state restore
+} bd
+%%EndResource
+%FOPBeginFontDict
+%%IncludeResource: font Courier-Bold
+%%IncludeResource: font Helvetica
+%%IncludeResource: font Courier-BoldOblique
+%%IncludeResource: font Courier-Oblique
+%%IncludeResource: font Times-Roman
+%%IncludeResource: font Helvetica-BoldOblique
+%%IncludeResource: font Helvetica-Bold
+%%IncludeResource: font Helvetica-Oblique
+%%IncludeResource: font Times-BoldItalic
+%%IncludeResource: font Courier
+%%IncludeResource: font Times-Italic
+%%IncludeResource: font Times-Bold
+%%BeginResource: font CMSY10
+%!PS-AdobeFont-1.0: mwb_cmsy10 1.0
+%%Title: mwb_cmsy10
+%Version: 1.0
+%%CreationDate: Thu Jan 27 10:30:23 2011
+%%Creator: Ben Hinkle
+%Copyright: Copyright (C) 2011 MathWorks, Inc
+% Generated by FontForge 20100429 (http://fontforge.sf.net/)
+%%EndComments
+
+10 dict begin
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0 ]readonly def
+/FontName /mwb_cmsy10 def
+/FontBBox {11 -216 943 727 }readonly def
+/PaintType 0 def
+/FontInfo 7 dict dup begin
+ /version (1.0) readonly def
+ /Notice (Copyright \050C\051 2011 MathWorks, Inc) readonly def
+ /FullName (mwb_cmsy10) readonly def
+ /FamilyName (mwb_cmsy10) readonly def
+ /Weight (Medium) readonly def
+ /ItalicAngle -14.035 def
+ /isFixedPitch false def
+end readonly def
+/Encoding 256 array
+ 0 1 255 { 1 index exch /.notdef put} for
+dup 0/.notdef                        put
+dup 2/multiply put
+dup 3/asteriskmath put
+dup 4/divide put
+dup 5/diamondmath put
+dup 6/plusminus put
+dup 7/minusplus put
+dup 8/circleplus put
+dup 9/circleminus put
+dup 10/circlemultiply put
+dup 11/circledivide put
+dup 12/circledot put
+dup 13/circlecopyrt put
+dup 14/openbullet put
+dup 15/bullet put
+dup 16/equivasymptotic put
+dup 17/equivalence put
+dup 18/reflexsubset put
+dup 19/reflexsuperset put
+dup 20/lessequal put
+dup 21/greaterequal put
+dup 22/precedesequal put
+dup 23/followsequal put
+dup 24/similar put
+dup 25/approxequal put
+dup 26/propersubset put
+dup 27/propersuperset put
+dup 28/lessmuch put
+dup 29/greatermuch put
+dup 30/precedes put
+dup 31/follows put
+dup 32/arrowleft put
+dup 127/spade put
+dup 128/arrowleft put
+dup  33/minus put
+dup  34/periodcentered put
+dup  35/multiply put
+dup  36/asteriskmath put
+dup  37/divide put
+dup  38/diamondmath put
+dup  39/plusminus put
+dup  40/minusplus put
+dup  41/circleplus put
+dup  42/circleminus put
+dup  43/circlemultiply put
+dup  44/circledivide put
+dup  45/circledot put
+dup  46/circlecopyrt put
+dup  47/openbullet put
+dup  48/bullet put
+dup  49/equivasymptotic put
+dup  50/equivalence put
+dup  51/reflexsubset put
+dup  52/reflexsuperset put
+dup  53/lessequal put
+dup  54/greaterequal put
+dup  55/precedesequal put
+dup  56/followsequal put
+dup  57/similar put
+dup  58/approxequal put
+dup  59/propersubset put
+dup  60/propersuperset put
+dup  61/lessmuch put
+dup  62/greatermuch put
+dup  63/precedes put
+dup  64/follows put
+dup  65/arrowleft put
+dup  66/spade put
+readonly def
+currentdict end
+currentfile eexec
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+cleartomark
+%%EndResource
+%%BeginResource: font CMSY10
+%!PS-AdobeFont-1.0: mwa_cmsy10 1.0
+%%Title: mwa_cmsy10
+%Version: 1.0
+%%CreationDate: Thu Jan 27 10:30:22 2011
+%%Creator: Ben Hinkle
+%Copyright: Copyright (C) 2011 MathWorks, Inc
+% Generated by FontForge 20100429 (http://fontforge.sf.net/)
+%%EndComments
+
+10 dict begin
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0 ]readonly def
+/FontName /mwa_cmsy10 def
+/FontBBox {-29 -960 1116 775 }readonly def
+/PaintType 0 def
+/FontInfo 7 dict dup begin
+ /version (1.0) readonly def
+ /Notice (Copyright \050C\051 2011 MathWorks, Inc) readonly def
+ /FullName (mwa_cmsy10) readonly def
+ /FamilyName (mwa_cmsy10) readonly def
+ /Weight (Medium) readonly def
+ /ItalicAngle -14.035 def
+ /isFixedPitch false def
+end readonly def
+/Encoding 256 array
+ 0 1 255 { 1 index exch /.notdef put} for
+dup 33/arrowright put
+dup 34/arrowup put
+dup 35/arrowdown put
+dup 36/arrowboth put
+dup 37/arrownortheast put
+dup 38/arrowsoutheast put
+dup 39/similarequal put
+dup 40/arrowdblleft put
+dup 41/arrowdblright put
+dup 42/arrowdblup put
+dup 43/arrowdbldown put
+dup 44/arrowdblboth put
+dup 45/arrownorthwest put
+dup 46/arrowsouthwest put
+dup 47/proportional put
+dup 48/prime put
+dup 49/infinity put
+dup 50/element put
+dup 51/owner put
+dup 52/triangle put
+dup 53/triangleinv put
+dup 54/negationslash put
+dup 55/mapsto put
+dup 56/universal put
+dup 57/existential put
+dup 58/logicalnot put
+dup 59/emptyset put
+dup 60/Rfractur put
+dup 61/Ifractur put
+dup 62/latticetop put
+dup 63/perpendicular put
+dup 64/aleph put
+dup 65/A put
+dup 66/B put
+dup 67/C put
+dup 68/D put
+dup 69/E put
+dup 70/F put
+dup 71/G put
+dup 72/H put
+dup 73/I put
+dup 74/J put
+dup 75/K put
+dup 76/L put
+dup 77/M put
+dup 78/N put
+dup 79/O put
+dup 80/P put
+dup 81/Q put
+dup 82/R put
+dup 83/S put
+dup 84/T put
+dup 85/U put
+dup 86/V put
+dup 87/W put
+dup 88/X put
+dup 89/Y put
+dup 90/Z put
+dup 91/union put
+dup 92/intersection put
+dup 93/unionmulti put
+dup 94/logicaland put
+dup 95/logicalor put
+dup 96/turnstileleft put
+dup 97/turnstileright put
+dup 98/floorleft put
+dup 99/floorright put
+dup 100/ceilingleft put
+dup 101/ceilingright put
+dup 102/braceleft put
+dup 103/braceright put
+dup 104/angbracketleft put
+dup 105/angbracketright put
+dup 106/bar put
+dup 107/bardbl put
+dup 108/arrowbothv put
+dup 109/arrowdblbothv put
+dup 110/backslash put
+dup 111/wreathproduct put
+dup 112/radical put
+dup 113/coproduct put
+dup 114/nabla put
+dup 115/integral put
+dup 116/unionsq put
+dup 117/intersectionsq put
+dup 118/subsetsqequal put
+dup 119/supersetsqequal put
+dup 120/section put
+dup 121/dagger put
+dup 122/daggerdbl put
+dup 123/paragraph put
+dup 124/club put
+dup 125/diamond put
+dup 126/heart put
+readonly def
+currentdict end
+currentfile eexec
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+cleartomark
+%%EndResource
+%%BeginResource: font CMSY10
+%!PS-AdobeFont-1.0: mwa_cmsy10 1.0
+%%Title: mwa_cmsy10
+%Version: 1.0
+%%CreationDate: Thu Jan 27 10:30:22 2011
+%%Creator: Ben Hinkle
+%Copyright: Copyright (C) 2011 MathWorks, Inc
+% Generated by FontForge 20100429 (http://fontforge.sf.net/)
+%%EndComments
+
+10 dict begin
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0 ]readonly def
+/FontName /mwa_cmsy10 def
+/FontBBox {-29 -960 1116 775 }readonly def
+/PaintType 0 def
+/FontInfo 7 dict dup begin
+ /version (1.0) readonly def
+ /Notice (Copyright \050C\051 2011 MathWorks, Inc) readonly def
+ /FullName (mwa_cmsy10) readonly def
+ /FamilyName (mwa_cmsy10) readonly def
+ /Weight (Medium) readonly def
+ /ItalicAngle -14.035 def
+ /isFixedPitch false def
+end readonly def
+/Encoding 256 array
+ 0 1 255 { 1 index exch /.notdef put} for
+dup 33/arrowright put
+dup 34/arrowup put
+dup 35/arrowdown put
+dup 36/arrowboth put
+dup 37/arrownortheast put
+dup 38/arrowsoutheast put
+dup 39/similarequal put
+dup 40/arrowdblleft put
+dup 41/arrowdblright put
+dup 42/arrowdblup put
+dup 43/arrowdbldown put
+dup 44/arrowdblboth put
+dup 45/arrownorthwest put
+dup 46/arrowsouthwest put
+dup 47/proportional put
+dup 48/prime put
+dup 49/infinity put
+dup 50/element put
+dup 51/owner put
+dup 52/triangle put
+dup 53/triangleinv put
+dup 54/negationslash put
+dup 55/mapsto put
+dup 56/universal put
+dup 57/existential put
+dup 58/logicalnot put
+dup 59/emptyset put
+dup 60/Rfractur put
+dup 61/Ifractur put
+dup 62/latticetop put
+dup 63/perpendicular put
+dup 64/aleph put
+dup 65/A put
+dup 66/B put
+dup 67/C put
+dup 68/D put
+dup 69/E put
+dup 70/F put
+dup 71/G put
+dup 72/H put
+dup 73/I put
+dup 74/J put
+dup 75/K put
+dup 76/L put
+dup 77/M put
+dup 78/N put
+dup 79/O put
+dup 80/P put
+dup 81/Q put
+dup 82/R put
+dup 83/S put
+dup 84/T put
+dup 85/U put
+dup 86/V put
+dup 87/W put
+dup 88/X put
+dup 89/Y put
+dup 90/Z put
+dup 91/union put
+dup 92/intersection put
+dup 93/unionmulti put
+dup 94/logicaland put
+dup 95/logicalor put
+dup 96/turnstileleft put
+dup 97/turnstileright put
+dup 98/floorleft put
+dup 99/floorright put
+dup 100/ceilingleft put
+dup 101/ceilingright put
+dup 102/braceleft put
+dup 103/braceright put
+dup 104/angbracketleft put
+dup 105/angbracketright put
+dup 106/bar put
+dup 107/bardbl put
+dup 108/arrowbothv put
+dup 109/arrowdblbothv put
+dup 110/backslash put
+dup 111/wreathproduct put
+dup 112/radical put
+dup 113/coproduct put
+dup 114/nabla put
+dup 115/integral put
+dup 116/unionsq put
+dup 117/intersectionsq put
+dup 118/subsetsqequal put
+dup 119/supersetsqequal put
+dup 120/section put
+dup 121/dagger put
+dup 122/daggerdbl put
+dup 123/paragraph put
+dup 124/club put
+dup 125/diamond put
+dup 126/heart put
+readonly def
+currentdict end
+currentfile eexec
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+%%Trailer
+%%Pages: 1
+%%EOF
index 82c0cc1..f0dc797 100644 (file)
@@ -19,7 +19,6 @@
 \usepackage{upquote} 
 \usepackage{color}
 \usepackage[usenames,dvipsnames]{xcolor}
-\usepackage{epstopdf}
 \usepackage{fancyhdr}
 \usepackage{courier}
 \usepackage{pifont}
@@ -141,7 +140,7 @@ Lausanne, le
 \begin{center}
 \textbf{Résumé}
 \end{center}
-Cet article a pour but la description et la mise en oeuvre d'une méthode automatique permettant l'établissement de la parasitémie de la malaria d'un patient à partir d'une ou plusieurs images microscopiques d'un frottis sanguin.
+Cet article a pour but la description et la mise en oeuvre d'une méthode automatique permettant l'établissement de la parasitémie d'un patient infecté par la malaria. Cette méthode se base sur une ou plusieurs images microscopiques d'un frottis sanguin.
 
 \vspace{5cm}
 
@@ -169,23 +168,17 @@ Cet article a pour but la description et la mise en oeuvre d'une méthode automa
 
 % TODO: Future dans l'intro?
 
-Le but de ce projet est d'établir une méthode complète et non-supervisée afin de dénombrer les erythrocytes et de les classer en deux catégories de cellules à savoir les saines et les infectées. Celle ci sera ensuite implémentée sous la forme d'un logiciel qui puisse être facilement utilisé par une personne avec un minimum de connaissance en informatique.
+Le but de ce projet est d'établir une méthode complète et non-supervisée afin de dénombrer les érythrocytes et de les classer en deux catégories de cellules à savoir les saines et les infectées. Cette méthode sera ensuite implémentée sous la forme d'un logiciel qui puisse être facilement utilisé par une personne avec un minimum de connaissance en informatique.
 
-Cette réalisation est faite en partenariat avec le laboratoire de parasitologie du CHUV 
+Cette réalisation est faite en partenariat avec le Dr. Guy Prod'hom de l'institut de microbiologie du \emph{CHUV} à Lausanne. % Ajouter du blabla
 
-Images 100x
-Images 50x
+% Problème du comptage manuel
 
-CHUV
+% \subsection{Images sources}
 
-\subection{Parasitémie de la malaria}
+% Mentionner certains problèmes comme les débris ou les plaquettes
 
-% Parlé du cycle de développement, à quel stade de développement on compte les rbc infecté, etc..
-
-
-\subsection{Images}
-
-Les images initiales correspondent à des photographies de sang infecté agrandies 100 fois. Une \emph{coloration de May-Grünwald Giemsa} est utilisée afin de faire ressortir les parasites avec une teinte particulière. Les éléments principaux composant les images sont : les globules rouges (figure~\ref{fig:medical-globules-rouges}), les globules blancs (figure~\ref{fig:medical-globules-blancs}), les plaquettes (figure~\ref{fig:medical-plaquettes}) et les différentes formes du parasite (figure~\ref{fig:medical-plasmodium}).
+Les images à disposition correspondent à des photographies de sang infecté agrandies 50 fois. Une \emph{coloration de May-Grünwald Giemsa} est utilisée afin de faire ressortir les parasites avec une teinte particulière. Les éléments principaux composant les images sont : les globules rouges (figure~\ref{fig:medical-globules-rouges}), les globules blancs (figure~\ref{fig:medical-globules-blancs}), les plaquettes (figure~\ref{fig:medical-plaquettes}) et les différentes formes du parasite (figure~\ref{fig:medical-plasmodium}).
 
 \begin{figure}[htbp]
        \centering
@@ -212,8 +205,6 @@ Les images initiales correspondent à des photographies de sang infecté agrandi
 
 Dans le cycle de vie du parasite \emph{Plasmodium}, responsable de la malaria, celui-ci passe par une étape d'alimentation active (trophozoïte) où il va se loger à l'intérieur des globules rouges. Puis il va subir la schizogonie (reproduction asexuée) et va se développer en schizonte. Ces étapes sont montrées par la figure~\ref{fig:medical-plasmodium}. 
 
-Notre but est de dénombrer les globules rouges sains ainsi que ceux infectés par des trophozoïtes immatures en début de croissance, comme montré par la figure~\ref{fig:medical-plasmodium_1}.
-
 \begin{figure}[htbp]
        \centering
        \begin{subfigure}[t]{0.2\textwidth}
@@ -243,584 +234,216 @@ Notre but est de dénombrer les globules rouges sains ainsi que ceux infectés p
        \label{fig:medical-plasmodium}
 \end{figure}
 
+%\subsection{Parasitémie de la malaria}
 
-\section{Outils utilisés}
-
-Cette section aborde brièvement la description des différents outils morphologiques ainsi que les logiciels qui ont permis de mettre en pratique le processus.
-
-
-\subsection{Outils morphologiques}
-
-Une grande partie du processus décrit dans ce document s'appuie sur la morphologie mathématique\footnote{Georges Matheron et Jean Serra - 1964}\cite{Talbot-Morphologie}\cite{Manzanera-Morphologie}, ici appliquée au traitement d'images.
-
-Nous passons en revue un certain nombre d'opérateurs qui sont utilisés dans le processus. Pour chacun d'entre eux, nous donnons une brève explication, une définition formelle et la fonction correspondante utilisée dans \emph{MATLAB}.
-
-
-\subsubsection{Définitions de base}
-
-Nous définissions tout d'abord l'image $E$ comme étant un sous-ensemble à deux dimensions de $\mathbb{Z}$ puis $B$ comme un sous-ensemble de $E$ que nous nommons \emph{élément structurant}. $B_x$ correspond à $B$ translaté de $x$ dont $x$ appartient à $E$. $\breve{B}$ est l'élément symétrique de $B$, si l'élément $B$ est symétrique,  alors $\breve{B} = B$.
-
-{\setlength{\abovedisplayskip}{0pt}
-\begin{flalign*}
-       E &= \mathbb{Z}^2 & \\
-       B_x &= \{ b + x \mid b \in B \} & \\
-       \breve{B} &= \{ -b \mid b \in B \} &
-\end{flalign*}
-
-\subsubsection{Dilatation et érosion}
-
-La dilatation est définie comme la somme de \emph{Minkowski} et nommée $\delta$. L'érosion est nommée $\epsilon$.
-
-{\setlength{\abovedisplayskip}{0pt}
-\begin{flalign*}
-       \delta_B(X) &= X \oplus B = \{ x \mid \breve{B}_x \cap{X} \neq \emptyset \} &\\
-       \epsilon_B(X) &= X \ominus B = \{x \mid B_x \subset X \} &
-\end{flalign*}
-
-Les deux fonctions \emph{MATLAB} associées sont \texttt{mmdil} et \texttt{mmero}.
-
-
-\subsubsection{Ouverture et fermeture}
-
-L'ouverture et la fermeture sont définies à partir de dilatation et d'érosion et sont nommées $\gamma$, respectivement $\phi$.
-
-{\setlength{\abovedisplayskip}{0pt}
-\begin{flalign*}
-       \gamma_B(X) &= X \circ B = \delta_B (\epsilon_B (X)) &\\
-       \phi_B(X) &= X \bullet B = \epsilon_B (\delta_B (X)) &
-\end{flalign*}
-
-Les deux fonctions \emph{MATLAB} associées sont \texttt{mmopen} et \texttt{mmclose}.
-
+L'objectif est de dénombrer les globules rouges sains ainsi que ceux infectés par des trophozoïtes immatures en début de croissance, comme montré par la figure~\ref{fig:medical-plasmodium_1}. Ce stade est appelé \emph{ring stage} car les parasites ont une forme d'anneau. Cet anneau, montré en détail par la figure~\ref{fig:noyau-cytoplasme}, comprend un noyau ainsi qu'un cytoplasme l'entourant.
 
-\subsubsection{Ouverture et fermeture par aire}
-
-L'ouverture et la fermeture par aire sont nommées $\gamma_{\lambda{}B}$ et $\phi_{\lambda{}B}$. Ces opérateurs permettent de raboter les bosses, respectivement de combler les trous ayant une aire égale ou inférieure à $\lambda$ et dont chaque élément est connexe à au moins un autre par l'élément structurant $B$.
-
-La figure~\ref{fig:fermeture-par-aire} illustre cela en montrant une vue en deux dimensions de deux fermetures ayant pour valeur $\lambda_1$ et $\lambda_2$.
+La parasitémie est établit en pourcentage comme étant le rapport entre nombre d'érythrocytes infecté et le nombre total d'érythrocyte. Le group d'experts \emph{UK NEQAS} (\emph{United Kingdom National External Quality Assessment Service}) recommande de considéré un minimum de 1000 erythrocytes. Le \emph{CDC} (\emph{Centers for Disease Control and Prevention}) recommande quant à lui au moins 500 érythrocytes si la parasitémie est supérieure à 10~\% et au moins 2000 érythrocytes si la parasitémie est inférieure à 0.1~\%.
 
 \begin{figure}[htbp]
        \centering
-       \includegraphics[width=0.5\linewidth]{img/schema_fermeture_par_aire.eps}
-       \caption{Vue en deux dimensions de deux fermetures par aire ayant des valeurs de $\lambda$ différentes}
-       \label{fig:fermeture-par-aire}
+       \includegraphics[width=0.5\linewidth]{img/RBC_noyau_cytoplasme.pdf}
+       \caption{Détail des deux parties constituantes d'un trophozoïte immature}
+       \label{fig:noyau-cytoplasme}
 \end{figure}
 
-Les deux fonctions \emph{MATLAB} associées sont \texttt{mmareaopen} et \texttt{mmareaclose}.
-
-
-\subsubsection{Reconstruction par dilatation}
-
-Une reconstruction par dilatation est réalisée sur une image $G$ à partir d'un marqueur $F$ et d'un élément structurant $B$. Elle consiste à l'application successive de dilatations sur $F$ à l'aide de $B$ en ne gardant que les éléments appartenant à $G$ jusqu'à idempotence.
-
-{\setlength{\abovedisplayskip}{0pt}
-\begin{flalign*}
-       F_0 &= F &\\
-       F_{n+1} &= (F_n \oplus B) \cap G &
-\end{flalign*}
-
-La fonction \emph{MATLAB} associée est \texttt{mminfrec}.
-
-
-\subsubsection{Ligne de partage des eaux}
-
-La ligne de partage des eaux (\emph{watersheds} en anglais) consiste à considérer une image comme une surface en trois dimensions où les valeurs d'intensité correspondent à des hauteurs puis à remplir progressivement les creux avec de l'eau jusqu'à que toute l'image soit entièrement remplie. Lorsque deux bassins se rejoignent pendant la montée des eaux, une digue est construite. L'ensemble des digues représente alors le résultat final.
-
-Dans notre cas, nous utilisons des marqueurs pour définir à partir de quels creux l'eau est injectée.
-
-La fonction \emph{MATLAB} associée est \texttt{mmcwatershed}.
-
-
-\subsubsection{Maximaux régionaux}
-
-Les maximaux régionaux d'une image sont définis comme suit. La première ligne correspond à la définition de la connectivité des éléments de $M$. $B$ est un élément structurant.
-
-{\setlength{\abovedisplayskip}{0pt}
-\begin{flalign*}
-       \begin{cases}
-               \forall p \in M & \text{si } f(p) = t, \quad \exists p' \in M, \quad p' \neq p, \quad p' \in (p \oplus B) \\
-               \forall p \in (M \oplus B) \setminus M & \text{si } f(p) < t 
-       \end{cases}&&
-\end{flalign*}
-La fonction \emph{MATLAB} associée est \texttt{mmregmax}. La figure \ref{fig:regmax_sample} montre un exemple d'application de cette fonction à l'aide d'un élément structurant carré de taille 5.
+Certains artefacts comme des débris ou des plaquettes peuvent venir se supperposer aux érythrocyte et être confondus avec des parasites. Ces cas peuvent être relativement fréquents et altérer significativement la parasitémie si comptés comme étant des cas positifs d'infection. La figure~\ref{fig:RBC-artefact} montre quelques exemple de faux positifs potentiels.
 
 \begin{figure}[htbp]
        \centering
-       \begin{subfigure}[b]{0.38\textwidth}
-               \includegraphics[width=\linewidth]{img/regmax_sample.eps}
-               \caption{Image à laquelle est appliquée la fonction \texttt{mmregmax}. Les pixels noires valent zéro}
-               \label{fig:regmax_sample_original}
+       \begin{subfigure}[t]{0.2\textwidth}
+               \includegraphics[width=\linewidth]{img/RBC-plaquette.jpg}
+               \caption{Thrombocyte}
+               \label{fig:RBC-artefact-plaquette}
        \end{subfigure} 
        ~
-       \begin{subfigure}[b]{0.15\textwidth}
-               \includegraphics[width=\linewidth]{img/regmax_sample_se.eps}
-               \caption{L'élément structurant : un carré de taille 5 symétrique}
-               \label{fig:regmax_sample_se}
+       \begin{subfigure}[t]{0.2\textwidth}
+               \includegraphics[width=\linewidth]{img/RBC-debris-1.jpg}
+               \caption{Débris}
+               \label{fig:RBC-artefact-debris-1}
        \end{subfigure}
        ~       
-       \begin{subfigure}[b]{0.38\textwidth}
-               \includegraphics[width=\linewidth]{img/regmax_sample_result.eps}
-               \caption{Le résultat de l'application de \texttt{mmregmax}. Les pixels noires valent zéro}
-               \label{fig:regmax_sample_result}
+       \begin{subfigure}[t]{0.2\textwidth}
+               \includegraphics[width=\linewidth]{img/RBC-debris-2.jpg}
+               \caption{Débris}
+               \label{fig:RBC-artefact-debris-2}
+       \end{subfigure}
+       ~       
+       \begin{subfigure}[t]{0.2\textwidth}
+               \includegraphics[width=\linewidth]{img/RBC-debris-3.jpg}
+               \caption{Débris}
+               \label{fig:RBC-artefact-debris-3}
        \end{subfigure}
-       \caption{Exemple d'application de la fonction \texttt{mmregmax}. Les deux pixels de valeur 5 forment une zone connexe : celle-ci est écartée car la dilaté du pixel de valeur 8 par l'élément structurant contient un des pixels de valeur 5}
-       \label{fig:regmax_sample}
+       \caption{Artefacts se supperposant à des érytrocytes}
+       \label{fig:RBC-artefact}
 \end{figure}
 
-\subsection{Outils logiciels}
-
-L'implémentation est réalisée à l'aide de \emph{MATLAB 2015a}. Les différentes fonctions morphologiques (préfixées par \texttt{mm}) proviennent de la bibliothèque nommée \emph{SDC Morphology Toolbox for MATLAB}\footnote{\url{http://mmorph.com/}}.
-
 
-\section{Approche générale}
 
-Le processus complet est illustré par la figure \ref{fig:processusComplet}. Les différents chiffres correspondent aux étapes décrites ci-après. Il y a deux phases principales : le traitement des éléments colorés (étapes 1 à 3) et la segmentation des globules rouges (étapes 4 et 5).
+\section{Aperçu du processus}
 
-La première étape (\ding{192}) consiste à détecter les éléments qui ont été colorés comprenant les parasites, les globules blancs et les plaquettes. Les parasites peuvent se trouver à l'extérieur de cellules, par exemple groupés sous la forme de schizontes.
-
-Les deuxième et troisième étapes (\ding{193} et \ding{194}) consistent à chercher et à extraire les globules blancs, respectivement les schizontes.
-
-La quatrième étape (\ding{195}) va segmenter les globules rouges.
-
-La cinquième étape (\ding{196}) consiste à reconstruire les globules rouges parasités à partir des parasites marqués de la première étape (\ding{192}) et de la segmentation de la quatrième étape (\ding{195}) .
-
-La sixième et dernière étape va compter le nombre de globules rouges total et de globules rouges infectés afin de calculer le rapport entre ces deux valeurs.
+La figure~\ref{fig:processusComplet} montre le processus complet du traitement d'une image de frottis sanguin. Celui ci peut-être divisé en trois groupe de traitement à savoir la ségmentation des érytrocytes, la ségmentation des parasites et finalement la classification des cellules.
 
 \begin{figure}[htbp]
        \centering
-       \includegraphics[width=1\linewidth]{img/schema_processus.eps}
+       \includegraphics[width=0.8\linewidth]{img/schema_processus_2.pdf}
        \caption{Le processus complet.}
        \label{fig:processusComplet}
 \end{figure}
 
+Dans le premier groupe, l'on va chercher à construire des ellipses qui correspondent le mieux aux bords des érythrocytes. Pour ce faire une estimation du rayon moyen est réalisée par granulométrie à l'aide d'une succession de fermeture par aire. La construction des ellipses utilise les pixels des bords des erytrocytes ainsi que son gradient, la méthode \emph{RANSAC} permet de créer un grand nombre d'ellipses candidates auquel un score est attribué. Finalement les ellipses sont élaguée en supprimant celles chevauchées par d'autres ellipses sont le score est supérieur.
 
-\section{Détail du processus}
+Dans le deuxième groupe, les zones colorées et sombre vont être extraites en comparant les valeurs d'intensité de l'image avec la moyenne d'intensité des éléments de l'avant-plan. Ces zones sombres correspondent aux leucocytes, aux plaquettes et à certains débris. Une fermeture morphoplogique, dont la taille de l'élément structurant est calculé en fonction du rayon moyen, va permettre de mettre en évidence le cytoplasme des parasites. Les noyaux sont, quant à eux, mis en évidence à l'aide d'une fermeture par aire.
 
-Le processus est actuellement réalisé sur des images réduites de 40~\% à l'aide d'une interpolation bicubique par rapport à la taille originale afin d'augmenter la vitesse de l'ensemble du processus. La taille des images traitées après réduction est de 1167 x 1556. Voir la figure \ref{fig:imgRGB}. 
+Le troisième groupe correspond à la mise en commun des informations du premier et du deuxième groupe afin de définir les erytrocytes et de les classer. La parasitémie est alors établit en calculant le rapport entre la population d'érythrocytes infecté et celle saine.
 
-La qualité du résultat final n'est pas très affectée par la diminution de la taille des images : sur l'ensemble des douze images présentées à la section~\ref{resultats}, nous obtenons un taux d'erreur moyen de 17.9~\% pour les images originales contre 18.2~\% pour les images réduites. Le taux mesuré de référence est de 16.9 \%. Le temps moyen est de 485~s pour les images originales contre 78~s pour les images réduites.
 
-\begin{figure}[htbp]
-       \centering
-       \includegraphics[width=0.8\linewidth]{img/imgRGB.png}
-       \caption{Image originale en entrée ($\mli{imgRGB}$)}
-       \label{fig:imgRGB}
-\end{figure}
-
-Tout au long de cette description nous lions certains résultats d'opération à des noms de variables issues du code \emph{MATLAB} (fourni en annexe). Ceci afin de pouvoir facilement retrouver le détail de ces opérations dans le code et de faciliter le suivi de l'enchaînement des opérations.
-
-\subsection{Détection des parasites}
-\label{detection-parasites}
+\section{Détail de la méthode}
 
-Cette opération a pour but de marquer les éléments colorés qui correspondent aux parasites, schizontes et globules blancs. Nous utilisons ici les composantes \emph{teinte} et \emph{saturation} de l'image.
+% filtrage préliminaire (taille d'un érythrocyte: 7.5 \micro~m)
 
-\subsubsection{Filtrage préliminaire}
+Les filtres gaussiens appliquées initialement vont permettre de supprimer une partie du bruit haute-fréquence. L'écart type de chaque filtre est choisi en fonction de la taille des objets. La taille d'un érytrocyte est d'environ 7.5~\textmu{m} et la taille d'un parasite d'environ 2.5~\textmu{m}. Des écarts types de 0.2~\textmu{m} et 0.15~\textmu{m} sont choisis pour pour la ségmentation des érytrocytes respectivement la ségmentation des parasites.
 
-Il est nécessaire, pour la suite du traitement, d'avoir les intensités les plus élevées pour les éléments colorés pour les deux composantes \emph{teinte} et \emph{valeur}. Pour se faire nous allons utiliser le négatif des deux composantes. De plus la composante \emph{teinte} est translatée de $100 / 255$ (141°) afin que les différents éléments de l'image, à savoir le fond, les cellules et les parties colorées, ne soient pas proches des valeurs extrêmes, voir l'histogramme montré par la figure \ref{fig:histogram_imgH}. $\mli{imgRGB} \rightarrow (\mli{imgH}, \mli{imgS})$.
-
-Les histogrammes des deux images $\mli{imgH}$ et $\mli{imgS}$ sont montrés par les figures \ref{fig:histogram_imgH} et \ref{fig:histogram_imgS}. L'histogramme de la teinte montre un premier sommet vers 80 correspondant au fond et un deuxième sommet vers 88 correspondant aux cellules, Les éléments colorés se trouvent aux environs de 120. L'histogramme de la saturation montre un premier sommet vers 128 correspondant aux cellules et un deuxième sommet vers 150 correspondant au fond : les éléments colorés se trouvent aux environs de 200.
-
-\begin{figure}[htbp]
-       \centering
-       \includegraphics[width=0.8\linewidth]{img/histogram_imgH.eps}
-       \caption{Histogramme de la teinte ($\mli{imgH}$)}
-       \label{fig:histogram_imgH}
-\end{figure}
-
-\begin{figure}[htbp]
-       \centering
-       \includegraphics[width=0.8\linewidth]{img/histogram_imgS.eps}
-       \caption{Histogramme de la saturation ($\mli{imgS}$)}
-       \label{fig:histogram_imgS}
-\end{figure}
+\subsection{Identification des érytrocytes}
 
-Afin de gommer le bruit et de rendre le fond plus homogène, un filtre médian 5 x 5 puis une fermeture par aire de 1000 sont appliqués à $\mli{imgH}$ et $\mli{imgS}$. $\mli{imgH} \rightarrow \mli{imgFiltered\{1\}}, \mli{imgS} \rightarrow \mli{imgFiltered\{2\}}$. Les figures \ref{fig:imgHFiltered} et \ref{fig:imgSFiltered} montrent les composantes \emph{teinte} et \emph{saturation} après filtrage.
+Le but de cette étape est d'identifier les érythrocytes. Le profil de ceux-ci est un disque biconcave et apparait de manière plus ou moins déformée sur les photographies. Ces déformations les font prendre une forme plus ou moins elliptique dont le rapport entre le grand axe et le petit axe n'excède pas 1.6 ($\pm 23~\%$ d'un rayon moyen).
 
-\begin{figure}[htbp]
-       \centering
-       \includegraphics[width=0.8\linewidth]{img/imgHFiltered.jpg}
-       \caption{Composante \emph{teinte} après filtrage ($\mli{imgFiltered\{1\}}$) représentée en fausse couleur}
-       \label{fig:imgHFiltered}
-\end{figure}
+Le résultat de cette recherche sera une liste d'ellipses pouvant se recouvrir partiellement. Les ellipses sont décrites par les paramètres montrés par la figure~\ref{fig:ellipse-parametres}.
 
 \begin{figure}[htbp]
        \centering
-       \includegraphics[width=0.8\linewidth]{img/imgSFiltered.jpg}
-       \caption{Composante \emph{saturation} après filtrage ($\mli{imgFiltered\{2\}}$) représentée en fausse couleur}
-       \label{fig:imgSFiltered}
+       \includegraphics[width=0.5\linewidth]{img/ellipse-parametres.pdf}
+       \caption{Paramètre d'une ellipse, $\alpha \in [0, \pi[$ est l'angle d'inclinaison du grand axe $a$, $a \geqslant b$ > 0}
+       \label{fig:ellipse-parametres}
 \end{figure}
 
 
 \subsubsection{Granulométrie}
 
-L'objectif ici est de déterminer le rayon moyen des globules rouges (souvent abrégé \emph{RBC} pour \emph{red blood cell}). Pour ce faire, nous réalisons une succession de fermetures à l'aide d'un élément structurant de forme octogonale sur la composante \emph{saturation} ($\mli{imgFiltered\{2\}}$). Il est à noter que les globules rouges ont une intensité moindre que le fond (voir la figure~\ref{fig:imgSFiltered}) : c'est pour cela que nous utilisons une fermeture et non une ouverture.  Nous évitons ici un élément structurant ayant la forme d'un disque pour des raisons de performance, une fermeture avec ce dernier demandant d'effectuer beaucoup plus de calculs. Une ouverture par aire de 1000 est appliquée au préalable sur la composante \emph{saturation} afin de niveler l'intensité des cellules.
-
-\begin{sloppypar} % Pour éviter que certaines formules inline débordent dans la marge.
-Nous partons d'un rayon initial de 1 pour arriver à un rayon maximal de $\mli{width}(\mli{imgRGB}) / 35$. Le rayon maximal est défini en fonction de la largeur de l'image, ce qui permet de ne pas être dépendant de sa résolution. À chaque itération une fermeture est effectuée suivi du calcul du volume relatif de l'image : $\mli{volume}(\mli{imgClosed}) \rightarrow A, 1 - A / \mli{volImg} \rightarrow N$ où $\mli{volImg}$ est le volume de $\mli{imgFiltered\{2\}}$. La différence de volume (spectre granulométrique) est calculée pour chaque itération $i$ comme suit : $\lvert N_{i + 1} - N_i \rvert$.
-\end{sloppypar}
-
-Nous définissons le rayon moyen des globules rouges ($\mli{RBCRadius}$) comme étant le rayon de l'élément structurant correspondant à la valeur maximale parmi le spectre granulométrique. Le rayon moyen du noyau des parasites est impossible à extraire des différences de volume : nous le calculons comme étant un cinquième de la taille des globules rouges : $\mli{RBCRadius} / 5 \rightarrow \mli{nucleiRadius}$. La figure \ref{fig:patternSpectrum} montre le graphique d'un spectre granulométrique. Dans ce cas $\mli{RBCRadius}$ est égal à 35 et $\mli{nucleiRadius}$ à 7.
-
-\begin{figure}[htbp]
-       \centering
-       \includegraphics[width=0.8\linewidth]{img/pattern_spectrum_granulometry.eps}
-       \caption{Spectre granulométrique}
-       \label{fig:patternSpectrum}
-\end{figure}
-
-\subsubsection{Segmentation des éléments colorés}
-
-Afin d'identifier les éléments colorés, une recherche des maximaux régionaux est réalisée sur les deux composantes \emph{teinte} et \emph{saturation} à l'aide d'un élément structurant octogonal de rayon $\mli{RBCRadius}$. Les deux résultats sont ensuite dilatés de $\mli{nucleiRadius}$ puis combinés avec une intersection.
-
-{\setlength{\abovedisplayskip}{0pt}
-\begin{flalign*}
-   \mli{M\{1\}} &= \mli{regmax}(\mli{imgFiltered\{1\}}, \mli{RBCRadius}) &\\
-   \mli{M\{2\}} &= \mli{regmax}(\mli{imgFiltered\{2\}}, \mli{RBCRadius}) &\\
-   \mli{MHS} &= (\mli{M\{1\}} \oplus \mli{nucleiRadius}) \cap (\mli{M\{2\}} \oplus \mli{nucleiRadius}) &
-\end{flalign*}
-
-Nous allons ensuite seuiller les deux composantes puis combiner les résultats avec une intersection. Le seuil de chaque image, $\mli{{\mu}H}$ et ${\mu}S$, est déterminé comme étant la moyenne des valeurs de l'image désignée par le masque $\mli{MHS}$.
-
-{\setlength{\abovedisplayskip}{0pt}
-\begin{flalign*}
-       \mli{MHS}_{vol} &= \displaystyle\sum_{p \in \mli{MHS}}p &\\
-       \mli{{\mu}H} &= \frac{\displaystyle\sum_{p \in \mli{MHS}} H(p)}{\mli{MHS}_{vol}} & où~H &= \mli{imgFiltered\{1\}} &\\
-       \mli{{\mu}S} &= \frac{\displaystyle\sum_{p \in \mli{MHS}} S(p)}{\mli{MHS}_{vol}} & où~S &= \mli{imgFiltered\{2\}} &
-\end{flalign*}
-
-La teinte et la saturation sont seuillées à l'aide des valeurs $\mli{{\mu}H}$, respectivement $\mli{{\mu}S}$. Nous obtenons alors les images $\mli{TH}$ et $\mli{TS}$ que nous combinons comme suit pour obtenir $\mli{THS}$. L'ouverture par aire permet d'éliminer les toutes petites traces qui représentent dans la majorité des cas du bruit. La figure \ref{fig:THS} montre $\mli{THS}$ en blanc imprimé sur l'image d'entrée.
-
-{\setlength{\abovedisplayskip}{0pt}
-\begin{flalign*}
-       \mli{THS} &= \gamma_{10}(\mli{TH} \cap \mli{TS}) &
-\end{flalign*}
-
-\begin{figure}[htbp]
-       \centering
-       \includegraphics[width=0.8\linewidth]{img/THS.jpg}
-       \caption{Les éléments colorés mis en évidence ($\mli{THS}$)}
-       \label{fig:THS}
-\end{figure}
-
-
-\subsection{Détection des globules blancs}
-
-Cette opération utilise l'image de marquage des éléments colorés ($\mli{THS}$) ainsi que la taille des globules rouges ($\mli{RBCRadius}$). Les globules blancs sont tout d'abord marqués à l'aide d'une érosion utilisant un élément structurant octogonal dont la taille correspond à 50 \% de celle des globules rouges ($0.5 \cdot \mli{RBCRadius}$). Puis une reconstruction par dilatation des éléments colorés en utilisant le marqueur précédent permet de ne garder que les globules blancs.
-
-Finalement nous lissons le résultat en réalisant une fermeture suivie d'une ouverture afin de fermer les petits trous qui pourraient rester. Le résultat est appelé $\mli{WBC}$ et est montré par la figure \ref{fig:WBC}.
-
-\begin{figure}[htbp]
-       \centering
-       \includegraphics[width=0.8\linewidth]{img/WBC.jpg}
-       \caption{Le gradient des globules blancs imprimé sur l'image originale ($\mli{WBC}$)}
-       \label{fig:WBC}
-\end{figure}
+L'objectif est de déterminer le rayon moyen des érytrocytes. Pour ce faire nous allons appliquer une série de fermetures par aire et calculer les différences des sommes des intensités de chaque pixel. Chaque aire est calculée comme étant celui d'un cercle de rayon donné. Les rayons des aires vont être compris autour d'un rayon estimé à partir de la résolution donnée initialement. Les bornes inférieure et supérieure correspondent au rayon estimé minoré de 50~\% respectivement au rayon estimé majoré de 50~\%.
 
+La fermeture suppose ici que les erythrocytes on un niveau d'intensité plus bas que le fond, ce qui est le cas pour la composante verte dans toutes les images étudiées.
 
-\subsection{Détection des schizontes}
-
-Pour détecter les schizontes, nous utilisons l'image de marquage des éléments colorés ($\mli{THS}$) ainsi que la taille des globules rouges ($\mli{RBCRadius}$). Nous allons ici plus loin que l'article et créons un graphe non-orienté dont les arcs représentent une \emph{distance de Hausdorff} égale ou plus petite à 1.3 fois le rayon des globules rouges entre deux sous-ensembles. Nous cherchons alors les sous-ensembles connexes de plus de $n$ éléments et les marquons comme étant des schizontes. Dans notre cas $n$ est égal à 6. Il est important de ne pas utiliser un nombre trop bas car des cellules poly-infectées pourraient être considérées comme des schizontes : la figure \ref{fig:poly-et-schizonte} illustre cela.
+Deux autres approches on également été essayées, à savoir la fermeture morphologique par un élément structurant en forme de disque et par un élément structurant de forme octogonale. Un des problèmes de ces deux approches est qu'elles ont tendance à sous-estimer légèrement le rayon moyen dans le cas d'éléments elliptiques. La figure~\ref{fig:granulometrie-ellipse-cercle-closing-comparaison} montre à gauche le résultat d'une granulometrie par aire (le cercle et l'ellipse ont la même aire) et à droite le résultat d'une granulometrie morphologique. Le rayon $r$ est plus proche de la moyenne entre les deux rayons de l'ellipse $a$ et $b$ que le rayon $r'$. Il est à noter que dans cet exemple $a = 2b$ et que donc $r = \sqrt{2}b = 1.41b$ ce qui n'est pas très éloigné de la moyenne $(a+b)/2 = (2+1)/2 = 1.5$.
 
 \begin{figure}[htbp]
        \centering
-       \begin{subfigure}[t]{0.3\textwidth}
-               \includegraphics[width=\linewidth]{img/poly-infected_cell.jpg}
-               \caption{Globule rouge poly-infecté, quatre sous-ensembles}
-               \label{fig:regmax_sample_original}
+       \begin{subfigure}[t]{0.4\textwidth}
+               \includegraphics[width=\linewidth]{img/granulometrie-ellipse-cercle-closing-area.pdf}
+               \caption{Cercle dont la surface est égale à celle de l'ellipse}
+               \label{fig:granulometrie-ellipse-cercle-closing-area}
        \end{subfigure} 
        ~
-       \begin{subfigure}[t]{0.3\textwidth}
-               \includegraphics[width=\linewidth]{img/schizonte.jpg}
-               \caption{Un schizonte, sept sous-ensembles}
-               \label{fig:schizonte}
-       \end{subfigure}
-       \caption{Comparaison entre le nombre de sous-ensembles d'une cellule poly-infectée et d'un schizonte}
-       \label{fig:poly-et-schizonte}
+       \begin{subfigure}[t]{0.4\textwidth}
+               \includegraphics[width=\linewidth]{img/granulometrie-ellipse-cercle-closing-morpho.pdf}
+               \caption{Cercle dont \emph{r} est égale au petit rayon \emph{a} de l'ellipse}
+               \label{fig:granulometrie-ellipse-cercle-closing-morpho}
+       \end{subfigure} 
+       \caption{Comparaison du rayon d'un cercle trouvé par fermeture par aire d'une ellipse (à gauche) et celui trouvé par une fermeture morphologique (à droite)}
+       \label{fig:granulometrie-ellipse-cercle-closing-comparaison}
 \end{figure}
 
-Pour savoir si deux sous-ensembles, $A$ et $B$, sont à une \emph{distance de Hausdorff} égale ou plus petite que $k$, il faut que $B \subseteq A \oplus \mli{SE_k}$ et que $A \subseteq B \oplus \mli{SE_k}$ où $\mli{SE_k}$ est un élément structurant ayant la forme d'un disque de diamètre $k$.
-
-\subsection{Segmentation des globules rouges}
-\label{segmentation-globules-rouges}
-
-Le but ici est de séparer les globules rouges : nous utilisons l'image initiale ($\mli{imgRGB}$), le rayon des globules rouges ($\mli{RBCRadius}$), les globules blancs ($\mli{WBC}$) et les schizontes ($\mli{schizonts}$). Pour ce faire nous utilisons dans un premier temps une segmentation par ligne de partage des eaux à l'aide d'une transformée en distance. Nous séparons ensuite les segments en deux groupes, un groupe dont chaque élément représente une unique cellule et un autre groupe dont chaque élément représente un composé de cellules. Les cellules des éléments composés sont ensuite séparées : le résultat est uni aux cellules uniques et les éléments touchant les bords sont enlevés afin d'obtenir le résultat final.
-
-Durant le processus, nous avons besoin de définir le rayon minimum d'un globule rouge ainsi que son aire. Le rayon minimum est défini comme étant 75~\% du rayon nominal, comme montré ci-dessous.
-
-{\setlength{\abovedisplayskip}{0pt}
-\begin{flalign*}
-       \mli{RBCRadiusMin} &= \lfloor 0.75 \cdot \mli{RBCRadius} \rfloor &\\
-       \mli{RBCRadiusMinArea} &= \pi \cdot \mli{RBCRadiusMin}^2 &
-\end{flalign*}
-
-
-\subsubsection{Filtrage préliminaire}
-
-Nous utilisons ici la composante verte de l'image à laquelle nous appliquons une ouverture par aire avec une zone de taille $\mli{RBCRadiusMinArea}$. Les globules blancs ($\mli{WBC}$) ainsi que les schizontes ($\mli{schizonts}$) sont \og{}soustraits\fg{} de l'image : le résultat est nommé $\mli{grayFlat}$ comme montré par la figure~\ref{fig:segmentation-grayflat}.
+Un autre problème de la granulométrie à l'aide de fermetures morphologique utilisant un disque comme élément structurant est sa complexité algorithmique quadratique en fonction du rayon du disque comme le montre la figure~\ref{fig:spectre-granulométrique-morpho-cercle}. La figure~\ref{fig:spectre-granulométrique-morpho-cercle} montre un exemple de progression du temps de calcul en fonction du rayon. Une solution à ce problème est l'utilisation d'une approximation du cercle, l'octogone, dont la dilaté ou l'érodé par un autre octogone crée un nouvel octogone. L'associativité de l'erosion ou de la dilatation, montré par l'équation~\ref{eq:dilatation-associatif}, permet la décomposition d'une dilatation ($e_{1} \oplus e_{2}$) en plusieurs dilatations successives ($\oplus~e_{1}$ puis $\oplus~e_{2}$). Cela permet de réduire la complexité algorithmique et ainsi de gagner du temps.
 
 {\setlength{\abovedisplayskip}{0pt}
-\begin{flalign*}
-       \mli{grayFlat} &= \gamma_{\mli{RBCRadiusMinArea}}(\mli{green}(\mli{imgRGB})) \setminus \mli{WBC} \setminus \mli{schizonts} &
-\end{flalign*}
-
-La figure~\ref{fig:segmentation-histogramme-grayflat} montre l'histogramme de $\mli{grayFlat}$. Un \emph{seuillage d'Otsu} lui est appliqué suivi d'une ouverture par un disque de rayon $\mli{RBCRadiusMin}$ afin d'adoucir les contours et d'enlever tous les éléments plus petits qu'un globule rouge.
-
-\begin{figure}[htbp]
-       \centering
-       \includegraphics[width=0.8\linewidth]{img/segmentation_input_grayflat.jpg}
-       \caption{Composante verte dont l'intensité des cellules a été homogénéisée à l'aide d'une ouverture par aire. Les globules blancs et les schizontes ont été peints en blanc ($\mli{grayFlat}$)}
-       \label{fig:segmentation-grayflat}
-\end{figure}
+\begin{flalign} \label{eq:dilatation-associatif}
+       f' = f \oplus (e_{1} \oplus e_{2}) = (f \oplus e_{1}) \oplus e_{2} &
+\end{flalign}
 
-\begin{figure}[htbp]
-       \centering
-       \includegraphics[width=0.8\linewidth]{img/histogram_segmentation_grayFlat.eps}
-       \caption{Histogramme de la composante verte de $\mli{imgRGB}$ après l'application d'une ouverture par aire et de la \og{}suppression\fg{} des globules blancs et des schizontes}
-       \label{fig:segmentation-histogramme-grayflat}
-\end{figure}
+L'implémentation utilisée ici, \emph{OpenCV}, ne réalise pas, à priori, cette optimisation lorsque l'élément structurant peut être décomposé. De ce fait le temps de calcul, montré par la figure~\ref{fig:spectre-granulométrique-morpho-octogone}, ne se différencie pas de l'utilisation d'un disque. De plus, l'octogone produit un spectre granulométrique (figure~\ref{fig:spectre-granulométrique-morpho-octogone}) moins harmonieux que pour les deux autres granulométries.
 
+Concernant les mesures du temps de calcul avec des images réelles, l'on constate que la fermeture par aire est beaucoup plus rapide : 8.4~s (figure~\ref{fig:temps-aire}) que la fermeture morphologique : 54~s (figures~\ref{fig:spectre-granulométrique-morpho-cercle} et~\ref{fig:spectre-granulométrique-morpho-octogone}). La première fermeture par aire, pour le premier rayon, prend un peu moins de 5~s puis les suivantes environ 100~ms. Cela s'explique par la recherche initiale des minima puis par l'agrandissement ceux-ci jusqu'à l'aire correspondant au premier rayon. Le détail de cet algorithme se trouve à la section~\ref{} % TODO: référence.
 
-\subsubsection{Calcul des distances et ligne de partage des eaux}
-
-Après le seuillage des globules rouges, certaines régions comprennent deux ou plusieurs globules rouges qu'il faut séparer. Pour ce faire nous appliquons une transformée en distance (\texttt{mmdist}) qui permet de donner une distance à chaque pixel blanc. Cette distance correspond à la plus courte distance euclidienne entre le pixel blanc et un pixel noir. Nous cherchons ensuite les maximaux régionaux de l'image en distance en utilisant un élément structurant octogonal dont le rayon est la moitié de $\mli{RBCRadius}$. Une dilatée d'un disque de rayon $\mli{RBCRadius} / 7$ est alors appliquée aux maximaux pour fusionner d'éventuelles zones très proches. L'algorithme de recherche de la ligne de partage des eaux est appliqué au négatif des distances en utilisant les maximaux dilatés comme marqueurs. Finalement l'intersection entre le négatif de la ligne de partage des eaux et de $\mli{thresholdOpened}$ correspond au résultat. La figure~\ref{fig:segmentation-watershed} montre les étapes de ce processus.
+Il faut remarquer que le spectre granulométrique par aire comporte un deuxième sommet au rayon 44 qui correspond à certains amas de cellules. Il est nécessaire d'éviter de traiter des images comprenant trop de cellules se touchant. En pratique, ce cas ne survient pas car l'image est alors trop peu lisible pour être exploitée.
 
 \begin{figure}[htbp]
-       \centering
-       \begin{subfigure}[t]{0.3\textwidth}
-               \includegraphics[width=\linewidth]{img/segmentation_watershed_1.jpg}
-               \caption{Image binaire initiale ($\mli{thresholdOpened}$)}
-               \label{fig:segmentation-watershed-1}
-       \end{subfigure} 
-       ~
-       \begin{subfigure}[t]{0.3\textwidth}
-               \includegraphics[width=\linewidth]{img/segmentation_watershed_2.jpg}
-               \caption{Superposition de trois images : les distances, les maximaux et sa dilaté}
-               \label{fig:segmentation-watershed-3}
+       \centering      
+       \begin{subfigure}[t]{0.48\textwidth}
+               \includegraphics[width=1\linewidth]{img/Granulometry-pattern-spectrum-area.pdf}
+               \caption{Spectre granulométrique d'une succession de fermetures par aire}
+               \label{fig:spectre-granulométrique-aire}
        \end{subfigure}
-       ~       
-       \begin{subfigure}[t]{0.3\textwidth}
-               \includegraphics[width=\linewidth]{img/segmentation_watershed_3.jpg}
-               \caption{Image après découpe de l'image initiale avec le résultat de la ligne de partage des eaux}
-               \label{fig:segmentation-watershed-3}
+       ~
+       \begin{subfigure}[t]{0.48\textwidth}
+               \includegraphics[width=1\linewidth]{img/Granulometry-time-area.pdf}
+               \caption{Temps en milliseconde pour chaque fermeture. Temps total : 8.4~s}
+               \label{fig:temps-aire}
        \end{subfigure}
-       \caption{Wastershed par transformé en distance}
-       \label{fig:segmentation-watershed}
+       \caption{Spectre granulométrique et temps en milliseconde pour chacune des fermetures par aire. Réalisé sur une image 2592 \texttimes 1944 contenant environ 700 érythrocytes}
+       \label{fig:Granulometry-area}
 \end{figure}
 
-
-\subsubsection{Identification et traitement des cellules composées}
-
-Afin d'identifier les zones composées de plusieurs cellules nous calculons le rapport grand-axe / petit-axe pour chacune des zones : si le rapport d'une zone est plus grand ou égal à 1.3, alors la zone est définie comme étant composée.
-
-La figure~\ref{fig:segmentation-composites} montre le processus de séparation d'une zone composée de plusieurs cellules : ce processus est décrit ci-après.
-
-Une ouverture à l'aide d'un disque de rayon $\mli{RBCRadiusMin}$ est appliquée à $\mli{gray}$ en utilisant les zones composées comme masque. Le résultat est montré par la figure \ref{fig:segmentation-composites-3}. Un gradient morphologique suivi d'une binarisation sont réalisés pour obtenir l'image de la figure \ref{fig:segmentation-composites-4}. Les trous d'une aire plus petite ou égale à la moitié de $\mli{RBCRadiusMinArea}$ sont remplis avec une ouverture par aire puis un amincissement est appliqué. Nous remplissons ensuite les trous de l'amincissement et mettons à zéro les pixels marqués par celui-ci. Nous érodons le résultat afin d'éviter que les cellules composées ne touchent les cellules uniques lors de l'union qui aura lieu après. Finalement une ouverture par aire de valeur $\mli{RBCRadiusMinArea} / 5$ est effectuée pour enlever d'éventuels artefacts, comme montré par la figure~\ref{fig:soustraction-thinning}.
-
-% L'exemple provient de 1401063467-0007-schizonte.png (x: 900, y: 50).
 \begin{figure}[htbp]
        \centering
-       \begin{subfigure}[t]{0.3\textwidth}
-               \includegraphics[width=\linewidth]{img/segmentation_composites_1.jpg}
-               \caption{Composante verte ($\mli{gray}$)}
-               \label{fig:segmentation-composites-1}
-       \end{subfigure} 
-       
-       \begin{subfigure}[t]{0.3\textwidth}
-               \includegraphics[width=\linewidth]{img/segmentation_composites_2.jpg}
-               \caption{Image binaire des éléments composés, obtenue à partir de $\mli{grayFlat}$}
-               \label{fig:segmentation-composites-2}
+       \begin{subfigure}[t]{0.48\textwidth}
+               \includegraphics[width=1\linewidth]{img/Granulometry-pattern-spectrum-morpho-circle.pdf}
+               \caption{Spectre granulométrique d'une succession de fermetures morphologiques avec un disque}
+               \label{fig:spectre-granulométrique-morpho-cercle}
        \end{subfigure}
-       ~       
-       \begin{subfigure}[t]{0.3\textwidth}
-               \includegraphics[width=\linewidth]{img/segmentation_composites_3.jpg}
-               \caption{Ouverture de \emph{(a)} en utilisant \emph{(b)} comme masque}
-               \label{fig:segmentation-composites-3}
-       \end{subfigure} 
        ~
-       \begin{subfigure}[t]{0.3\textwidth}
-               \includegraphics[width=\linewidth]{img/segmentation_composites_4.jpg}
-               \caption{Gradient morphologique}
-               \label{fig:segmentation-composites-4}
-       \end{subfigure} 
-       
-       \begin{subfigure}[t]{0.3\textwidth}
-               \includegraphics[width=\linewidth]{img/segmentation_composites_5.jpg}
-               \caption{Remplissage des trous}
-               \label{fig:segmentation-composites-5}
-       \end{subfigure}
-       ~       
-       \begin{subfigure}[t]{0.3\textwidth}
-               \includegraphics[width=\linewidth]{img/segmentation_composites_6.jpg}
-               \caption{Amincissement}
-               \label{fig:segmentation-composites-6}
+       \begin{subfigure}[t]{0.48\textwidth}
+               \includegraphics[width=1\linewidth]{img/Granulometry-time-morpho-circle.pdf}
+               \caption{Temps en milliseconde pour chaque fermeture. Temps total : 54~s}
+               \label{fig:temps-morpho-cercle}
        \end{subfigure}
-       ~
-       \begin{subfigure}[t]{0.3\textwidth}
-               \includegraphics[width=\linewidth]{img/segmentation_composites_7.jpg}
-               \caption{Fermeture par aire puis mise à zéro des pixels marqués par \emph{(f)}}
-               \label{fig:segmentation-composites-7}
-       \end{subfigure}
-       \caption{Processus de séparation des cellules composées}
-       \label{fig:segmentation-composites}
+       \caption{Spectre granulométrique et temps en milliseconde pour chacune des fermetures morphologique avec un disque comme élément structurant. Réalisé sur une image 2592 \texttimes 1944 contenant environ 700 érythrocytes}
+       \label{fig:Granulometry-morpho-circle}
 \end{figure}
 
 \begin{figure}[htbp]
        \centering
-       \includegraphics[width=0.8\linewidth]{img/splitting_composites.jpg}
-       \caption{Résultat de l'amincissement soustrait de son remplissage. De petits artefacts, ici marqués en jaune, restent présents. S'ils ne sont pas enlevés, ceux-ci seront comptés comme des cellules à part entière lors de la labellisation finale}
-       \label{fig:soustraction-thinning}
-\end{figure}
-
-
-\subsubsection{Résultat final}
-
-Les deux résultats précédents, c'est-à-dire les cellules uniques et les zones composées de plusieurs cellules sont unies. Les zones touchant les bords sont supprimées afin d'obtenir le résultat final. Son gradient imprimé sur l'image initiale est montré par la figure~\ref{fig:segmentation_finale}. Le résultat est nommé $\mli{redCells}$.
-
-\begin{figure}[htbp]
-       \centering
-       \includegraphics[width=0.8\linewidth]{img/segmentation_finale.jpg}
-       \caption{Le gradient du résultat de la segmentation des globules rouges imprimé sur l'image initiale ($\mli{redCells}$)}
-       \label{fig:segmentation_finale}
+       \begin{subfigure}[t]{0.48\textwidth}
+               \includegraphics[width=1\linewidth]{img/Granulometry-pattern-spectrum-morpho-octagon.pdf}
+               \caption{Spectre granulométrique d'une succession de fermetures morphologiques avec un octogone}
+               \label{fig:spectre-granulométrique-morpho-octogone}
+       \end{subfigure}
+       ~
+       \begin{subfigure}[t]{0.48\textwidth}
+               \includegraphics[width=1\linewidth]{img/Granulometry-time-morpho-octagon.pdf}
+               \caption{Temps en milliseconde pour chaque fermeture. Temps total : 54~s}
+               \label{fig:temps-morpho-octogone}
+       \end{subfigure}
+       \caption{Spectre granulométrique et temps en milliseconde pour chacune des fermetures morphologique avec un octogone comme élément structurant. Réalisé sur une image 2592 \texttimes 1944 contenant environ 700 érythrocytes}
+       \label{fig:Granulometry-morpho-octagon}
 \end{figure}
 
-\subsection{Identification des globules rouges infectés}
+\subsubsection{Calcul du gradient}
 
-À partir du résultat de la segmentation des globules rouges obtenu à la section~\ref{segmentation-globules-rouges} et des éléments colorés obtenus à la section~\ref{detection-parasites} nous pouvons reconstruire les cellules infectées par une succession de dilatations des éléments colorés puis de masquages par les globules rouges. Ceci est réalisé à l'aide de la fonction \texttt{mminfrec}. Le résultat est nommé $\mli{infectedRedCells}$.
+\subsubsection{Recherche des bords des érythrocytes}
 
-La figure~\ref{fig:globules_rouges_infectes} montre les globules rouges infectés.
+\subsubsection{Recherche d'ellipses}
 
-\begin{figure}[htbp]
-       \centering
-       \includegraphics[width=0.8\linewidth]{img/globules_rouges_infectes.jpg}
-       \caption{Le gradient des globules rouges imprimé sur l'image originale ($\mli{infectedRedCells}$)}
-       \label{fig:globules_rouges_infectes}
-\end{figure}
+% RANSAC + scoring + pruning
 
 
+\subsection{Segmentation des parasites}
 
-\subsection{Calcul du taux d'infection}
+\subsubsection{Extraction des zones colorées sombres}
 
-Les deux images $\mli{redCells}$ et $\mli{infectedRedCells}$ sont labellisées afin d'obtenir le nombre total de globules rouges ($\mli{nbRedCells}$) et le nombre de globules rouges infectés ($\mli{nbInfectedRedCells}$). Le pourcentage d'infection est le rapport de ces deux valeurs : $100 \cdot \mli{nbInfectedRedCells} / \mli{nbRedCells}$.
+\subsubsection{Segmentation des cytoplasmes}
 
+\subsubsection{Segmentation des noyaux}
 
-\section{Résultats}
-\label{resultats}
+\subsection{Classification des cellules}
 
-\subsection{Qualité des images en entrée}
+\subsubsection{Suppression des cellules non-conformes}
 
-La première étape, décrite à la section~\ref{detection-parasites} et qui consiste à marquer les éléments colorés, est très sensible à la qualité de l'image initiale. Suivant les réglages du microscope et du logiciel de capture associé, les éléments colorés peuvent ressortir avec une plus grande ou une plus faible saturation par rapport au fond et aux globules rouges. De plus la teinte des éléments colorés peut être similaire à la celle du fond : la figure~\ref{fig:problemes-teinte} montre une image dont la teinte est inexploitable.
+\subsubsection{Attribution d'une classe}
 
-\begin{figure}[htbp]
-       \centering
-       \begin{subfigure}[t]{0.4\textwidth}
-               \includegraphics[width=\linewidth]{img/problemes_teinte_input.jpg}
-               \caption{Image en entrée}
-               \label{fig:problemes-teinte-input}
-       \end{subfigure} 
-       ~
-       \begin{subfigure}[t]{0.4\textwidth}
-               \includegraphics[width=\linewidth]{img/problemes_teinte_composante.jpg}
-               \caption{Composante teinte}
-               \label{fig:problemes-teinte-composante}
-       \end{subfigure}
-       
-       \begin{subfigure}[t]{0.6\textwidth}
-               \includegraphics[width=\linewidth]{img/histogram_probleme_teinte.eps}
-               \caption{Histogramme de la composante \emph{teinte} de l'image complète. Les éléments colorés se situent entre l'intensité 190 et 200 (mis en évidence en rouge) et se confondent avec le fond}
-               \label{fig:problemes-teinte-histogramme}
-       \end{subfigure}
-       \caption{}
-       \label{fig:problemes-teinte}
-\end{figure}
 
+\section{Implémentation}
 
-\subsection{Méthode}
 
-Les résultats se basent sur l'analyse d'un ensemble d'images provenant d'un même patient : le taux d'infection devrait donc être homogène entre les différentes images. Pour mesurer la qualité de nos détections, un dénombrement réalisé à la main est effectué, puis celui-ci est comparé à nos détections automatisées. La différence entre le résultat issu de la détection et la réalité est alors calculée et nommée \emph{erreur}.
 
-Finalement, nous calculons les moyennes et les écarts types des erreurs, et des deux taux, celui de référence et celui détecté.
 
-\subsection{Échantillon n°1412151257}
+\section{Résultats}
 
-Les tableaux \ref{tab:resultats60} et \ref{tab:resultats100} montrent les résultats du processus appliqué à des images réduites de 40~\%, respectivement à des images originales.
+% Comparaison avec MA et le PA.
 
-\begin{table}[htbp]
-\setlength{\tabcolsep}{3pt}
-\centering
-\begin{tabular}{ r || r | r | r || r | r | r || r || r }
-        & \multicolumn{3}{ c| }{Mesuré} & \multicolumn{3}{ c| }{Détecté} & & \\ \cline{2-8}
-       N° & Total & Infectés & Taux & Total & Infectés & Taux & Erreur & t~[s] \\ \cline{1-9}
-       0013 & 150 & 22 & 14.7~\% & 153 & 23 & 15.0~\% &  2.5~\% & 68 \\
-       0014 & 132 & 18 & 13.6~\% & 131 & 18 & 13.7~\% &  0.8~\% & 65 \\
-       0036 & 133 & 20 & 15.0~\% & 135 & 21 & 15.6~\% &  3.4~\% & 70 \\
-       0038 & 116 & 19 & 16.4~\% & 117 & 19 & 16.2~\% & -0.9~\% & 68 \\
-       0039 & 136 & 26 & 19.1~\% & 137 & 27 & 19.7~\% &  3.1~\% & 79 \\
-       0040 & 128 & 19 & 14.8~\% & 128 & 18 & 14.1~\% & -5.3~\% & 78 \\
-       0041 & 120 & 26 & 21.7~\% & 121 & 27 & 22.3~\% &  3.0~\% & 89 \\
-       0042 & 135 & 26 & 19.3~\% & 144 & 30 & 20.8~\% &  8.2~\% & 82 \\
-       0043 & 128 & 20 & 15.6~\% & 132 & 27 & 20.5~\% & 30.9~\% & 80 \\
-       0044 & 133 & 25 & 18.8~\% & 135 & 25 & 18.5~\% & -1.5~\% & 86 \\
-       0045 & 136 & 22 & 16.2~\% & 136 & 28 & 20.6~\% & 27.3~\% & 84 \\
-       0046 & 127 & 29 & 22.8~\% & 128 & 27 & 21.1~\% & -7.6~\% & 85
-\end{tabular}
-\caption{Résultats sur les images réduites de 40~\%}
-\label{tab:resultats60}
-\end{table}
-
-
-\begin{table}[htbp]
-\setlength{\tabcolsep}{3pt}
-\centering
-\begin{tabular}{ r || r | r | r || r | r | r || r || r }
-        & \multicolumn{3}{ c| }{Mesuré} & \multicolumn{3}{ c| }{Détecté} & & \\ \cline{2-8}
-       N° & Total & Infectés & Taux & Total & Infectés & Taux & Erreur & t~[s] \\ \cline{1-9}
-       0013 & 150 & 22 & 14.7~\% & 153 & 22 & 14.4~\% & -2.0~\% & 429 \\
-       0014 & 132 & 18 & 13.6~\% & 136 & 19 & 14.0~\% &  2.5~\% & 462 \\
-       0036 & 133 & 20 & 15.0~\% & 135 & 21 & 15.6~\% &  3.4~\% & 431 \\
-       0038 & 116 & 19 & 16.4~\% & 118 & 19 & 16.1~\% & -1.7~\% & 447 \\
-       0039 & 136 & 26 & 19.1~\% & 136 & 27 & 19.9~\% &  3.8~\% & 463 \\
-       0040 & 128 & 19 & 14.8~\% & 131 & 18 & 13.7~\% & -7.4~\% & 458 \\
-       0041 & 120 & 26 & 21.7~\% & 122 & 27 & 22.1~\% &  2.1~\% & 526 \\
-       0042 & 135 & 26 & 19.3~\% & 142 & 27 & 19.0~\% & -1.3~\% & 483 \\
-       0043 & 128 & 20 & 15.6~\% & 128 & 25 & 19.5~\% & 25.0~\% & 515 \\
-       0044 & 133 & 25 & 18.8~\% & 137 & 25 & 18.2~\% & -2.9~\% & 505 \\
-       0045 & 136 & 22 & 16.2~\% & 144 & 30 & 20.8~\% & 28.8~\% & 538 \\
-       0046 & 127 & 29 & 22.8~\% & 126 & 27 & 21.4~\% & -6.2~\% & 557
-\end{tabular}
-\caption{Résultats sur les images originales (non-réduites)}
-\label{tab:resultats100}
-\end{table}
-
-Le taux moyen d'infection réel est de 17.3~\% et le taux détecté avec des images réduites de 40~\% est de 18.2~\%. Ces données sont présentées dans le tableau~\ref{tab:resultats}.
-
-\begin{table}[htbp]
-\setlength{\tabcolsep}{3pt}
-\centering
-\begin{tabular}{ l | r | r | r }
-       & Réel & 40~\% & 100~\% \\ \cline{2-4}
-       Taux de cellules infectées moyen détecté & 17.3~\% & 18.2~\% & 17.9~\% \\
-%      Écart type sur le taux de cellules infectées moyen détecté & 2.9~\% & 3.1~\% & 3.0~\% \\ 
-       \cline{1-4}
-       Erreur moyenne & & 5.3~\% & 3.7~\% \\
-%      Écart type sur l'erreur & & 11.9~\% & 11.4~\% \\ 
-       \cline{1-4}
-       Temps moyen & & 78~s & 485~s
-\end{tabular}
-\caption{Résultats résumant les tables \ref{tab:resultats60} et \ref{tab:resultats100}. La colonne 40~\% correspond aux images réduites. La colonne 100~\% correspond aux images originales}
-\label{tab:resultats}
-\end{table}
 
 
 \section{Conclusion}
 
-Par la mise en pratique de la méthode décrite dans \emph{Analysis of infected blood cell images using morphological operators}\cite{DiRuberto-Analysis-infected-blood-morphological} nous avons pu démontrer sa pertinence et son efficacité. Quelques modifications ont dû être apportées telles que les critères de détection des schizontes ou encore l'utilisation d'une transformée en distance pour la définition des marqueurs de l'algorithme de la ligne de partage des eaux lors de la segmentation des globules rouges.
-
-De plus, une des difficultés majeures a été d'obtenir une image initiale ayant des composantes \emph{teinte} et \emph{saturation} satisfaisantes. Cette première étape de détection des éléments colorés reste très dépendante de la qualité et des caractéristiques de l'image d'entrée.
-
-Néanmoins, une fois de bons réglages effectués sur le système de capture d'image du microscope, les résultats du processus, en terme de taux d'infection, correspondent à ce qui est attendu. Sur le cas analysé (n°1412151257), à partir de douze captures, nous obtenons une différence de taux de 5.3~\% pour les images réduites et de 3.7~\% pour les images non-réduites.
-
-
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 \bibliography{main}