From: Greg Burri <greg.burri@gmail.com>
Date: Wed, 10 Jun 2015 22:33:43 +0000 (+0200)
Subject: Mise à jour du rapport.
X-Git-Url: http://git.euphorik.ch/index.cgi?a=commitdiff_plain;h=0267756d03a8eec3cc9e72ff5791e5733f5ffe66;p=malaria.git

Mise à jour du rapport.
---

diff --git a/Documentation/2010-Ma.pdf b/Documentation/2010-Ma.pdf
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diff --git a/Documentation/2011-Purwar.pdf b/Documentation/2011-Purwar.pdf
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diff --git a/Documentation/Malaria/2013-Bailey-2.pdf b/Documentation/Malaria/2013-Bailey-2.pdf
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diff --git a/Documentation/Malaria/2013-Manser.pdf b/Documentation/Malaria/2013-Manser.pdf
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diff --git a/rapport/img/globules_rouges_infectes.jpg b/rapport/img/globules_rouges_infectes.jpg
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 0 1 255 { Encoding exch /.notdef put } for
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 Encoding 49 /one put
 Encoding 50 /two put
 Encoding 51 /three put
 Encoding 52 /four put
 Encoding 53 /five put
-/CharStrings 6 dict dup begin
+Encoding 83 /S put
+Encoding 84 /T put
+Encoding 97 /a put
+Encoding 98 /b put
+Encoding 99 /c put
+Encoding 100 /d put
+Encoding 101 /e put
+Encoding 103 /g put
+Encoding 105 /i put
+Encoding 108 /l put
+Encoding 109 /m put
+Encoding 110 /n put
+Encoding 111 /o put
+Encoding 114 /r put
+Encoding 115 /s put
+Encoding 116 /t put
+Encoding 117 /u put
+Encoding 233 /eacute put
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 /one 1 def
 /two 2 def
 /four 3 def
 /three 4 def
 /five 5 def
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 end readonly def
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 showpage
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          xlink:href="process_schizonte.jpg"
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+      <image
+         sodipodi:absref="D:\PA\rapport\img\process_cellules_infectées.jpg"
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+           id="flowPara5866-7-5">Dénombrement des cellules infectées</flowPara></flowRoot>    </g>
+    <path
+       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1.74999988;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2Mend-5)"
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          r="11.594452"
@@ -688,5 +689,29 @@
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            sodipodi:role="line">5</tspan></text>
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+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:17.5px;line-height:125%;font-family:Garamond;-inkscape-font-specification:'Garamond, Normal';text-align:center;letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;text-anchor:middle;fill:#ffffff;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
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+       y="60.335453"
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+       sodipodi:linespacing="125%"><tspan
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+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:17.5px;line-height:125%;font-family:Garamond;-inkscape-font-specification:'Garamond, Normal';text-align:center;letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;text-anchor:middle;fill:#ffffff;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
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+       y="348.35944"
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+         sodipodi:role="line"
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diff --git a/rapport/main.bib b/rapport/main.bib
index abf976c..a30d667 100644
--- a/rapport/main.bib
+++ b/rapport/main.bib
@@ -15,4 +15,10 @@
    author = "Talbot, Hugues",
    title = "Introduction à la Morphologie Mathématique, Théorie et applications",
    year = "2010"
+}
+
+@misc {Manzanera-Morphologie,
+   author = "Manzanera, Antoine",
+   title = "Cours de morphologie mathématique",
+   year = "2005"
 }
\ No newline at end of file
diff --git a/rapport/main.tex b/rapport/main.tex
index b68bb02..c295911 100644
--- a/rapport/main.tex
+++ b/rapport/main.tex
@@ -156,8 +156,6 @@ Cet article a pour but la description et la réalisation d'une méthode automati
 %%%%%%%%%%%%%%%%%%
 \section{Introduction}
 
-% TODO : utiliser \mli{} pour tous les variables à plus d'un caractère.
-
 Le but de ce projet est de dénombrer de manière automatisée les globules rouges infectés par la malaria. Le nombre total de globules rouges est également recensé afin de pouvoir établir un taux d'infection. Cela est réalisé à partir d'images microscopiques fournies par Dr. Guy Prod'hom du laboratoire de parasitologie du \emph{CHUV} à Lausanne.
 
 Pour ce faire, nous nous basons sur l'article \emph{Analysis of infected blood cell images using morphological operators}\cite{DiRuberto-Analysis-infected-blood-morphological} qui décrit une approche principalement basée sur des opérations morphologiques afin d'établir la parasitémie d'images d'échantillons de sang infecté par la malaria.
@@ -166,7 +164,7 @@ L'ensemble du processus décrit ci-après est implémenté sous \emph{MATLAB} :
 
 \subsection{Images}
 
-Les images initiales correspondent à des photographies de sang infecté agrandies 100 fois. Un colorant est utilisé 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 décrit ci-après et montré par la figure~\ref{fig:medical-plasmodium}.
+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. Ces éléments sont montrés par les figures \ref{fig:medical-elements-sang} et \ref{fig:medical-plasmodium}.
 
 \begin{figure}[htbp]
 	\centering
@@ -229,35 +227,137 @@ Notre but est de dénombrer les globules rouges sains ainsi que ceux infectés p
 
 Cette section aborde la description des différents outils morphologiques ainsi que les logiciels qui ont permis de mettre en pratique le processus.
 
+\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/}}.
+
+
 \subsection{Outils morphologique}
 
-<TODO>
+Une grande partie du processus décrit dans ce document s'appuie sur la morphologie mathématique\footnote{Georges Matheron et Jean Serra - 1964} appliquée, ici, au traitement d'images.
 
-La morphologie mathématique est un
+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}.
 
-* mmdil
-* mmero
-* mmopen
-* mmclose
-* mmareaopen (areaOpen)
-* mmareaclose
-* Regmax (maximaux locaux)
-* mminfrec reconstruction par dilatation
-* Watersheds (ligne de partage des eaux)
 
-* gamma\_lambda à la place de areaOpen
+\subsubsection{Définitions de base}
 
+Nous définissions tout d'abord l'image $E$ comme étant un sous-ensemble à deux dimension 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 $\breve{B} = B$.
 
-\subsection{Outils logiciels}
+{\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 nommé $\delta$. 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}.
 
-<TODO> Description de la structuration du projet MATLAB, du toolbox morphologique, etc...
+
+\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}.
+
+
+\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 $\lambda_2$.
+
+\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}
+	\label{fig:fermeture-par-aire}
+\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 de la montée des eaux une digue est construite, l'ensemble des digue 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*}
+	\forall p \in M, \exists p' \in M, p \in (p' \oplus B) && \\
+	\begin{cases}
+		\forall p \in M & \text{si } f(p) = t \\
+		\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.
+
+\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}
+	\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}
+	\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}
+	\end{subfigure}
+	\caption{Example d'application de la fonction \texttt{mmregmax}}
+	\label{fig:regmax_sample}
+\end{figure}
 
 
 \section{Approche générale}
 
-Le processus complet est illustré par la figure \ref{fig:processusComplet}. Les différents chiffres correspondent aux étapes décrites ci après.
+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).
 
-La première étape (\ding{192}) consiste à détecter les éléments qui ont été spécifiquement 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.
+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.
 
@@ -334,7 +434,7 @@ Afin de gommer le bruit et de rendre le fond plus homogène, un filtre médian 5
 
 \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{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.
+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 un 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 : $\lvert N_{i + 1} - N_i \rvert$.
@@ -349,61 +449,32 @@ Nous définissons le rayon moyen des globules rouges ($\mli{RBCRadius}$) comme 
 	\label{fig:patternSpectrum}
 \end{figure}
 
-\subsubsection{Extrêmes régionaux}
-
-Pour la recherche des extrêmes régionaux nous utilisons la fonction \texttt{mmregmax}.
-
-<TODO> : Formaliser mmregmax et dire que ça ne correspond pas à la définition du bouquin.
- 
-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.
-
-\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}
-	\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}
-	\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}
-	\end{subfigure}
-	\caption{Example d'application de la fonction \texttt{mmregmax}}
-	\label{fig:regmax_sample}
-\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}
+\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}
+\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}
+\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}
+\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} &= \mli{areaOpen}(\mli{TH} \cap \mli{TS}, 10) &
-\end{flalign}
+\begin{flalign*}
+	\mli{THS} &= \gamma_{10}(\mli{TH} \cap \mli{TS}) &
+\end{flalign*}
 
 \begin{figure}[htbp]
 	\centering
@@ -453,15 +524,15 @@ Pour savoir si deux sous-ensembles, $A$ et $B$, sont à une \emph{distance de Ha
 \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 (\emph{watersheds} en anglais) 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.
+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}
+\begin{flalign*}
 	\mli{RBCRadiusMin} &= \lfloor 0.75 \cdot \mli{RBCRadius} \rfloor &\\
 	\mli{RBCRadiusMinArea} &= \pi \cdot \mli{RBCRadiusMin}^2 &
-\end{flalign}
+\end{flalign*}
 
 
 \subsubsection{Filtrage préliminaire}
@@ -469,9 +540,9 @@ Durant le processus, nous avons besoin de définir le rayon minimum d'un globule
 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}.
 
 {\setlength{\abovedisplayskip}{0pt}
-\begin{flalign}
-	\mli{grayFlat} &= \mli{areaOpen}(\mli{green}(\mli{imgRGB})) \setminus \mli{WBC} \setminus \mli{schizonts} &
-\end{flalign}
+\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.
 
@@ -618,7 +689,7 @@ Les deux images $\mli{redCells}$ et $\mli{infectedRedCells}$ sont labellisées a
 
 \subsection{Qualité des images en entrée}
 
-La première étape, décrite à la section~\ref{detection-parasites}, qui consiste à marquer les éléments colorés est très sensible à la qualité de l'image initiale. Suivant les réglages du microscope 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 peuvent être similaire à la celle du fond : La figure~\ref{fig:problemes-teinte} montre une image dont la teinte est inexploitable.
+La première étape, décrite à la section~\ref{detection-parasites}, 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 peuvent être similaire à la celle du fond : La figure~\ref{fig:problemes-teinte} montre une image dont la teinte est inexploitable.
 
 \begin{figure}[htbp]
 	\centering
@@ -701,7 +772,7 @@ Les tableaux \ref{tab:resultats60} et \ref{tab:resultats100} montrent les résul
 \label{tab:resultats100}
 \end{table}
 
-Le taux moyen d'infection réel est de 17.3~\%, le taux détecté avec des images réduites
+Le taux moyen d'infection réel est de 17.3~\%, 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}
@@ -733,6 +804,19 @@ Le taux moyen d'infection réel est de 17.3~\%, le taux détecté avec des image
 \begin{appendices}
 \section{Code MATLAB} \label{app:matlab}
 
+\subsection{Structuration}
+
+Le script principal est \texttt{Main.m}. Il débute avec les paramètres suivants :
+
+\begin{itemize}
+	\item \texttt{interactive} : si \texttt{true} alors le chemin vers une image est demandé sinon une ou plusieurs images sont définies dans \texttt{imagePathInput} ?
+	\item \texttt{defaultDir} : Le dossier contenant les images.
+	\item \texttt{scaleFactor} : La taille des images traitée, 1 correspond à la taille original.
+	\item \texttt{invertSaturation} : Doit être mis à \texttt{true} si les éléments colorés sont moins saturé que le fond
+\end{itemize}
+
+Les autres fichiers \emph{MATLAB} correspondent aux fonctions, principalement appelées à partir de \texttt{Main.m}.
+
 \subsection{\texttt{Main.m}}
 \lstinputlisting{../src/Main.m} 
 
diff --git a/src/SegmentationOfRedCells.m b/src/SegmentationOfRedCells.m
index 0fd2285..b658ee6 100644
--- a/src/SegmentationOfRedCells.m
+++ b/src/SegmentationOfRedCells.m
@@ -79,7 +79,7 @@ function [segmentedCells] = SegmentationOfRedCells(imgRGB, RBCRadius, WBC, schiz
     compositesThinningFlooded = mmero(compositesThinningFlooded); % We erode by a 3x3 cross to avoid the composites to touch the singles when fusionning later.
     compositesThinningFlooded = mmareaopen(compositesThinningFlooded, RBCRadiusMinArea / 5); % We remove little artefacts.
     imwrite(compositesThinningFlooded, '../output/red cells segmentation - composites thinning flooded.png')
-
+ 
     segmentedCells = mmedgeoff(or(compositesThinningFlooded, watershedSingles));
     
     WriteImageGradient(segmentedCells, '../output/red cells segmentation - individual.png', [255, 255, 255]);
diff --git a/src/test_regmax.m b/src/test_regmax.m
index a18d705..0900b4b 100644
--- a/src/test_regmax.m
+++ b/src/test_regmax.m
@@ -43,7 +43,7 @@ end
 
 colormap(gray);
 axis image
-s
+
 grid on
 
 ax = gca;