\chapter{Transformation L*a*b}
\label{appendix:lab_transformation}

L*a*b est otbenu en utilisant la transformation XYZ de la matrice d'image (espace CIZ XYZ). Les équations (\ref{eq:Lab_L}), (\ref{eq:Lab_a}), (\ref{eq:Lab_b}) détaillent la manière dont chaque canal est calculé.

\begin{equation}
L = 116 f_y - 16
\label{eq:Lab_L}
\end{equation}

\begin{equation}
a = 500(f_x - f_y)
\label{eq:Lab_a}
\end{equation}

\begin{equation}
b = 200(f_y - f_z)
\label{eq:Lab_b}
\end{equation}

où

\vspace{1mm}

\begin{equation}
    f_x = \left\{
      \begin{array}{lr}
        \sqrt[3]{x_r} & \text{if } x_r > \epsilon \\
    	{{\kappa x_r + 16} \over {116}} & \text{otherwise}
      \end{array}
    \right.
\label{eq:Lab_fx}
\end{equation}


\begin{equation}
    f_y = \left\{
      \begin{array}{lr}
        \sqrt[3]{y_r} & \text{if } y_r > \epsilon \\
    	{{\kappa y_r + 16} \over {116}} & \text{otherwise}
      \end{array}
    \right.
\label{Lab_fy}
\end{equation}

\begin{equation}
    f_z = \left\{
      \begin{array}{lr}
        \sqrt[3]{z_r} & \text{if } z_r > \epsilon \\
    	{{\kappa z_r + 16} \over {116}} & \text{otherwise}
      \end{array}
    \right.
\label{Lab_fz}
\end{equation}

% default display	
\hspace{18mm} $x_r = {{X} \over {X_r}}$,\hspace{2mm} $y_r = {{Y} \over {Y_r}}$, \hspace{2mm} $z_r = {{Z} \over {Z_r}}$

\begin{equation}
    \epsilon  = \left\{
      \begin{array}{lr}
        {0.008856} & \text{Actual CIE standard} \\
    	{216 / 24389} & \text{Intent of the CIE standard}
      \end{array}
\right.
\label{Lab_espsilon}
\end{equation}

\begin{equation}
    \kappa   = \left\{
      \begin{array}{lr}
        {903.3} & \text{Actual CIE standard} \\
    	{24389 / 27} & \text{Intent of the CIE standard}
      \end{array}
    \right.
\label{Lab_kappa}
\end{equation}