Commit 4318a2f1 authored by Turnhout, M.C. van's avatar Turnhout, M.C. van
Browse files

update/align notation

parent a385008e
......@@ -184,7 +184,7 @@ For computational reasons it is best when the absorption column of the complemen
\end{equation}
Note that this column still needs to be normalized to $\col{\hat{k}}_3$. Also note that you can find the angle between two columns $n$ and $m$ by taking the inverse cosine of the inner dot product of the two columns (when the columns are normalised):
\begin{equation}
\varphi_{nm} =\acos\left(\colt{\hat{k}}_n \cdot \col{\hat{k}}_m\right) \label{angledot}
\varphi_{nm} =\acos\left(\colt{\hat{k}}_n \cdot \col{\hat{k}}_m\right) = \acos\left( \hat{k}_{R_n}\hat{k}_{R_m} + \hat{k}_{G_n}\hat{k}_{G_m} +\hat{k}_{B_n}\hat{k}_{B_m} \right) \label{angledot}
\end{equation}
\subsubsection{`Normalised' channel absorption}
......@@ -192,7 +192,7 @@ Note that this column still needs to be normalized to $\col{\hat{k}}_3$. Also no
The Java code by Ruifrok that was adapted by Landini for the `colour deconvolution' ImageJ plugin \cite{Landini2004,Landini2020,Landini2020a} contained an alternative (otherwise undocumented) method to obtain a third complementary dye.
This method attempts to normalise total absorption of all three dyes across a channel for each of the R, G and B channels and sets channel absorption for the complementary dye to 0 when this is not possible:
This method attempts to normalise total absorption of all three dyes across a channel for each of the R, G and B channels and sets channel absorption for the complementary dye to 0 when this is not possible ($n = R, G, B$):
\begin{equation}
\hat{A}_{n_3} = \begin{cases} \sqrt{1 - k_{n_1}^2 - k_{n_2}^2} & \text{~for~} k_{n_1}^2 + k_{n_2}^2 \leq 1 \\
0 & \text{~for~} k_{n_1}^2 + k_{n_2}^2 > 1\end{cases} \label{k3norm}
......
......@@ -167,7 +167,7 @@ Our calculations are done with the $\mat{K}$ matrix with (normalised) absorption
To find the angle between two columns, you can take the inverse cosine of `the inner dot product' of the two columns (equation \ref{angledot}). So the absorption angle between the flags yellow dye and green dye is
\begin{align}
\varphi_{yg} & = \acos \left(\colt{\hat{k}}_y \cdot \col{\hat{k}}_g\right) \notag \\
& = \acos \left(k_{y_1} \cdot k_{g_1} + k_{y_2} \cdot k_{g_2} + k_{y_3} \cdot k_{g_3} \right) \notag \\
& = \acos \left(k_{R_y} \cdot k_{R_g} + k_{G_y} \cdot k_{G_g} + k_{B_y} \cdot k_{B_g} \right) \notag \\
& = \acos \left( 0.0007\cdot 0.9631 + 0.0357\cdot 0.0860 + 0.9994 \cdot 0.2549\right) \notag \\
& = \acos\left(0.2585\right) \notag \\
& = 75\,\degree
......@@ -261,7 +261,7 @@ The Java code by Ruifrok that was adapted by Landini for the `colour deconvoluti
\col{\hat{A}}_c = \begin{bmatrix*}[r] \sqrt{1 - 0.0007^2 - 0.9631^2} \\ \sqrt{1 - 0.0357^2 - 0.0860^2}\\0\end{bmatrix*} & = \begin{bmatrix*}[r] 0.2690 \\ 0.9957\\0\end{bmatrix*}\\
\col{\hat{k}}_c = \frac{\col{\hat{A}}_c }{\abs{\col{\hat{A}}_c }} & = \begin{bmatrix*}[r] 0.2609 \\ 0.9654 \\ 0\end{bmatrix*}
\end{align}
This is not an `ideal perpendicular' column: the angles with yellow $\varphi_{yc} = 88$\,\degree\ and green $\varphi_{yc} = 70$\,\degree\ are smaller than 90\,\degree . In return, however, this pure dye has a `proper colour' in RGB with values:
This is not an `ideal perpendicular' column: the angles with yellow $\varphi_{yc} = 88$\,\degree\ and green $\varphi_{gc} = 70$\,\degree\ are smaller than 90\,\degree . In return, however, this pure dye has a `proper colour' in RGB with values:
\begin{equation}
\col{\hat{R}}_c = 256\cdot \mathrm{e}^{\begin{bmatrix*}[r] 0.2609 \\ 0.9654\\ 0\end{bmatrix*} \cdot 1} - 1 = \begin{bmatrix*}[r] 196\\ 96 \\ 255 \end{bmatrix*} \label{flagJA2rRc}
\end{equation}
......
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