### update/align notation

parent a385008e
 ... @@ -184,7 +184,7 @@ For computational reasons it is best when the absorption column of the complemen ... @@ -184,7 +184,7 @@ For computational reasons it is best when the absorption column of the complemen \end{equation} \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): 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} \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} \end{equation} \subsubsection{Normalised' channel absorption} \subsubsection{Normalised' channel absorption} ... @@ -192,7 +192,7 @@ Note that this column still needs to be normalized to $\col{\hat{k}}_3$. Also no ... @@ -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. 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} \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 \\ \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} 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 ... @@ -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 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} \begin{align} \varphi_{yg} & = \acos \left(\colt{\hat{k}}_y \cdot \col{\hat{k}}_g\right) \notag \\ \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( 0.0007\cdot 0.9631 + 0.0357\cdot 0.0860 + 0.9994 \cdot 0.2549\right) \notag \\ & = \acos\left(0.2585\right) \notag \\ & = \acos\left(0.2585\right) \notag \\ & = 75\,\degree & = 75\,\degree ... @@ -261,7 +261,7 @@ The Java code by Ruifrok that was adapted by Landini for the colour deconvoluti ... @@ -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{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*} \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} \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} \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} \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} \end{equation} ... ...
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