Commit 918adda1 authored by Turnhout, M.C. van's avatar Turnhout, M.C. van
Browse files

add some figures

parent 7c828ec9
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%% To include the image in your LaTeX document, write
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%% \includegraphics{<filename>.pdf}
%% To scale the image, write
%% \def\svgwidth{<desired width>}
%% \input{<filename>.pdf_tex}
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%% \includegraphics[width=<desired width>]{<filename>.pdf}
%%
%% Images with a different path to the parent latex file can
%% be accessed with the `import' package (which may need to be
%% installed) using
%% \usepackage{import}
%% in the preamble, and then including the image with
%% \import{<path to file>}{<filename>.pdf_tex}
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%% Images with a different path to the parent latex file can
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%% installed) using
%% \usepackage{import}
%% in the preamble, and then including the image with
%% \import{<path to file>}{<filename>.pdf_tex}
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......@@ -76,8 +76,10 @@ The actual (non-zero) value for `the amount of dye' that we find is precisely th
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\caption{Deconvolved (pure) dye amounts for the three colour flag. For each dye we find a binary image with dye amounts of either zero (black) or the length of the dye's absorption column $\abs{{\hat{A}}}$ before normalisation to ${\hat{k}}$ (white). With \textbf{(a)} the amount of yellow is either zero or $\hat{a}_y = 5.5487$ (equation \ref{flagJAabsy}); \textbf{(b)} the amount of green is either zero or $\hat{a}_g = 5.7575$ (equation \ref{flagJAabsg}); and \textbf{(c)} the amount of yellow is either zero or $\hat{a}_b = 9.6045$ (equation \ref{flagJAabsb}). \label{flagJAas}}
\def\svgwidth{0.47\linewidth}\includesvg{pics/flagJAab}}\hfill
\subfloat[\label{flagJAaygb}]{%
\def\svgwidth{0.47\linewidth}\includesvg{pics/flagJAaygb}}\\
\caption{Deconvolved (pure) dye amounts for the three colour flag. For each dye we find a binary image with dye amounts of either zero (black) or the length of the dye's absorption column $\abs{{\hat{A}}}$ before normalisation to ${\hat{k}}$ (white). With \textbf{(a)} the amount of yellow is either zero or $\hat{a}_y = 5.5487$ (equation \ref{flagJAabsy}); \textbf{(b)} the amount of green is either zero or $\hat{a}_g = 5.7575$ (equation \ref{flagJAabsg}); \textbf{(c)} the amount of yellow is either zero or $\hat{a}_b = 9.6045$ (equation \ref{flagJAabsb}); and \textbf{(d)} a combined image with absolute absorption values. \label{flagJAas}}
\end{figure}
......@@ -182,11 +184,19 @@ To do so, we need to calculate the normalised cross product of the normalised ye
\col{\hat{A}}_c & = \col{\hat{k}}_y \times \col{\hat{k}}_g = \begin{bmatrix*}[r] -0.0769 \\ 0.9623 \\ -0.0343\end{bmatrix*}\\
\col{\hat{k}}_c & = \frac{\col{\hat{A}}_c }{\abs{\col{\hat{A}}_c }} = \begin{bmatrix*}[r] -0.0796 \\ 0.9962\\ -0.0355\end{bmatrix*}
\end{align}
Note the negative absorption coefficients: this column represents` a dye' that produces, instead of absorbs, red and blue. As a result the `RGB' values for this dye (normalised amount of 1) come out as:
Note the negative absorption coefficients: this column represents `a dye' that produces, instead of absorbs, red and blue. As a result the `RGB' values for this dye (normalised amount of 1) come out as:
\begin{equation}
\col{\hat{R}}_c = 256\cdot \mathrm{e}^{\begin{bmatrix*}[r] 0.0796 \\ -0.9962\\ 0.0355\end{bmatrix*} \cdot 1} - 1 = \begin{bmatrix} 277\\ 95 \\ 265 \end{bmatrix}
\end{equation}
In other words: this third `dye colour' is not a real `colour' and cannot be (properly) represented in RGB space.
In other words: this third `dye colour' is not a real `colour' and cannot be (properly) represented in RGB space. It is a column that (precisely) covers the part of the 3D absorption space that is not covered by the other two columns, it is `the remainder' for use in deconvolution.
\begin{figure}
\subfloat[\label{arrowsflag}]{%
\def\svgwidth{0.47\linewidth}\includesvg{pics/arrowsflag}}\hfill
\subfloat[\label{arrowsRGB}]{%
\def\svgwidth{0.47\linewidth}\includesvg{pics/arrowsRGB}}\\
\caption{Absorption columns in 3D space.\label{arrowsfig}}
\end{figure}
\subsection{Deconvolution}
......@@ -196,7 +206,8 @@ In other words: this third `dye colour' is not a real `colour' and cannot be (pr
\subsection{Adding a third `Ruifrok' absorption column}
\begin{equation}
\col{\hat{A}}_c = \begin{bmatrix} \sqrt{1 - k_{y_1}^2 - k_{g_1}^2} \\[1em] \sqrt{1 - k_{y_2}^2 - k_{g_2}^2} \\[1em] \sqrt{1 - k_{y_3}^2 - k_{g_3}^2}\end{bmatrix},\, \col{\hat{A}}_c \leq 0
\hat{A}_{c_n} = \begin{cases} \sqrt{1 - k_{y_n}^2 - k_{g_n}^2} & \text{~for~} k_{y_n}^2 + k_{g_n}^2 \leq 1 \\
0 & \text{~for~} k_{y_n}^2 + k_{g_n}^2 > 1\end{cases}
\end{equation}
......
......@@ -4,7 +4,8 @@ clear all; close all;
im = imread('pics/Flag_of_Jamaica.png');
% image colours
dyes = [254 209 0; % yellow
0 155 58]'; % green
0 155 58;
0 0 0]'; % green
% do the deconvolve
[amounts, P, Q, R, RGB, A, K, iOD] = cld_decon(dyes, im, 'rgb');
......
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