From 2844f03285199bf2bd604c7ce40b08620a7c6ea0 Mon Sep 17 00:00:00 2001
From: bobarna <barnabas.borcsok@gmail.com>
Date: Tue, 25 Jan 2022 14:13:00 +0100
Subject: [PATCH] Work on RTE function
---
img/RTE_illustration.tex | 71 +++++++++++++
img/differential_volume.pdf | Bin 8415 -> 8420 bytes
img/differential_volume.pdf_tex | 5 +-
img/emission_illustration.tex | 62 ++++++++++++
img/in_scattering_illustration.tex | 89 ++++++++++++++++
img/interactions.tex | 9 +-
img/summing_up_the_losses.tex | 120 ++++++++++++++++++++++
main.tex | 156 +++++++++++++++++++++++++++--
8 files changed, 501 insertions(+), 11 deletions(-)
create mode 100644 img/RTE_illustration.tex
create mode 100644 img/emission_illustration.tex
create mode 100644 img/in_scattering_illustration.tex
create mode 100644 img/summing_up_the_losses.tex
diff --git a/img/RTE_illustration.tex b/img/RTE_illustration.tex
new file mode 100644
index 0000000..9545ad1
--- /dev/null
+++ b/img/RTE_illustration.tex
@@ -0,0 +1,71 @@
+\begin{tikzpicture}
+% 1 = 20p
+ \def\XSTART{-40pt}
+ \def\YSTART{60pt}
+ \def\YDIAONE{35pt}
+ \def\XONESTART{10pt}
+ \def\XONEDELTA{30pt} %width of cylinders
+ \def\DD{-10pt} %
+ \def\GAP{50pt} %
+
+ \def\LABELYSHIFT{-50pt}
+
+ %FIRST cylinder
+ \def\YEND{\YSTART+\YCURVE+\YDIATWO}
+ \def\XONEEND{\XONESTART+\XONEDELTA}
+ \def\YONEMIDDLE{\YSTART+\YDIAONE/2}
+
+ %SECOND cylinder
+ \def\XTWOSTART{\XONESTART+\XONEDELTA+\GAP}
+ \def\XTWOEND{\XONEEND+\GAP+\XONEDELTA}
+
+ %THIRD cylinder
+ \def\XTHREESTART{\XONESTART+\XONEDELTA+\GAP+\XONEDELTA+\GAP}
+ \def\XTHREEEND{\XONEEND+\GAP+\XONEDELTA+\GAP+\XONEDELTA}
+
+ % FOURTH cylinder
+ \def\XFOURSTART{\XONESTART+\XONEDELTA+\GAP+\XONEDELTA+\GAP+\XONEDELTA+\GAP}
+ \def\XFOUREND{\XONEEND+\GAP+\XONEDELTA+\GAP+\XONEDELTA+\GAP+\XONEDELTA}
+
+ \tikzset{
+ partial ellipse/.style args={#1:#2:#3}{
+ insert path={+ (#1:#3) arc (#1:#2:#3)}
+ },
+ dimen/.style={<->,>=latex,thin,
+ every rectangle node/.style={fill=white,midway,font=\sffamily}},
+ }
+
+
+ % SECOND ARROW
+ \draw[-{Triangle[width=20pt,length=10pt, color=orange]},
+ line width=8pt, color=orange]
+ ({\XONEEND+0.5},\YONEMIDDLE) -- ({\XONEEND+\GAP-0.2},\YONEMIDDLE);
+
+ % FIRST CYLINDER
+ \draw [fill=gray] (\XONESTART,\YSTART) coordinate (BA)
+ rectangle (\XONEEND,{\YSTART+\YDIAONE}) coordinate (BB);
+ \draw [fill=lightgray](\XONESTART,\YONEMIDDLE)
+ ellipse ({\YDIAONE/6} and {\YDIAONE/2});
+ \draw [fill=gray,dashed](\XONEEND,\YONEMIDDLE)
+ ellipse ({\YDIAONE/6} and {\YDIAONE/2});
+ \draw (\XONEEND,\YONEMIDDLE)
+ [partial ellipse=-90:90:{\YDIAONE/6} and {\YDIAONE/2}];
+
+ % \node[yshift=\LABELYSHIFT] at ($0.5*(BA)+0.5*(BB)$) {\textit{Absorption}};
+
+ %nabla omega below
+ \draw ($(BB)-(0,\YDIAONE)$) -- ++(0,\DD) coordinate (D1) -- +(0,5pt);
+ \draw (BA) -- ++(0,\DD) coordinate (D2) -- +(0,5pt);
+ \draw [dimen] (D1) -- (D2) node {$\nabla\boldsymbol\omega$};
+
+ % INCOMING ARROW
+ \draw[-{Triangle[width=25pt,length=15pt, color=orange]},
+ line width=15pt, color=orange]
+ node [above=85pt ] {$\textcolor{black}{L(\textbf{x},\boldsymbol\omega)}$}
+ (\XSTART,\YONEMIDDLE) -- (\XONESTART-0.2,\YONEMIDDLE);
+
+
+ %ground
+ %\draw [fill=gray] (0,0) rectangle (\XEND,\GROUND);
+
+\end{tikzpicture}%
diff --git a/img/differential_volume.pdf b/img/differential_volume.pdf
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diff --git a/img/differential_volume.pdf_tex b/img/differential_volume.pdf_tex
index 7c72df6..f2183e8 100644
--- a/img/differential_volume.pdf_tex
+++ b/img/differential_volume.pdf_tex
@@ -63,10 +63,11 @@
\put(0.93851821,0.28735976){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$\boldsymbol{\omega}$\end{tabular}}}}%
\put(0,0){\includegraphics[width=\unitlength,page=4]{differential_volume.pdf}}%
\put(0.34651173,0.21053985){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$\nabla\boldsymbol{\omega}$\end{tabular}}}}%
- \put(0.36648021,0.11863622){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$\sigma_a \nabla\boldsymbol{\omega} = P\{collision\}$\end{tabular}}}}%
+ \put(0.376347,0.11863622){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$\sigma_a \nabla\boldsymbol{\omega} = P\{collision\}$\end{tabular}}}}%
\put(0,0){\includegraphics[width=\unitlength,page=5]{differential_volume.pdf}}%
- \put(0.36794645,0.08222732){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}"absorption coefficient"\end{tabular}}}}%
+ \put(0.4205693,0.0690718){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}"absorption coefficient"\end{tabular}}}}%
\put(0.08069971,0.17857448){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$Area = 1$\end{tabular}}}}%
\put(0,0){\includegraphics[width=\unitlength,page=6]{differential_volume.pdf}}%
+ \put(0.42456928,0.03827682){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$[\sigma_a] = \frac{1}{m}$\end{tabular}}}}%
\end{picture}%
\endgroup%
diff --git a/img/emission_illustration.tex b/img/emission_illustration.tex
new file mode 100644
index 0000000..2710340
--- /dev/null
+++ b/img/emission_illustration.tex
@@ -0,0 +1,62 @@
+\begin{tikzpicture}
+% 1 = 20p
+ \def\XSTART{-40pt}
+ \def\YSTART{60pt}
+ \def\YDIAONE{35pt}
+ \def\XONESTART{10pt}
+ \def\XONEDELTA{30pt} %width of cylinders
+ \def\DD{-10pt} %
+ \def\GAP{50pt} %
+
+ \def\LABELYSHIFT{-50pt}
+
+ %FIRST cylinder
+ \def\YEND{\YSTART+\YCURVE+\YDIATWO}
+ \def\XONEEND{\XONESTART+\XONEDELTA}
+ \def\YONEMIDDLE{\YSTART+\YDIAONE/2}
+
+ %SECOND cylinder
+ \def\XTWOSTART{\XONESTART+\XONEDELTA+\GAP}
+ \def\XTWOEND{\XONEEND+\GAP+\XONEDELTA}
+
+ %THIRD cylinder
+ \def\XTHREESTART{\XONESTART+\XONEDELTA+\GAP+\XONEDELTA+\GAP}
+ \def\XTHREEEND{\XONEEND+\GAP+\XONEDELTA+\GAP+\XONEDELTA}
+
+ % FOURTH cylinder
+ \def\XFOURSTART{\XONESTART+\XONEDELTA+\GAP+\XONEDELTA+\GAP+\XONEDELTA+\GAP}
+ \def\XFOUREND{\XONEEND+\GAP+\XONEDELTA+\GAP+\XONEDELTA+\GAP+\XONEDELTA}
+
+ \tikzset{
+ partial ellipse/.style args={#1:#2:#3}{
+ insert path={+ (#1:#3) arc (#1:#2:#3)}
+ },
+ dimen/.style={<->,>=latex,thin,
+ every rectangle node/.style={fill=white,midway,font=\sffamily}},
+ }
+
+ % OUTGOING ARROW (EMISSION)
+ \draw[-{Triangle[width=25pt,length=15pt, color=orange]},
+ line width=15pt, color=orange]
+ (\XFOUREND+0.5,\YONEMIDDLE) -- (\XFOUREND+\GAP,\YONEMIDDLE);
+
+ %FOURTH CYLINDER
+ \draw [fill=gray] (\XFOURSTART,\YSTART) coordinate (BA)
+ rectangle (\XFOUREND,{\YSTART+\YDIAONE}) coordinate (BB);
+ \draw [fill=lightgray](\XFOURSTART,\YONEMIDDLE)
+ ellipse ({\YDIAONE/6} and {\YDIAONE/2});
+ \draw [fill=gray,dashed](\XFOUREND,\YONEMIDDLE)
+ ellipse ({\YDIAONE/6} and {\YDIAONE/2});
+ \draw (\XFOUREND,\YONEMIDDLE)
+ [partial ellipse=-90:90:{\YDIAONE/6} and {\YDIAONE/2}];
+
+ % \node[yshift=\LABELYSHIFT] at ($0.5*(BA)+0.5*(BB)$) {\textit{Emission}};
+
+ % FIFTH ARROW (IN-SCATTERING)
+ \draw[-{Triangle[width=20pt,length=10pt, color=orange]},
+ line width=8pt, color=orange]
+ (\XTHREEEND+0.5,\YONEMIDDLE) -- (\XTHREEEND+\GAP-0.2,\YONEMIDDLE);
+
+
+
+\end{tikzpicture}%
diff --git a/img/in_scattering_illustration.tex b/img/in_scattering_illustration.tex
new file mode 100644
index 0000000..0c75a2a
--- /dev/null
+++ b/img/in_scattering_illustration.tex
@@ -0,0 +1,89 @@
+\begin{tikzpicture}
+% 1 = 20p
+ \def\XSTART{-40pt}
+ \def\YSTART{60pt}
+ \def\YDIAONE{35pt}
+ \def\XONESTART{10pt}
+ \def\XONEDELTA{30pt} %width of cylinders
+ \def\DD{-10pt} %
+ \def\GAP{50pt} %
+
+ \def\LABELYSHIFT{-50pt}
+
+ %FIRST cylinder
+ \def\YEND{\YSTART+\YCURVE+\YDIATWO}
+ \def\XONEEND{\XONESTART+\XONEDELTA}
+ \def\YONEMIDDLE{\YSTART+\YDIAONE/2}
+
+ %SECOND cylinder
+ \def\XTWOSTART{\XONESTART+\XONEDELTA+\GAP}
+ \def\XTWOEND{\XONEEND+\GAP+\XONEDELTA}
+
+ %THIRD cylinder
+ \def\XTHREESTART{\XONESTART+\XONEDELTA+\GAP+\XONEDELTA+\GAP}
+ \def\XTHREEEND{\XONEEND+\GAP+\XONEDELTA+\GAP+\XONEDELTA}
+
+ % FOURTH cylinder
+ \def\XFOURSTART{\XONESTART+\XONEDELTA+\GAP+\XONEDELTA+\GAP+\XONEDELTA+\GAP}
+ \def\XFOUREND{\XONEEND+\GAP+\XONEDELTA+\GAP+\XONEDELTA+\GAP+\XONEDELTA}
+
+ \tikzset{
+ partial ellipse/.style args={#1:#2:#3}{
+ insert path={+ (#1:#3) arc (#1:#2:#3)}
+ },
+ dimen/.style={<->,>=latex,thin,
+ every rectangle node/.style={fill=white,midway,font=\sffamily}},
+ }
+
+ % FIFTH ARROW (IN-SCATTERING)
+ \draw[-{Triangle[width=20pt,length=10pt, color=orange]},
+ line width=8pt, color=orange]
+ (\XTHREEEND+0.5,\YONEMIDDLE) -- (\XTHREEEND+\GAP-0.2,\YONEMIDDLE);
+
+
+ % INSCATTERING ARROWS AROUND THIRD CYLINDER
+ % above
+ \draw[-{Triangle[width=8pt,length=8pt, color=orange!60]},
+ line width=3pt, color=orange!60]
+ ({\XTHREESTART+\XONEDELTA/2},{\YONEMIDDLE+40pt}) --
+ ({\XTHREESTART+\XONEDELTA/2},{\YONEMIDDLE+\YDIAONE/1.5});
+ \draw[-{Triangle[width=8pt,length=8pt, color=orange!60]},
+ line width=3pt, color=orange!60]
+ ({\XTHREESTART+\XONEDELTA/2-20pt},{\YONEMIDDLE+35pt}) --
+ ({\XTHREESTART+\XONEDELTA/2-10pt},{\YONEMIDDLE+\YDIAONE/1.5});
+ \draw[-{Triangle[width=8pt,length=8pt, color=orange!60]},
+ line width=3pt, color=orange!60]
+ ({\XTHREESTART+\XONEDELTA/2+20pt},{\YONEMIDDLE+35pt}) --
+ ({\XTHREESTART+\XONEDELTA/2+10pt},{\YONEMIDDLE+\YDIAONE/1.5});
+ % below
+ \draw[-{Triangle[width=8pt,length=8pt, color=orange!60]},
+ line width=3pt, color=orange!60]
+ ({\XTHREESTART+\XONEDELTA/2},{\YONEMIDDLE-40pt}) --
+ ({\XTHREESTART+\XONEDELTA/2},{\YONEMIDDLE-\YDIAONE/1.5});
+ \draw[-{Triangle[width=8pt,length=8pt, color=orange!60]},
+ line width=3pt, color=orange!60]
+ ({\XTHREESTART+\XONEDELTA/2-20pt},{\YONEMIDDLE-35pt}) --
+ ({\XTHREESTART+\XONEDELTA/2-10pt},{\YONEMIDDLE-\YDIAONE/1.5});
+ \draw[-{Triangle[width=8pt,length=8pt, color=orange!60]},
+ line width=3pt, color=orange!60]
+ ({\XTHREESTART+\XONEDELTA/2+20pt},{\YONEMIDDLE-35pt}) --
+ ({\XTHREESTART+\XONEDELTA/2+10pt},{\YONEMIDDLE-\YDIAONE/1.5});
+
+ \draw [fill=gray] (\XTHREESTART,\YSTART) coordinate (BA)
+ rectangle (\XTHREEEND,{\YSTART+\YDIAONE}) coordinate (BB);
+ \draw [fill=lightgray](\XTHREESTART,\YONEMIDDLE)
+ ellipse ({\YDIAONE/6} and {\YDIAONE/2});
+ \draw [fill=gray,dashed](\XTHREEEND,\YONEMIDDLE)
+ ellipse ({\YDIAONE/6} and {\YDIAONE/2});
+ \draw (\XTHREEEND,\YONEMIDDLE)
+ [partial ellipse=-90:90:{\YDIAONE/6} and {\YDIAONE/2}];
+
+ % \node[yshift=\LABELYSHIFT] at ($0.5*(BA)+0.5*(BB)$) {\textit{In-scattering}};
+
+ % THIRD ARROW (OUT-SCATTERING)
+ \draw[-{Triangle[width=8pt,length=10pt, color=orange]},
+ line width=3pt, color=orange]
+ (\XTWOEND+0.5,\YONEMIDDLE) -- ({\XTWOEND+\GAP-0.2},\YONEMIDDLE);
+
+
+\end{tikzpicture}%
diff --git a/img/interactions.tex b/img/interactions.tex
index fb8b634..e154cff 100644
--- a/img/interactions.tex
+++ b/img/interactions.tex
@@ -140,7 +140,10 @@
\draw (\XTWOEND,\YONEMIDDLE)
[partial ellipse=-90:90:{\YDIAONE/6} and {\YDIAONE/2}];
- \node[yshift=\LABELYSHIFT] at ($0.5*(BA)+0.5*(BB)$) {\textit{Out-scattering}};
+ \node[yshift=\LABELYSHIFT] at ($0.5*(BA)+0.5*(BB)$) {
+ \textit{Out-scattering}
+ };
+
% SECOND ARROW
\draw[-{Triangle[width=20pt,length=10pt, color=orange]},
@@ -157,7 +160,9 @@
\draw (\XONEEND,\YONEMIDDLE)
[partial ellipse=-90:90:{\YDIAONE/6} and {\YDIAONE/2}];
- \node[yshift=\LABELYSHIFT] at ($0.5*(BA)+0.5*(BB)$) {\textit{Absorption}};
+ \node[yshift=\LABELYSHIFT] at ($0.5*(BA)+0.5*(BB)$) {
+ \textit{Absorption}
+ };
%nabla omega below
\draw ($(BB)-(0,\YDIAONE)$) -- ++(0,\DD) coordinate (D1) -- +(0,5pt);
diff --git a/img/summing_up_the_losses.tex b/img/summing_up_the_losses.tex
new file mode 100644
index 0000000..c52c465
--- /dev/null
+++ b/img/summing_up_the_losses.tex
@@ -0,0 +1,120 @@
+\begin{tikzpicture}
+% 1 = 20p
+ \def\XSTART{-40pt}
+ \def\YSTART{60pt}
+ \def\YDIAONE{35pt}
+ \def\XONESTART{10pt}
+ \def\XONEDELTA{30pt} %width of cylinders
+ \def\DD{-10pt} %
+ \def\GAP{50pt} %
+
+ \def\LABELYSHIFT{-50pt}
+
+ %FIRST cylinder
+ \def\YEND{\YSTART+\YCURVE+\YDIATWO}
+ \def\XONEEND{\XONESTART+\XONEDELTA}
+ \def\YONEMIDDLE{\YSTART+\YDIAONE/2}
+
+ %SECOND cylinder
+ \def\XTWOSTART{\XONESTART+\XONEDELTA+\GAP}
+ \def\XTWOEND{\XONEEND+\GAP+\XONEDELTA}
+
+ %THIRD cylinder
+ \def\XTHREESTART{\XONESTART+\XONEDELTA+\GAP+\XONEDELTA+\GAP}
+ \def\XTHREEEND{\XONEEND+\GAP+\XONEDELTA+\GAP+\XONEDELTA}
+
+ % FOURTH cylinder
+ \def\XFOURSTART{\XONESTART+\XONEDELTA+\GAP+\XONEDELTA+\GAP+\XONEDELTA+\GAP}
+ \def\XFOUREND{\XONEEND+\GAP+\XONEDELTA+\GAP+\XONEDELTA+\GAP+\XONEDELTA}
+
+ \tikzset{
+ partial ellipse/.style args={#1:#2:#3}{
+ insert path={+ (#1:#3) arc (#1:#2:#3)}
+ },
+ dimen/.style={<->,>=latex,thin,
+ every rectangle node/.style={fill=white,midway,font=\sffamily}},
+ }
+
+ % THIRD ARROW (OUT-SCATTERING)
+ \draw[-{Triangle[width=8pt,length=10pt, color=orange]},
+ line width=3pt, color=orange]
+ (\XTWOEND+0.5,\YONEMIDDLE) -- ({\XTWOEND+\GAP-0.2},\YONEMIDDLE);
+
+ % OUTSCATTERING ARROWS AROUND SECOND CYLINDER
+ % above
+ \draw[-{Triangle[width=8pt,length=8pt, color=orange!60]},
+ line width=3pt, color=orange!60]
+ ({\XTWOSTART+\XONEDELTA/2},{\YONEMIDDLE+\YDIAONE/1.5}) --
+ ({\XTWOSTART+\XONEDELTA/2},{\YONEMIDDLE+40pt});
+ \draw[-{Triangle[width=8pt,length=8pt, color=orange!60]},
+ line width=3pt, color=orange!60]
+ ({\XTWOSTART+\XONEDELTA/2-10pt},{\YONEMIDDLE+\YDIAONE/1.5}) --
+ ({\XTWOSTART+\XONEDELTA/2-20pt},{\YONEMIDDLE+35pt});
+ \draw[-{Triangle[width=8pt,length=8pt, color=orange!60]},
+ line width=3pt, color=orange!60]
+ ({\XTWOSTART+\XONEDELTA/2+10pt},{\YONEMIDDLE+\YDIAONE/1.5}) --
+ ({\XTWOSTART+\XONEDELTA/2+20pt},{\YONEMIDDLE+35pt});
+ % below
+ \draw[-{Triangle[width=8pt,length=8pt, color=orange!60]},
+ line width=3pt, color=orange!60]
+ ({\XTWOSTART+\XONEDELTA/2},{\YONEMIDDLE-\YDIAONE/1.5}) --
+ ({\XTWOSTART+\XONEDELTA/2},{\YONEMIDDLE-40pt});
+ \draw[-{Triangle[width=8pt,length=8pt, color=orange!60]},
+ line width=3pt, color=orange!60]
+ ({\XTWOSTART+\XONEDELTA/2-10pt},{\YONEMIDDLE-\YDIAONE/1.5}) --
+ ({\XTWOSTART+\XONEDELTA/2-20pt},{\YONEMIDDLE-35pt});
+ \draw[-{Triangle[width=8pt,length=8pt, color=orange!60]},
+ line width=3pt, color=orange!60]
+ ({\XTWOSTART+\XONEDELTA/2+10pt},{\YONEMIDDLE-\YDIAONE/1.5}) --
+ ({\XTWOSTART+\XONEDELTA/2+20pt},{\YONEMIDDLE-35pt});
+
+ % SECOND CYLINDER
+ \draw [fill=gray] (\XTWOSTART,\YSTART) coordinate (BA)
+ rectangle (\XTWOEND,{\YSTART+\YDIAONE}) coordinate (BB);
+ \draw [fill=lightgray](\XTWOSTART,\YONEMIDDLE)
+ ellipse ({\YDIAONE/6} and {\YDIAONE/2});
+ \draw [fill=gray,dashed](\XTWOEND,\YONEMIDDLE)
+ ellipse ({\YDIAONE/6} and {\YDIAONE/2});
+ \draw (\XTWOEND,\YONEMIDDLE)
+ [partial ellipse=-90:90:{\YDIAONE/6} and {\YDIAONE/2}];
+
+ \node[yshift=\LABELYSHIFT] at ($0.5*(BA)+0.5*(BB)$) {
+ \textit{Out-scattering}
+ };
+
+
+ % SECOND ARROW
+ \draw[-{Triangle[width=20pt,length=10pt, color=orange]},
+ line width=8pt, color=orange]
+ ({\XONEEND+0.5},\YONEMIDDLE) -- ({\XONEEND+\GAP-0.2},\YONEMIDDLE);
+
+ % FIRST CYLINDER
+ \draw [fill=gray] (\XONESTART,\YSTART) coordinate (BA)
+ rectangle (\XONEEND,{\YSTART+\YDIAONE}) coordinate (BB);
+ \draw [fill=lightgray](\XONESTART,\YONEMIDDLE)
+ ellipse ({\YDIAONE/6} and {\YDIAONE/2});
+ \draw [fill=gray,dashed](\XONEEND,\YONEMIDDLE)
+ ellipse ({\YDIAONE/6} and {\YDIAONE/2});
+ \draw (\XONEEND,\YONEMIDDLE)
+ [partial ellipse=-90:90:{\YDIAONE/6} and {\YDIAONE/2}];
+
+ \node[yshift=\LABELYSHIFT] at ($0.5*(BA)+0.5*(BB)$) {
+ \textit{Absorption}
+ };
+
+ %nabla omega below
+ \draw ($(BB)-(0,\YDIAONE)$) -- ++(0,\DD) coordinate (D1) -- +(0,5pt);
+ \draw (BA) -- ++(0,\DD) coordinate (D2) -- +(0,5pt);
+ \draw [dimen] (D1) -- (D2) node {$\nabla\boldsymbol\omega$};
+
+ % INCOMING ARROW
+ \draw[-{Triangle[width=25pt,length=15pt, color=orange]},
+ line width=15pt, color=orange]
+ node [above=85pt ] {$\textcolor{black}{L(\textbf{x},\boldsymbol\omega)}$}
+ (\XSTART,\YONEMIDDLE) -- (\XONESTART-0.2,\YONEMIDDLE);
+
+
+ %ground
+ %\draw [fill=gray] (0,0) rectangle (\XEND,\GROUND);
+
+\end{tikzpicture}%
diff --git a/main.tex b/main.tex
index 714c8c2..1320209 100644
--- a/main.tex
+++ b/main.tex
@@ -130,8 +130,9 @@
\end{figure}
\end{frame}
+
\begin{frame}{Possible interactions}{between the volume and the light
- traveling through the medium (1/2)}
+ traveling through the medium}
\begin{figure}[ht]
\centering
\input{img/interactions.tex}
@@ -139,23 +140,164 @@
\end{figure}
\end{frame}
-%\section{Second Section}
-\begin{frame}{Possible interactions}{between the volume and the light
- traveling through the medium (2/2)}
+\begin{frame}{Summing up the losses}
\begin{figure}[ht]
\centering
- \input{img/interactions.tex}
+ \input{img/summing_up_the_losses.tex}
\label{fig:interactions}
\end{figure}
+
+\begin{columns}[t, onlytextwidth]
+ \column{.49\textwidth}
+ $\sigma_a$ : Absorption coefficient \\
+ $\sigma_s$ : Scattering coefficient \\
+ $\sigma_a + \sigma_s = \sigma_t $ : Extinction coefficient
+ \column{.49\textwidth}
+ We lose
+ $\sigma_t(\bx)L(\bx, \bomega)$
+ radiance \\ due to \textit{absorption} and \textit{out-scattering}.
+
+\end{columns}
+\vspace{3mm}
+
+
+\end{frame}
+
+\begin{frame}{In-scattered radiance}
+\begin{figure}[ht]
+ \centering
+ % \scalebox{.7}{
+ \input{img/in_scattering_illustration.tex}
+ % }
+\end{figure}
+$$ L_s(\bx, \bomega) = \int_{S^2} f_p(\bomega, \bomega') L_i(\bx, \bomega')
+ d\bomega' $$
+\end{frame}
+
+
+\begin{frame}{Emission}
+\begin{figure}[ht]
+ \centering
+ % \scalebox{.7}{
+ \input{img/emission_illustration.tex}
+ % }
+\end{figure}
+$$L_e(\bx, \bomega)$$
+
+$$\sigma_a(\boldsymbol{x})L_e(\bx, \bomega)$$
+\end{frame}
+
+%\section{Second Section}
+\begin{frame}{Assembling all the parts}
+ \begin{figure}[ht]
+ \centering
+ \input{img/interactions.tex}
+ \label{fig:interactions}
+ \end{figure}
\begin{itemize}
\item Loses $\sigma_a L(x, \omega)$ due to absorption
\item Loses $\sigma_s L(x, \omega)$ due to out-scattering
\item Gains $\sigma_s L_i(x, \omega)$ due to in-scattering
- \item Gains $\sigma_e L(x, \omega)$ due to emission
+ \item Gains $\sigma_a L(x, \omega)$ due to emission
\end{itemize}
\end{frame}
-\begin{frame}{Properties of the medium}{Collision coefficients}
+
+\begin{frame}{RTE -- Radiative Transfer Equation}
+ The change in radiance $L$ traveling along direction $\boldsymbol{\omega}$
+ through a differential volume element at point $\boldsymbol{x}$.
+ \begin{figure}[ht]
+ \centering
+ \scalebox{.7}{
+ \input{img/RTE_illustration.tex}
+ }
+ \end{figure}
+ \begin{equation}
+ \label{eq:RTE}
+ (\bomega \nabla)L(\bx,\bomega) =
+ - \sigma_t(\bx)L(\bx,\bomega)
+ + \sigma_s(\bx)L_s(\bx,\bomega) + \sigma_a(\bx)L_e(\bx,\bomega)
+ \end{equation}
+\end{frame}
+
+\begin{frame}{RTE -- Radiative Transfer Equation}
+ The change in radiance $L$ traveling along direction $\boldsymbol{\omega}$
+ through a differential volume element at point $\boldsymbol{x}$.
+ \begin{figure}[ht]
+ \centering
+ \scalebox{.7}{
+ \input{img/RTE_illustration.tex}
+ }
+ \end{figure}
+ \begin{equation}
+ \label{eq:RTE}
+ (\bomega \nabla)L(\bx,\bomega) =
+ - \sigma_t(\bx)L(\bx,\bomega)
+ + \sigma_s(\bx)L_s(\bx,\bomega) + \sigma_a(\bx)L_e(\bx,\bomega)
+ \end{equation}
+ \centering
+ \vfill
+ \textbf{Let's integrate it!}
+\end{frame}
+
+\begin{frame}{Transmittance}{Integrating the loss of radiance}
+\begin{figure}[ht]
+ \centering
+ \scalebox{.7}{
+ \input{img/RTE_illustration.tex}
+ }
+ \begin{align}
+ \begin{aligned}
+ L(\bx + \nabla\bomega,\bomega) &= L(\bx,\bomega)
+ - \sigma_t(\bx)L(\bx,\bomega)\nabla\bomega \\
+ \frac{\nabla L(\bx,\bomega)}{\nabla\bomega} &=
+ - \sigma_t(\bx)L(\bx,\bomega) \\
+ \int_{L(x,\omega)}^{L(x+y\omega)} \frac{dL}{L} &= -\int_0^y
+ \sigma(x+\bomega')d\bomega'\\
+ ln(L(x+y\omega,\omega)) - ln(L(x,\omega)) = - \int_0^y \sigma(x+\bomega')
+ \end{aligned}
+ \end{align}
+
+\end{figure}
+\end{frame}
+
+\begin{frame}{RTE -- Radiative Transfer Equation}
+ {The integral version}
+ \vfill
+ \begin{figure}[ht]
+ \centering
+ \scalebox{.7}{
+ \input{img/RTE_illustration.tex}
+ }
+ \end{figure}
+ \vfill
+ \begin{equation}
+ L(\bx, \bomega) = \int_0^\infty
+ %T(\bx, \by)
+ e^{-\int_0^y{\sigma_t(\bx-s\bomega)}ds}
+ \Big[
+ \sigma_s(\by)L_s(\by, \bomega) + \sigma_a(\by)L_e(\by, \bomega)
+ \Big]
+ d\by
+ \end{equation}
+ \vfill
+\end{frame}
+
+\begin{frame}{VRE -- The Volume Rendering Equation}
+ \begin{equation}
+ L(\bx, \bomega) = \int_{0}^{z}
+ T(\bx, \by)
+ \big[
+ \sigma_a(\by)L_e(\by, \bomega) +
+ \sigma_s(\by)L_s(\by, \bomega)
+ \big] dy
+ +
+ T(\bx, \textbf{z})L(\textbf{z},\bomega)
+ \end{equation}
+
+\end{frame}
+
+\begin{frame}{The volume rendering equation}
\begin{figure}[ht]
\centering
\incfig{vre}
--
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