diff --git a/img/ray-marching.pdf_tex b/img/ray-marching.pdf_tex
index a7f5f1c1a801a811d8355bd2a9b420cc7230f43e..3cc9bf519a723fd4f30986cbfd81358a26db54ef 100644
--- a/img/ray-marching.pdf_tex
+++ b/img/ray-marching.pdf_tex
@@ -60,6 +60,7 @@
\put(0.23576592,0.24879687){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$\Delta s$\end{tabular}}}}%
\put(0,0){\includegraphics[width=\unitlength,page=3]{ray-marching.pdf}}%
\put(0.07170968,0.14867355){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$T_0 = 1.0$\end{tabular}}}}%
- \put(0.28602403,0.01358111){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$T_{i+1} = T_i - \Delta s \cdot \sigma_t(\bm{x}_i) $\end{tabular}}}}%
+ \put(0.28602403,0.01358111){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$T_{i}
+ = T_{i-1}\cdot e^{- \sigma_t(\bm{x}_i)\Delta s} $\end{tabular}}}}%
\end{picture}%
\endgroup%
diff --git a/main.tex b/main.tex
index 7afde70f0d93216584303de91a71fc77b1379148..4e6334e34e6eafd9de87b51ca5cc5dfe98c808c1 100644
--- a/main.tex
+++ b/main.tex
@@ -249,8 +249,14 @@ $$\sigma_a(\boldsymbol{x})L_e(\bx, \bomega)$$
\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)
+ \underbrace{ - \sigma_t(\bx)L(\bx,\bomega)}
+ _{Extinction}
+ +
+ \underbrace{\sigma_s(\bx)L_s(\bx,\bomega)}
+ _{In-scattering}
+ +
+ \underbrace{\sigma_a(\bx)L_e(\bx,\bomega)}
+ _{Emission}
\end{equation}
\end{frame}
@@ -266,8 +272,14 @@ $$\sigma_a(\boldsymbol{x})L_e(\bx, \bomega)$$
\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)
+ \underbrace{ - \sigma_t(\bx)L(\bx,\bomega)}
+ _{Extinction}
+ +
+ \underbrace{\sigma_s(\bx)L_s(\bx,\bomega)}
+ _{In-scattering}
+ +
+ \underbrace{\sigma_a(\bx)L_e(\bx,\bomega)}
+ _{Emission}
\end{equation}
\centering
\vfill
@@ -280,20 +292,32 @@ $$\sigma_a(\boldsymbol{x})L_e(\bx, \bomega)$$
\scalebox{.6}{
\input{img/RTE_illustration.tex}
}
+ % \begin{align}
+ % \begin{aligned}
+ % % \vert\nabla\bomega\vert &= \\
+ % L(\bx + \nabla\bomega,\bomega) &=
+ % L(\bx,\bomega) - \sigma_t(\bx)L(\bx,\bomega)\nabla\bomega \\
+ % L(\bx + dx) &=
+ % L(\bx) - L(\bx)\sigma_t(\bx)dx\bigg|_{dx=\nabla\bomega,
+ % L(\bx)=L(\bx, \bomega)} \\
+ % \frac{L(\bx + dx) - L(\bx)}{dx} = \Aboxed{\frac{dL(\bx)}{dx} &=
+ % -L(\bx)\sigma_t(\bx)} \text{ ("exponential extinction")}\\
+ % \int_{L(x)}^{L(x+S)} \frac{1}{L} dL &= -\int_0^S \sigma_t(\bx)dx\\
+ % ln(L(\bx+S)) - ln(L(\bx)) &= - \int_0^S \sigma_t(\bx) dx
+ % \end{aligned}
+ % \end{align}
\begin{align}
\begin{aligned}
- % \vert\nabla\bomega\vert &= \\
- L(\bx + \nabla\bomega,\bomega) &=
- L(\bx,\bomega) - \sigma_t(\bx)L(\bx,\bomega)\nabla\bomega \\
L(\bx + dx) &=
L(\bx) - L(\bx)\sigma_t(\bx)dx\bigg|_{dx=\nabla\bomega,
L(\bx)=L(\bx, \bomega)} \\
- \frac{L(\bx + dx) - L(\bx)}{dx} = \Aboxed{\frac{dL(\bx)}{dx} &=
+ \Aboxed{\frac{dL(\bx)}{dx} &=
-L(\bx)\sigma_t(\bx)} \text{ ("exponential extinction")}\\
\int_{L(x)}^{L(x+S)} \frac{1}{L} dL &= -\int_0^S \sigma_t(\bx)dx\\
ln(L(\bx+S)) - ln(L(\bx)) &= - \int_0^S \sigma_t(\bx) dx
\end{aligned}
\end{align}
+
\end{figure}
\end{frame}
@@ -426,7 +450,7 @@ Perfectly importance sample with $t' = -ln(1-\zeta)/\sigma_t$
\end{equation}
\begin{equation}
- \sigma_a + \sigma_s = 1;
+ \frac{\sigma_a + \sigma_s}{\sigma_t} = 1;
P_a = \frac{\sigma_a}{\sigma_t}; P_s = \frac{\sigma_a}{\sigma_t}
\end{equation}
@@ -462,8 +486,8 @@ Perfectly importance sample with $t' = -ln(1-\zeta)/\sigma_t$
\begin{equation}
L(\bm{x}, \bm{\omega}) = \int_{t=0}^{d} p(t)
- \Big[ \frac{\sigma_a}{\sigma_t} L_e(\bm{x_t}, \omega)
- + \frac{\sigma_s}{\sigma_t} L_s(\bm{x_t}, \bomega)
+ \Big[ P_a L_e(\bm{x_t}, \omega)
+ + P_s L_s(\bm{x_t}, \bomega)
\Big]dt + L_d(\bm{x_d}, \bm{\omega})
\end{equation}
\end{frame}
@@ -482,15 +506,15 @@ Perfectly importance sample with $t' = -ln(1-\zeta)/\sigma_t$
T_{\bar\sigma}(\bm{x},\bm{y})
\Big[
\underbrace{\mystrut{2ex}
- \sigma_s(\by)L_s(\by, \bomega)
+ P_s(\by)L_s(\by, \bomega)
}_{\text{in-scatter}}
+
\underbrace{\mystrut{2ex}
- \sigma_a(\by)L_e(\by, \bomega)
+ P_a(\by)L_e(\by, \bomega)
}_{\text{emission}}
+
\underbrace{\mystrut{2ex}
- \sigma_n(\by)L(\by, \bomega)
+ P_n(\by)L(\by, \bomega)
}_{\text{null-collision}}
\Big]
d\by
@@ -526,7 +550,61 @@ Perfectly importance sample with $t' = -ln(1-\zeta)/\sigma_t$
\item Data access usually dominates the render time \\
$\implies$ data structures are key for achieving good performance
\item Volume data can quickly grow into hundreds of gigabytes for production
+ \begin{itemize}
+ \item For example, peak storage needed for a single shot of the
+ movie Soul was 80 TBs.
+ \end{itemize}
\end{itemize}
+ \begin{figure}
+ \centering
+ \includegraphics[width=0.4\textwidth]{kd-tree-example.png}
+ \label{fig:kd-tree-example}
+ \caption{Heterogeneous volume with a spike in density results in high
+ $\bar\sigma$ everywhere. Using kd-trees lowers the number of evaluations
+ needed. \textit{Source: Production Volume Rendering SIGGRAPH 2017 Course by
+ Fong et. al.}}
+\end{figure}
+\end{frame}
+
+\begin{frame}{Remaning challenges and open problems}
+ \begin{itemize}
+ \item Joint handling of surfaces and volumes
+ \begin{itemize}
+ \item Unifying the different techniques
+ \end{itemize}
+ \item Machine Learning
+ \begin{itemize}
+ \item Vast cost of data access and tracking particles
+ high-albedo volumes (resulting in lots of scattering) --
+ e.g. clouds
+ \end{itemize}
+ \end{itemize}
+\end{frame}
+
+\begin{frame}{Summary}
+\begin{columns}[t, onlytextwidth]
+ \column{.49\textwidth}
+ \begin{itemize}
+ \item Problem statement and model of volume and light propagating
+ through it
+ \item Interaction between light ray and volume
+ \item Formula for getting the radiance $L(x, \bomega)$ to make it
+ applicable to usual ray tracing methods
+ \item Subtasks needed
+ \begin{itemize}
+ \item Distance sampling
+ \item Transmittance estimation
+ \end{itemize}
+ \item Optimization
+ \item Remaining challenges and open problems
+ \end{itemize}
+ \column{.49\textwidth}
+ \begin{figure}[ht]
+ \centering
+ \def\svgwidth{\columnwidth}
+ \import{./img/}{propagation-illustration.pdf_tex}
+ \end{figure}
+\end{columns}
\end{frame}
\maketitle