From 3ca7aa988760943e90a475be9bc2765104d7d4bf Mon Sep 17 00:00:00 2001
From: bobarna <barnabas.borcsok@gmail.com>
Date: Sun, 30 Jan 2022 19:46:20 +0100
Subject: [PATCH] Finish presentation
---
img/propagation-illustration.pdf | Bin 0 -> 2129 bytes
img/propagation-illustration.pdf_tex | 58 +++++++
img/tracking-only-boundary.pdf | Bin 0 -> 5217 bytes
img/tracking-only-boundary.pdf_tex | 68 +++++++++
img/tracking.pdf | Bin 9580 -> 9580 bytes
img/tracking.pdf_tex | 2 +-
img/tracking.svg | 216 ++++++++++-----------------
main.tex | 140 ++++++++++++++---
8 files changed, 327 insertions(+), 157 deletions(-)
create mode 100644 img/propagation-illustration.pdf
create mode 100644 img/propagation-illustration.pdf_tex
create mode 100644 img/tracking-only-boundary.pdf
create mode 100644 img/tracking-only-boundary.pdf_tex
diff --git a/img/propagation-illustration.pdf b/img/propagation-illustration.pdf
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diff --git a/img/propagation-illustration.pdf_tex b/img/propagation-illustration.pdf_tex
new file mode 100644
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--- /dev/null
+++ b/img/propagation-illustration.pdf_tex
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+%% Creator: Inkscape inkscape 0.92.5, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'propagation-illustration.pdf' (pdf, eps, ps)
+%%
+%% To include the image in your LaTeX document, write
+%% \input{<filename>.pdf_tex}
+%% instead of
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+%%
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diff --git a/img/tracking-only-boundary.pdf_tex b/img/tracking-only-boundary.pdf_tex
new file mode 100644
index 0000000..e4ad5bc
--- /dev/null
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+%%
+%% To include the image in your LaTeX document, write
+%% \input{<filename>.pdf_tex}
+%% instead of
+%% \includegraphics{<filename>.pdf}
+%% To scale the image, write
+%% \def\svgwidth{<desired width>}
+%% \input{<filename>.pdf_tex}
+%% instead of
+%% \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}
+%% Alternatively, one can specify
+%% \graphicspath{{<path to file>/}}
+%%
+%% For more information, please see info/svg-inkscape on CTAN:
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diff --git a/img/tracking.pdf b/img/tracking.pdf
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diff --git a/img/tracking.pdf_tex b/img/tracking.pdf_tex
index 8133276..87e43c7 100644
--- a/img/tracking.pdf_tex
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diff --git a/img/tracking.svg b/img/tracking.svg
index 7b4b70d..8c82c29 100644
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xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
- width="210mm"
- height="297mm"
- viewBox="0 0 210 297"
+ width="127.66727mm"
+ height="48.680305mm"
+ viewBox="0 0 127.66727 48.680305"
version="1.1"
id="svg6879"
inkscape:version="0.92.5 (2060ec1f9f, 2020-04-08)"
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inkscape:stockid="DotM"
orient="auto"
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+ refX="0"
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style="overflow:visible"
inkscape:isstock="true">
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<marker
inkscape:stockid="DotL"
orient="auto"
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inkscape:stockid="DotS"
orient="auto"
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+ refY="0"
+ refX="0"
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inkscape:stockid="Arrow2Sstart"
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inkscape:pageshadow="2"
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inkscape:document-units="mm"
inkscape:current-layer="layer1"
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inkscape:window-width="1916"
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inkscape:window-x="0"
inkscape:window-y="38"
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inkscape:groupmode="layer"
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- y="56.410553"
+ x="110.14149"
+ y="53.953709"
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+ x="110.14149"
+ y="53.953709"
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diff --git a/main.tex b/main.tex
index 580a1cd..7afde70 100644
--- a/main.tex
+++ b/main.tex
@@ -42,9 +42,9 @@
\newcommand\todoi[1]{\textit{\textcolor{blue}{TODO: #1}}}
\newcommand\tcb[1]{\textit{\textcolor{blue}{#1}}}
-\newcommand\bx[0]{\textbf{x}}
-\newcommand\by[0]{\textbf{y}}
-\newcommand\bomega[0]{\boldsymbol{\omega}}
+\newcommand\bx[0]{\bm{x}}
+\newcommand\by[0]{\bm{y}}
+\newcommand\bomega[0]{\bm{\omega}}
\usepackage{mathtools}
@@ -128,12 +128,11 @@
\end{frame}
\begin{frame}{Propagation of light in a medium}
- \begin{figure}[ht]
- \centering
- \def\svgwidth{\columnwidth}
- \includegraphics[width=10cm]{img/propagation-illustration.png}
- % \label{fig:propagation-illustration}
- \end{figure}
+ \begin{figure}[ht]
+ \centering
+ \def\svgwidth{0.7\columnwidth}
+ \import{./img/}{propagation-illustration.pdf_tex}
+ \end{figure}
\end{frame}
\begin{frame}{Change of radiance in a differential volume}
@@ -170,10 +169,14 @@
We lose
$\sigma_t(\bx)L(\bx, \bomega)$
radiance \\ due to \textit{absorption} and \textit{out-scattering}.
-
\end{columns}
-\vspace{3mm}
-
+\vfill
+\begin{columns}[t, onlytextwidth]
+ \column{.49\textwidth}
+ $\sigma_t\implies$Homogeneous
+ \column{.49\textwidth}
+ $\sigma_t(\bx)\implies$Heterogeneous
+\end{columns}
\end{frame}
@@ -400,18 +403,113 @@ $$ T(t) = e^{-\tau(t)} = e^{-\int_0^t \sigma_t(\bx-s\bomega)ds} $$
\end{itemize}
\end{frame}
-\begin{frame}{In homogeneous volumes}
- $$T(t) = e^{-\int_0^t \sigma_t(\bx-s\bomega)ds} = e^{-\int_0^t \sigma_t ds}
- = e^{-\sigma_t t}$$
-\end{frame}
+\begin{frame}{Tracking}{In homogeneous volumes}
+ \begin{itemize}
+ \item Simulate how a photon bounces around inside a volume
+ \item Explicitly modeling absorption and scattering effects
+ \end{itemize}
+\begin{align}
+ T(t) = e^{-\int_0^t \sigma_t(\bx-s\bomega)ds} = e^{-\int_0^t \sigma_t ds}
+ \Aboxed{= e^{-\sigma_t t} = T(t)}
+\end{align}
+
+PDF $p(t) = \sigma_t e^{-\sigma_t t}$ (by normalizing)
+
+Perfectly importance sample with $t' = -ln(1-\zeta)/\sigma_t$
+\hfill $\zeta \in [0,1)$
+\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]dt + L_d(\bm{x_d}, \bm{\omega})
+\end{equation}
-\begin{frame}{Regular Tracking}{In homogeneous volumes}
+\begin{equation}
+ \sigma_a + \sigma_s = 1;
+ P_a = \frac{\sigma_a}{\sigma_t}; P_s = \frac{\sigma_a}{\sigma_t}
+\end{equation}
+
+\end{frame}
+
+\begin{frame}{Closed-Form tracking}{In homogeneous volumes}
\begin{figure}[ht]
\centering
- \def\svgwidth{\columnwidth}
+ \def\svgwidth{0.7\columnwidth}
\import{./img/}{tracking.pdf_tex}
\end{figure}
+
+\begin{equation}
+ L(\bm{x}, \bm{\omega}) = \int_{t=0}^{d} p(t)
+ \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}
+
+\begin{frame}{Regular tracking}{In heterogeneous volumes}
+ What happens if the volume is \textbf{not homogeneous}?
+ \hspace{5.4em}$\implies$$\sigma_t(\bm{x})$
+
+ $\implies$ apply closed-form tracking to homogeneous sub-parts
+ \hspace{2em}$\implies$$\sigma_t$
+
+\begin{figure}[ht]
+ \centering
+ \def\svgwidth{0.3\columnwidth}
+ \import{./img/}{tracking-only-boundary.pdf_tex}
+\end{figure}
+
+\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]dt + L_d(\bm{x_d}, \bm{\omega})
+\end{equation}
+\end{frame}
+
+\begin{frame}{Delta tracking}{Introducing null-collisions}
+\begin{enumerate}
+ \item Problem: the volume is heterogeneous
+ \item Idea: \textbf{Increase the number of interactions} to make it
+ homogeneous,\\
+ but \textbf{reject} some of the interactions $\implies$
+ \textbf{null-collisions}
+\end{enumerate}
+
+\begin{equation}
+ L(\bx, \bomega) = \int_0^\infty
+ T_{\bar\sigma}(\bm{x},\bm{y})
+ \Big[
+ \underbrace{\mystrut{2ex}
+ \sigma_s(\by)L_s(\by, \bomega)
+ }_{\text{in-scatter}}
+ +
+ \underbrace{\mystrut{2ex}
+ \sigma_a(\by)L_e(\by, \bomega)
+ }_{\text{emission}}
+ +
+ \underbrace{\mystrut{2ex}
+ \sigma_n(\by)L(\by, \bomega)
+ }_{\text{null-collision}}
+ \Big]
+ d\by
+\end{equation}
+
+\vfill
+\begin{equation}
+ T_{\bar\sigma}(\bm{x},\bm{y}) =
+ e^{-\int_0^y \sigma_s(\bm{s}) + \sigma_a(\bm{s}) + \sigma_n(\bm{s}) d\bm{s}}
+\end{equation}
+
+\begin{equation}
+ \sigma_n(\bx) = \bar\sigma - \sigma_t(\bx)
+\end{equation}
+
+\begin{equation}
+ \bar\sigma = \sigma_s(\bm{x}) + \sigma_a(\bm{x}) + \sigma_n(\bm{x})
+\end{equation}
+
\end{frame}
\begin{frame}{Transmittance Estimation}{Ray Marching}
@@ -423,6 +521,12 @@ $$ T(t) = e^{-\tau(t)} = e^{-\int_0^t \sigma_t(\bx-s\bomega)ds} $$
\end{frame}
\begin{frame}{Acceleration Data Structures}
+ \begin{itemize}
+ \item Spatially-varying properties
+ \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
+ \end{itemize}
\end{frame}
\maketitle
--
GitLab