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Update on Overleaf.

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uxwmp 2023-06-20 22:37:05 +00:00 • committed by node
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75
handout/diagrams/geometry_term_diff.tex
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@ -1,52 +1,31 @@
\begin{center}
\begin{minipage}{0.4\linewidth}
\begin{figure}
\centering
\begin{minipage}{0.5\linewidth}
\begin{tikzpicture}[domain=-0.5:2]
\draw[very thin,color=gray] (-0.1,-0.6) grid (2.1,2.1);
\subfloat[Visibility of a point $x$ with respect to $\omega$]{
\begin{tikzpicture}[domain=-0.5:2]
\draw[very thin,color=gray] (-0.1,-0.6) grid (2.1,2.1);
\draw[->] (-0.2,-0.5) -- (2.2,-0.5) node[right] {$\omega$};
\draw[->] (0,-0.7) -- (0,2.2) node[above] {$V(x,\omega)$};
\draw[->] (-0.2,-0.5) -- (2.2,-0.5) node[right] {$\omega$};
\draw[->] (0,-0.7) -- (0,2.2) node[above] {$V(x,\omega)$};
\draw (0,0) -- (1,0) plot coordinates {(0,0) (1,0)}[color=red];
\draw (1,1) -- (2,1) plot coordinates {(1,1) (2,1)}[color=red];
\draw (0,0) node[left] {$0$};
\draw (0,1) node[left] {$1$};
\draw (1,-0.7) node[below] {$\omega_0$};
\end{tikzpicture}
\end{minipage}
\hspace{3mm}
\begin{minipage}{0.4\linewidth}
\caption{Visibility of a point $x$ with respect to $\omega$.}
\label{fig:visibility}
\end{minipage}
\end{figure}
\end{minipage}
% second diagram
\begin{minipage}{0.5\linewidth}
\begin{figure}
\centering
\begin{minipage}{0.5\linewidth}
\begin{tikzpicture}[domain=-0.5:2]
\draw[very thin,color=gray] (-0.1,-0.6) grid (2.1,2.1);
\draw (0,0) -- (1,0) plot coordinates {(0,0) (1,0)}[color=red];
\draw (1,1) -- (2,1) plot coordinates {(1,1) (2,1)}[color=red];
\draw (0,0) node[left] {$0$};
\draw (0,1) node[left] {$1$};
\draw (1,-0.7) node[below] {$\omega_0$};
\end{tikzpicture}
}
\hfil
% second diagram
\subfloat[Differentiation of the left graph with respect to $\omega$]{
\begin{tikzpicture}[domain=-0.5:2]
\draw[very thin,color=gray] (-0.1,-0.6) grid (2.1,2.1);
\draw[->] (-0.2,-0.5) -- (2.2,-0.5) node[right] {$\omega$};
\draw[->] (0,-0.7) -- (0,2.2) node[above] {$\frac{\partial}{\partial\omega}V(x,\omega)$};
\draw[->] (-0.2,-0.5) -- (2.2,-0.5) node[right] {$\omega$};
\draw[->] (0,-0.7) -- (0,2.2) node[above] {$\frac{\partial}{\partial\omega}V(x,\omega)$};
\draw (0,0) -- (2,0) plot coordinates {(0,0) (2,0)}[color=red];
\draw (0,0) node[left] {$0$};
\draw (0,1) node[left] {$1$};
\draw (0,2) node[left] {$\infty$};
\draw (1,2) node[color=red] {$\bullet$};
\draw (1,-0.7) node[below] {$\omega_0$};
\end{tikzpicture}
\end{minipage}
\begin{minipage}{0.4\linewidth}
\caption{Differentiation of the left graph with respect to $\omega$.}
\caption{Differentiation of the left graph with respect to $\omega$.}
\label{fig:dirac-delta-spike}
\end{minipage}
\end{figure}
\end{minipage}
\end{center}
\draw (0,0) -- (2,0) plot coordinates {(0,0) (2,0)}[color=red];
\draw (0,0) node[left] {$0$};
\draw (0,1) node[left] {$1$};
\draw (0,2) node[left] {$\infty$};
\draw (1,2) node[color=red] {$\bullet$};
\draw (1,-0.7) node[below] {$\omega_0$};
\end{tikzpicture}
}

22
handout/handout.tex
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@ -38,6 +38,8 @@
\usepackage{tikz}
\usetikzlibrary{calc,patterns,angles,quotes,shapes,arrows, positioning,overlay-beamer-styles}
\usepackage{subfig}
\usepackage{xcolor}
\usepackage{graphicx}
@ -68,28 +70,34 @@ To formalize the problem of photo realistic image synthesis, an equation has bee
\notag
\label{eq:rendering_integral}
\end{equation}
\caption{The rendering equation capturing physical light transport. It assigns a value to the the Intensity of light transported from a point $x$ to a point $x^\prime$. The geometry term $g$ will be discussed later. The term $\epsilon$ accounts for the emissivity of the point $x$. The integral term represents all light scattered from any other point in the scene towards the point $x$. The integral domain $S$ contains all points in the scene.}
\caption{The rendering equation capturing physical light transport. It assigns a value to the the Intensity of light transported from a point $x$ to a point $x^\prime$. The geometry term $g$ will be discussed later. The term $\epsilon$ accounts for the emissivity of the material at point $x$. The integral term represents all light scattered from any other point in the scene towards the point $x$. The integral domain $S$ contains all points in the scene.}
\end{figure}
This equation (equation \ref{eq:rendering_integral}) is now widely recognized as ``the rendering integral``. It is not analytically solvable and is thus most commonly solved using Monte-Carlo integration - i.e. ray tracing. Ray tracing works by backtracking light rays from the light sources in the scene and thus simulate physically realistic lighting.\\
This equation (equation \ref{eq:rendering_integral}) is now widely recognized as ``the rendering integral``. It can not be solved analytically and is thus most commonly approximated using Monte-Carlo integration - i.e. ray tracing. Ray tracing works by backtracking light rays from the light sources in the scene and thus simulate physically realistic lighting.\\
To do this, rays are cast from the camera, through each pixel in the camera frustum. The intersection point with the scene geometry $x$ is calculated for each ray and material properties (e.g. color, emissivity etc.) are taken into account to calculate the pixel color. From this point more rays are drawn towards each light source. If the light source is visible, its light contributes to the pixel color as well. To account for indirect lighting the ray ``bounces around``, yielding a color for some of the points scattering lights towards $x$. This approximates the integral term in equation \ref{eq:rendering_integral}.\\
Differentiable ray tracing is the task of calculating the gradient of this process with respect to \emph{any} scene parameter.
\section{Problems with differentiability}
The geometry term $g(x,x^\prime)$ in equation \ref{eq:rendering_integral} is the main problem when it comes do differentiating the rendering integral. This term is 1 iff $x$ is visible from $x^\prime$, 0 otherwise.
The geometry term $g(x,x^\prime)$ in equation \ref{eq:rendering_integral} is the main problem when it comes to differentiating the rendering integral. This term is 1 iff $x$ is visible from $x^\prime$ and 0 otherwise.
\begin{figure}[h]
\centering
\include{handout/diagrams/diagramm_occlusion}
\caption{A simple occlusion scenario. An infinitesimal change in ray angle $d\omega$ is sufficient to determine whether the wall is visible or not.}
\label{fig:occlusion_rays}
\end{figure}
As illustrated in figure \ref{fig:occlusion_rays} an infinitesimal angle change $d\omega$ can lead to the blocker obstructing the wall or the wall being visible. The geometry term is thus a Heaviside step function which when differentiated yields a dirac delta functional.
\begin{figure}
\include{handout/diagrams/geometry_term_diff}
\caption{Caption}
\label{fig:enter-label}
\centering
\input{handout/diagrams/geometry_term_diff.tex}
\caption{The visibility of a point and the differentiation}
\label{fig:visibility_term}
\end{figure}
As illustrated in figure \ref{fig:occlusion_rays} an infinitesimal angle change $d\omega$ can lead to the blocker obstructing the wall or the wall being visible. The geometry term is thus a Heaviside step function which when differentiated yields a Dirac delta functional (see figure \ref{fig:visibility_term}). Since the Dirac delta functional only differs from 0 in $\omega_0$ one point, the probability of sampling it when using Monte-Carlo integration is 0.
The key observation is that these discontinuities only occur at the edges of meshes.
\section{Proposed solution: Edge sampling}
% BIBLIOGRAPHY
\nocite{*} % List all entries of the .bib file, even those not cited in the main body
\printbibliography