Update on Overleaf.

presentation/modules/this_method.tex

@@ -19,7 +19,7 @@
     \begin{block}{Idea}
         \begin{itemize}
             \item Traditional sampling for continuous regions
-            \item Edge sampling the discontinuous part
+            \item Edge sampling for the discontinuous part
         \end{itemize}
     \end{block}
 \end{frame}
@@ -30,22 +30,73 @@
 \end{frame}
 
 \begin{frame}{Edge sampling - half spaces}
-    
+    \begin{minipage}{0.5\linewidth}
+        \input{presentation/diagrams/halfspaces}
+    \end{minipage}
+    \begin{minipage}{0.45\linewidth}
+        \begin{block}{Principle}
+            \begin{itemize}
+                \item Edge divides scene into two halfspaces
+                \item Calculate color in either halfspace
+                \item Calculate gradient based on color difference
+                \item Occlusion $\implies$ No color difference $\implies\nabla=0$
+            \end{itemize}
+        \end{block}
+    \end{minipage}
 \end{frame}
 
-\subsection{conclusion - what can this method do?}
-% talk about limitations here!
+\begin{frame}[t]{Edge sampling - math background}
+        Color:
+        \only<1>{
+        \[
+            I = \underbrace{\iint}_{\text{Screen space domain}}
+            \underbrace{f(x,y;\Phi)}_{\text{Scene function; Parameter set } \Phi}dxdy
+        \]
+        }
+
+        \only<2-> {
+        \[
+            I = \iint f(x,y;\Phi)dxdy
+        \]
+        }
+
+        \only<2> {
+        Use half spaces to rewrite integral:
+        \[
+            \sum_i\iint
+            \underbrace{\theta(}_{\text{step function}}
+            \underbrace{\alpha_i(x,y)}_{\text{Edge equation formed by the triangle edge}})
+            \underbrace{f_i(x,y)}_{\text{half space (may contain heaviside step functions itself)}}
+        \]
+        }
+    
+        \only<3->{
+        Use half spaces to rewrite integral:
+        \[
+            \sum_i\iint\theta(\alpha_i(x,y))f_i(x,y)
+        \]
+
+        Calculate gradient:
+        \[
+            \nabla\iint\theta(\alpha_i(x,y))f_i(x,y)dxdy = 
+            \underbrace{\iint\delta(\alpha_i(x,y))\nabla\alpha_i(x,y)f_i(x,y)dxdy}_{\text{Edges (can be differentiated)}} + 
+            \underbrace{\iint\nabla f_i(x,y)\theta(\alpha_i(x,y))dxdy}_{\text{Original pixel integral}}
+        \]
+        }
+        
+\end{frame}
 
 \begin{frame}{Inverse rendering - Results in this paper}
     \begin{block}{Inverse rendering here}
         \begin{itemize}
-            \item Fit camera pose, material parameters and light source intensity
+            \item Parameters: Camera pose, material parameters, light source intensity
             \item Scene: Strong indirect illumination and non lambertian materials
-            \item Initial guess: Assign almost all objects a white color, arbitrary camera pose
-            \item 177 parameters in total
-            \item Absolute difference as loss function and ADAM optimizer
-            \item Start at a resulution of $64\times 64$ and linearly increase to $512\times 512$ in 8 steps\\
-            $\implies$ Avoid getting stuck in local minima of the loss function
+            \item Initial guess: Almost all objects white, arbitrary camera pose
+            \item 177 parameters
+            \item Loss function: Absolute difference
+            \item ADAM optimizer
+            \item Start resolution of $64\times 64$, linearly increase to $512\times 512$ in 8 steps\\
+            $\implies$ Avoid local minima of the loss function
         \end{itemize}
     \end{block}
 \end{frame}
@@ -251,3 +302,27 @@
   \\ 
   All media in this section taken from \cite{ACM:diffable_raytracing}
 \end{frame}
+
+\subsection{Conclusion - What can this method do?}
+\begin{frame}{Conclusion}
+    \setbeamercovered{transparent}
+    \begin{block}{Possibilities}
+        \begin{itemize}
+            \item Differentiate with respect to \emph{any} scene parameter
+            \item Arbitrary non-dirac-materials
+            \item Global illumination, occlusion, shadows etc...
+            \item Automatic differentiation (PyTorch) $\rightarrow$ Inverse rendering
+        \end{itemize}
+    \end{block}
+
+    \pause
+    
+    \begin{block}{Some Limitations}
+        \begin{itemize}
+            \item Performance
+            \item Time domain (animations)
+            \item Interpenetrating meshes
+            \item Non-differentiable shaders
+        \end{itemize}
+    \end{block}
+\end{frame}