improc ass4: Updated separability section in report.

improc/ass4/report/report.tex

@@ -18,29 +18,145 @@
 \section{Analytical Local Structure}
 
 \subsection{Derivatives}
+\label{sub:derivatives}
 
 We have been given the following function:
 $$f(x, y) = A sin(Vx) + B cos(Wy)$$
 
-The partial derivatives $f_x, f_y, f_{xx}, f_{xy}$ and $f_{yy}$ can be derived as follows:
+The partial derivatives $f_x, f_y, f_{xx}, f_{xy}$ and $f_{yy}$ can be
+derived as follows:
+
 \begin{table}[H]
 	\begin{tabular}{rl}
-		$f_x$    & $= A \frac{\delta}{\delta x} sin(Vx) + B \frac{\delta}{\delta x} cos(Wy)$ \\
+		$f_x$    & $= \frac{\delta f}{\delta x}$ \\
+		         & $= A \frac{\delta}{\delta x} sin(Vx) + B \frac{\delta}{\delta x} cos(Wy)$ \\
 	    	     & $= A cos(Vx) \cdot V + B \cdot 0$ \\
 	    	     & $= AV cos(Vx)$ \\
 	    	     & \\
-		$f_y$    & $= A \frac{\delta}{\delta y} sin(Vx) + B \frac{\delta}{\delta y} cos(Wy)$ \\
+		$f_y$    & $= \frac{\delta f}{\delta y}$ \\
+		         & $= A \frac{\delta}{\delta y} sin(Vx) + B \frac{\delta}{\delta y} cos(Wy)$ \\
 	    	     & $= A \cdot 0 - B sin(Wy) \cdot W$ \\
 	    	     & $= -BW sin(Wy)$ \\
 	    	     & \\
-		$f_{xx}$ & $= AV \frac{\delta}{\delta x} cos(Vx)$ \\
+		$f_{xx}$ & $= \frac{\delta f_x}{\delta x}$ \\
+		         & $= AV \frac{\delta}{\delta x} cos(Vx)$ \\
 		         & $= -AV^2 sin(Vx)$ \\
 	    	     & \\
-		$f_{xy}$ & $= AV \frac{\delta}{\delta y} cos(Vx) = 0$ \\
+		$f_{xy}$ & $= \frac{\delta f_x}{\delta y} = AV \frac{\delta}{\delta y} cos(Vx) = 0$ \\
 	    	     & \\
-		$f_{yy}$ & $= -BW \frac{\delta}{\delta y} sin(Wy)$ \\
+		$f_{yy}$ & $= \frac{\delta f_y}{\delta y}$ \\
+		         & $= -BW \frac{\delta}{\delta y} sin(Wy)$ \\
 		         & $= -BW^2 cos(Wy)$ \\
 	\end{tabular}
 \end{table}
 
+\pagebreak
+
+\subsection{Plots}
+
+The following plots show $f(x, y)$ and its first and second derivatives. The
+image on the left shows $f_x$ and $f_y$. The image on the right shows $f_{xx}$
+and $f_{yy}$ in a quiver plot over $f(x, y)$. The arrows point towards the
+largest increase of gray value, which means that the derivations in chapter
+\ref{sub:derivatives} are correct.
+
+\begin{figure}[H]
+	\hspace{-2cm}
+	\includegraphics[scale=.8]{samples.pdf}
+	\caption{Plots of $f(x, y)$ and its first and second derivatives.}
+\end{figure}
+
+\section{Gaussian Convolution}
+
+\subsection{Implementation}
+
+All Gaussian functions are implemented in the file \emph{gauss.py}. The
+\texttt{Gauss} function fills a 2D array with the values of the 2D Gaussian
+function\footnote{\label{footnote:gaussian-filter}
+\url{http://en.wikipedia.org/wiki/Gaussian_filter}}:
+
+$$g_{2D}(x, y) = \frac{1}{2 \pi \sigma^2} e^{-\frac{x^2 + y^2}{2 \sigma^2}}$$
+
+This function converges to zero, but never actually equals zero. The filter's
+size is therefore chosen to be $\lceil 6 * \sigma \rceil$ in each direction by
+convention (since values for $x,y > 3 * \sigma$ are negligible). Finally,
+because the sum of the filter should be equal to 1, it is divided by its own
+sum.
+
+The result of the \texttt{Gauss} function is shown in figure
+\ref{fig:gauss-diff}. The subplots respectively show the original image, the
+Gaussian kernel and the convolved image.
+
+\begin{figure}[H]
+	\label{fig:gauss-diff}
+	\hspace{-5cm}
+	\includegraphics[scale=.6]{gauss_diff_5.pdf}
+	\caption{The result of \texttt{python gauss.py diff 5}.}
+\end{figure}
+
+\subsection{Measuring Performance}
+
+We've timed the runtime of the \texttt{Gauss} function for
+$\sigma = 1,2,3,5,7,9,11,15,19$, the results are in figure
+\ref{fig:times-2d}. The graph shows a computational complexity of
+$\mathcal{O}(\sigma^2)$.
+
+\begin{figure}[H]
+	\label{fig:times-2d}
+	\center
+	\includegraphics[scale=.5]{gauss_times_2d.pdf}
+	\caption{The result of \texttt{python gauss.py timer 2d 5} (so, each
+	         timing has been repeated 5 times and then averaged).}
+\end{figure}
+
+\section{Separable Gaussian Convolution}
+
+\subsection{Implementation}
+
+The \texttt{Gauss1} function uses the 1D Gaussian
+function\ref{footnote:gaussian-filter}:
+
+$$g_{1D}(x) = \frac{1}{\sqrt{2 \pi} \cdot \sigma} e^{-\frac{x^2}{2 \sigma^2}}$$
+
+This function returns a 1D array of kernel values, which is used by the
+function \texttt{convolve1d}. Using the separability property, the following
+code snippets produce the same result:
+
+\begin{verbatim}
+W = Gauss1(s)
+G = convolve1d(F, W, axis=0, mode='nearest')
+G = convolve1d(G, W, axis=1, mode='nearest')
+\end{verbatim}
+
+as opposed to:
+
+\begin{verbatim}
+G = convolve(F, Gauss(s), mode='nearest')
+\end{verbatim}
+
+The timing results of the first code snippet are displayed in figure
+\ref{fig:times-1d}. The graphs shows that \texttt{Gauss1} has a computational
+complexity of $\mathcal{O}(\sigma)$, which is much faster than the 2D
+convolution (certainly for higher scales).
+
+\begin{figure}[H]
+	\label{fig:times-1d}
+	\center
+	\includegraphics[scale=.5]{gauss_times_1d.pdf}
+	\caption{The result of \texttt{python gauss.py timer 1d 50}.}
+\end{figure}
+
+\section{Gaussian Derivatives}
+
+\subsection{Separability}
+
+We can show analytically that all derivatives of the 2D Gaussian function
+are separable as well:
+
+\begin{table}[H]
+	\begin{tabular}{rl}
+		$ $ & $= $ \\
+	\end{tabular}
+\end{table}
+
 \end{document}
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