fixed merge conflict

improc/ass4/gauss.py

@@ -2,7 +2,7 @@
 from numpy import zeros, arange, meshgrid, array, matrix
 from math import ceil, exp, pi, sqrt
 from matplotlib.pyplot import imread, imshow, plot, xlabel, ylabel, show, \
-        subplot, xlim, savefig
+        subplot, xlim, savefig, axis
 from mpl_toolkits.mplot3d import Axes3D
 from scipy.ndimage import convolve, convolve1d
 from time import time
@@ -164,19 +164,24 @@ if __name__ == '__main__':
             exit_with_usage()
 
         s = float(argv[2])
-
-        subplot(331)
+        subplot(331, title='Fs')
         imshow(gD(F, s, 0, 0), cmap='gray')
-        subplot(334)
+        axis('off')
+        subplot(334, title='Fsx')
         imshow(gD(F, s, 1, 0), cmap='gray')
-        subplot(335)
+        axis('off')
+        subplot(335, title='Fsy')
         imshow(gD(F, s, 0, 1), cmap='gray')
-        subplot(337)
+        axis('off')
+        subplot(337, title='Fsxx')
         imshow(gD(F, s, 2, 0), cmap='gray')
-        subplot(338)
+        axis('off')
+        subplot(338, title='Fsxy')
         imshow(gD(F, s, 1, 1), cmap='gray')
-        subplot(339)
+        axis('off')
+        subplot(339, title='Fsyy')
         imshow(gD(F, s, 0, 2), cmap='gray')
+        axis('off')
     else:
         exit_with_usage()
 

improc/ass4/report/report.tex

-\documentclass[10pt,a4paper]{article}
-
-\usepackage[english]{babel}
-\usepackage[utf8]{inputenc}
-\usepackage{amsmath,hyperref,graphicx,booktabs,float}
-
-% Paragraph indentation
-\setlength{\parindent}{0pt}
-\setlength{\parskip}{1ex plus 0.5ex minus 0.2ex}
-
-\title{Image processing 4: Local Structure}
-\author{Sander van Veen \& Tadde\"us Kroes \\ 6167969 \& 6054129}
-
-\begin{document}
-
-\maketitle
-
-\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:
-
-\begin{table}[H]
-	\begin{tabular}{rl}
-		$f_x$    & $= \frac{\delta f}{\delta x}$ \\
-		         & $= A \frac{\delta}{\delta x} sin(Vx) + B \frac{\delta}{\delta x} cos(Wy)$ \\
-	    	     & $= A cos(Vx) \cdot V + B \cdot 0$ \\
-	    	     & $= AV cos(Vx)$ \\
-	    	     & \\
-		$f_y$    & $= \frac{\delta f}{\delta y}$ \\
-		         & $= A \frac{\delta}{\delta y} sin(Vx) + B \frac{\delta}{\delta y} cos(Wy)$ \\
-	    	     & $= A \cdot 0 - B sin(Wy) \cdot W$ \\
-	    	     & $= -BW sin(Wy)$ \\
-	    	     & \\
-		$f_{xx}$ & $= \frac{\delta f_x}{\delta x}$ \\
-		         & $= AV \frac{\delta}{\delta x} cos(Vx)$ \\
-		         & $= -AV^2 sin(Vx)$ \\
-	    	     & \\
-		$f_{xy}$ & $= \frac{\delta f_x}{\delta y} = AV \frac{\delta}{\delta y} cos(Vx) = 0$ \\
-	    	     & \\
-		$f_{yy}$ & $= \frac{\delta f_y}{\delta y}$ \\
-		         & $= -BW \frac{\delta}{\delta y} sin(Wy)$ \\
-		         & $= -BW^2 cos(Wy)$ \\
-	\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-2d}. The subplots respectively show the original image, the
-Gaussian kernel and the convolved image.
-
-\begin{figure}[H]
-	\hspace{-5cm}
-	\includegraphics[scale=.6]{gauss_2d_5.pdf}
-	\caption{The result of \texttt{python gauss.py 2d 5}.}
-	\label{fig:gauss-2d}
-\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]
-	\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).}
-	\label{fig:times-2d}
-\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 the 1D convolution has a
-computational complexity of $\mathcal{O}(\sigma)$, which is much faster than
-the 2D convolution (certainly for higher scales).
-
-\begin{figure}[H]
-	\center
-	\includegraphics[scale=.5]{gauss_times_1d.pdf}
-	\caption{The result of \texttt{python gauss.py timer 1d 50}.}
-	\label{fig:times-1d}
-\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}{rll}
-		$ \frac{\delta}{\delta x} \frac{\delta}{\delta y} G_{2D}(x, y)$
-		& $= \frac{\delta}{\delta x} (\frac{\delta}{\delta y} (G_{1D}(x)
-            \cdot G_{1D}(y)))$ & $G_{2D}(x, y) = G_{1D}(x) \cdot G_{1D}(y)$ is
-            given \\
-		& $= \frac{\delta}{\delta x} (G_{1D}(x) \cdot
-		    \frac{\delta}{\delta y} G_{1D}(y))$ & because $G_{1D}(x)$ is
-		    constant with respect to $y$ \\
-		& $= \frac{\delta}{\delta x} G_{1D}(x) \cdot
-            \frac{\delta}{\delta y} G_{1D}(y)$ & \\
-	\end{tabular}
-\end{table}
-
-The separability property is used in the \texttt{gD} function, by calling the
-\texttt{convolve1d} function separately for each direction. This implementation
-yields the 2-jet of the cameraman image in figure \ref{fig:jet}.
-
-\begin{figure}[H]
-	\center
-	\includegraphics[scale=.4]{jet_3.pdf}
-	\caption{The result of \texttt{python gauss.py jet 3}.}
-	\label{fig:jet}
-\end{figure}
-
-\section{Canny Edge detector}
-
-The Canny Edge Detector is implemented in the file \emph{canny.py}. For the
-algorithm, we used the Wiki
-page\footnote{\url{http://en.wikipedia.org/wiki/Canny\_edge\_detector}}. The
-different Wiki sections are marked by comments with similar descriptions. Since
-the Wiki page is self-explanatory, we will not discuss the algorithm itself in
-this report.
-
-The program usage is as follows:
-
-\begin{verbatim}
-python canny.py SCALE [ LOWER_THRESHOLD HIGHER_THRESHOLD ]
-\end{verbatim}
-
-The scale is obviously used for finding the intensity gradient, and the
-thresholds are used in the "Hysterisis thresholding" part. Note that the
-thresholds are optional, because the assignment instructs to create a function
-\texttt{canny(F, s)} without any arguments for thresholds. Therefore, we were
-not sure whether to implement this section. If the thresholds are specified,
-the resulting plot will contain an additional image containing a binary image
-of edges. The thresholds can be specified in the range 0-255, they are scaled
-down by the program to match the image's color range.  An example execution of
-edge detection on the cameraman image using a scale of 2, a lower threshold of
-20 and a higher threshold of 60, can be viewed in figure \ref{fig:canny}.
-
-\begin{figure}[H]
-	\hspace{-5cm}
-	\includegraphics[scale=.6]{canny_2_20_60.pdf}
-	\caption{The result of \texttt{python canny.py 2 20 60}.}
-	\label{fig:canny}
-\end{figure}
-
-\end{document}