Changed a few minor things in report.

docs/report.tex

@@ -3,188 +3,130 @@
 \usepackage{amsmath}
 \usepackage{hyperref}
 \usepackage{graphicx}
+\usepackage{float}
 
-% Document properties
+\title{Using local binary patterns to read license plates in photographs}
+
+% Paragraph indentation
 \setlength{\parindent}{0pt}
 \setlength{\parskip}{1ex plus 0.5ex minus 0.2ex}
-\date{\today}
-\title{Using local binary patterns to read license plates in photographs}
-\author{
-    Gijs van der Voort\\
-    Richard Torenvliet\\
-    Jayke Meijer\\
-    Tadde\"us Kroes\\
-    Fabi\"en Tesselaar
-}
-
-% Front page / toc
+
 \begin{document}
 \maketitle
-\thispagestyle{empty}
-\newpage
+
+\section*{Project members}
+Gijs van der Voort\\
+Richard Torenvliet\\
+Jayke Meijer\\
+Tadde\"us Kroes\\
+Fabi\'en Tesselaar
+
 \tableofcontents
-\newpage
+\pagebreak
 
+\setcounter{secnumdepth}{1}
 
 \section{Problem description}
 
-License plates are used for uniquely identifying motorized vehicles and are made
-to be read by humans from great distances and in all kinds of weather
+License plates are used for uniquely identifying motorized vehicles and are
+made to be read by humans from great distances and in all kinds of weather
 conditions.
 
 Reading license plates with a computer is much more difficult. Our dataset
 contains photographs of license plates from various angles and distances. This
 means that not only do we have to implement a method to read the actual
 characters, but given the location of the license plate and each individual
-character, we must make sure we transform each character to a standard form.
-This has to be done or else the local binary patterns will never match!
+character, we must make sure we transform each character to a standard form. 
 
 Determining what character we are looking at will be done by using Local Binary
 Patterns. The main goal of our research is finding out how effective LBP's are
 in classifying characters on a license plate.
 
-
-\section{The process}
-
-The process with which we extract license places from photographs consists of
-multiple steps listed below. All these steps will be explained in detail further
-on in this report.
+In short our program must be able to do the following:
 
 \begin{enumerate}
-    \item Extract character images from a license plate photograph using the
-          location points in the XML files from our dataset.
-    \item Reduce the noise in a character image using a Gaussian filter.
-    \item Transforming a character image to a normal form.
-    \item Create a LBP histogram vector for a character image.
-    \item Match the a feature vector with a learning set using a SVM.
-    \item Verify the match given by the SVM against our dataset.
+    \item Extracting characters using the location points in the xml file.
+    \item Reduce noise where possible to ensure maximum readability.
+    \item Transforming a character to a normal form.
+    \item Creating a local binary pattern histogram vector.
+    \item Matching the found vector with a learning set.
+    \item And finally it has to check results with a real data set.
 \end{enumerate}
 
+\section{Language of choice}
 
-\section{The dataset}
-
-The dataset consists of photographs of license plates from various angles and
-distances. The photographs are all 8-bit gray-scale JPEG images. With every
-photograph there is a .info file. These files, consisting of XML data, contain
-information about the photographed license plate like the country, information
-about the image, the location of the license plate and the location of the
-characters in  the license plate.
-
-
-\section{Implementation}
-
-
-\subsection{Used programming language}
-
-Although the actual purpose of this research is to see if LBP is capable of
-recognizing license plate characters. We know that LBP is a fast algorithm thus
-an advantage had to be its speed compared with other license plate recognition
-implementations. The uncertainty of whether LBP's could get some results made us
-pick Python.
-
-Python is a very flexible programming language: there are a lot of existing
-modules and frameworks most of which are made in C, the higher order of the
-language makes programming applications quick and because it is fairly easy to
-transform a python module to a C based module, our system could be easily
-converted to a faster C implementation if our results are positive.
+The actual purpose of this project is to check if LBP is capable of recognizing
+license plate characters. We knew the LBP implementation would be pretty
+simple. Thus an advantage had to be its speed compared with other license plate 
+recognition implementations, but the uncertainty of whether we could get some
+results made us pick Python. We felt Python would not restrict us as much in 
+assigning tasks to each member of the group. In addition, when using the
+correct modules to handle images, Python can be decent in speed.
 
+\section{Theory}
 
-\subsection{Character extraction}
+Now we know what our program has to be capable of, we can start with the
+defining what problems we have and how we want to solve these.
 
+\subsection{Extracting a letter and resizing it}
 
-\subsubsection{Reading the INFO file}
+Rewrite this section once we have implemented this properly.
+%NO LONGER VALID!
+%Because we are already given the locations of the characters, we only need to
+%transform those locations using the same perspective transformation used to
+%create a front facing license plate. The next step is to transform the
+%characters to a normalized manner. The size of the letter W is used as a
+%standard to normalize the width of all the characters, because W is the widest
+%character of the alphabet. We plan to also normalize the height of characters,
+%the best manner for this is still to be determined.
 
-The XML reader will return a 'license plate' object when given an XML file. The
-license plate holds a list of, up to six, NormalizedImage characters and from
-which country the plate is from. The reader is currently assuming the XML file
-and image name are corresponding. Since this was the case for the given dataset.
-This can easily be adjusted if required.
-
-To parse the XML file, the minidom module is used. So the XML file can be
-treated as a tree, where one can search for certain nodes. In each XML file it
-is possible that multiple versions exist, so the first thing the reader will do
-is retrieve the current and most up-to-date version of the plate. The reader
-will only get results from this version.
-
-Now we are only interested in the individual characters so we can skip the
-location of the entire license plate. Each character has a single character
-value, indicating what someone thought what the letter or digit was and four
-coordinates to create a bounding box. To make things not to complicated a
-Character class and Point class are used. They act pretty much as associative
-lists, but it gives extra freedom on using the data. If less then four points
-have been set the character will not be saved.
-
-When four points have been gathered the data from the actual image is being
-requested. For each corner a small margin is added (around 3 pixels) so that no
-features will be lost and minimum amounts of new features will be introduced by
-noise in the margin.
-
-In the next section you can read more about the perspective transformation that
-is being done. After the transformation the character can be saved: Converted to
-gray-scale, but nothing further. This was used to create a learning set. If it
-does not need to be saved as an actual image it will be converted to a
-NormalizedImage. When these actions have been completed for each character the
-license plate is usable in the rest of the code.
+%\begin{enumerate}
+%    \item Crop the image in such a way that the character precisely fits the
+%          image.
+%    \item Scale the image to a standard height.
+%    \item Extend the image on either the left or right side to a certain width.
+%\end{enumerate}
 
+%The resulting image will always have the same size, the character contained
+%will always be of the same height, and the character will always be positioned
+%at either the left of right side of the image.
 
-\subsubsection{Perspective transformation}
+\subsection{Transformation}
 
-Once we retrieved the corner points of the character, we feed those to a module
-that extracts the (warped) character from the original image, and creates a new
-image where the character is cut out, and is transformed to a rectangle.
+A simple perspective transformation will be sufficient to transform and resize
+the characters to a normalized format. The corner positions of characters in
+the dataset are supplied together with the dataset.
 
-
-
-\subsection{Noise reduction}
+\subsection{Reducing noise}
 
 Small amounts of noise will probably be suppressed by usage of a Gaussian
 filter. A real problem occurs in very dirty license plates, where branches and
 dirt over a letter could radically change the local binary pattern. A question
 we can ask ourselves here, is whether we want to concentrate ourselves on these
 exceptional cases. By law, license plates have to be readable. However, the
-provided dataset showed that this does not means they always are.
-
-
-\subsubsection{Camera noise and small amounts of dirt}
-
-The dirt on the license plate can be of different sizes. We can reduce the
-smaller amounts of dirt in the same way as we reduce normal noise, by applying a
-Gaussian blur to the image. This is the next step in our program.
-
-The Gaussian filter we use comes from the \texttt{scipy.ndimage} module. We use
-this function instead of our own function, because the standard functions are
-most likely more optimized then our own implementation, and speed is an
-important factor in this application.
-
-
-\subsubsection{Larger amounts of dirt}
-
-Larger amounts of dirt are not going to be resolved by using a Gaussian filter.
-We rely on one of the characteristics of the Local Binary Pattern, only looking
-at the difference between two pixels, to take care of these problems.\\ Because
-there will probably always be a difference between the characters and the dirt,
-and the fact that the characters are very black, the shape of the characters
-will still be conserved in the LBP, even if there is dirt surrounding the
-character.
-
-
-\subsection{Building a feature vector}
-
-
-\subsubsection{Creating LBP's}
-
-The LBP algorithm that we implemented is a square variant of LBP, the same that
-is introduced by Ojala et al (1994). Wikipedia presents a different form where
-the pattern is circular.
-
+provided dataset showed that this does not mean they always are. We will have
+to see how the algorithm performs on these plates, however we have good hopes
+that our method will get a good score on dirty plates, as long as a big enough
+part of the license plate remains readable.
+
+\subsection{Local binary patterns}
+Once we have separate digits and characters, we intent to use Local Binary
+Patterns (Ojala, Pietikäinen \& Harwood, 1994) to determine what character
+or digit we are dealing with. Local Binary
+Patterns are a way to classify a texture based on the distribution of edge
+directions in the image. Since letters on a license plate consist mainly of
+straight lines and simple curves, LBP should be suited to identify these.
+
+\subsubsection{LBP Algorithm}
+The LBP algorithm that we implemented can use a variety of neighbourhoods,
+including the same square pattern that is introduced by Ojala et al (1994),
+and a circular form as presented by Wikipedia.
 \begin{itemize}
-
 \item Determine the size of the square where the local patterns are being
-registered. For explanation purposes let the square be 3 x 3.
-
-\item The gray-scale value of the middle pixel is used as threshold. Every value
-of the pixel around the middle pixel is evaluated. If it's value is greater than
-the threshold it will be become a one else a zero.
+registered. For explanation purposes let the square be 3 x 3. \\
+\item The grayscale value of the middle pixel is used as threshold. Every
+value of the pixel around the middle pixel is evaluated. If it's value is
+greater than the threshold it will be become a one else a zero.
 
 \begin{figure}[h!]
 \center
@@ -193,14 +135,14 @@ the threshold it will be become a one else a zero.
 \end{figure}
 
 Notice that the pattern will be come of the form 01001110. This is done when a
-the value of the evaluated pixel is greater than the threshold, shift the bit by
-the n(with i=i$_{th}$ pixel evaluated, starting with $i=0$).
+the value of the evaluated pixel is greater than the threshold, shift the bit
+by the n(with i=i$_{th}$ pixel evaluated, starting with $i=0$).
 
 This results in a mathematical expression:
 
-Let I($x_i, y_i$) an Image with gray-scale values and $g_n$ the gray-scale value
+Let I($x_i, y_i$) an Image with grayscale values and $g_n$ the grayscale value
 of the pixel $(x_i, y_i)$. Also let $s(g_i, g_c)$ (see below) with $g_c$ =
-gray-scale value of the center pixel and $g_i$ the gray-scale value of the pixel
+grayscale value of the center pixel and $g_i$ the grayscale value of the pixel
 to be evaluated.
 
 $$
@@ -217,7 +159,7 @@ The outcome of this operations will be a binary pattern.
 
 \item Given this pattern, the next step is to divide the pattern in cells. The
 amount of cells depends on the quality of the result, so trial and error is in
-order. Starting with dividing the pattern in to cells of size 16.
+order. Starting with dividing the pattern in to cells of size 16. 
 
 \item Compute a histogram for each cell.
 
@@ -230,88 +172,177 @@ order. Starting with dividing the pattern in to cells of size 16.
 \item Consider every histogram as a vector element and concatenate these. The
 result is a feature vector of the image.
 
-\item Feed these vectors to a support vector machine. This will 'learn' which
-vector indicates what vector is which character.
+\item Feed these vectors to a support vector machine. This will ''learn'' which
+vector indicates what vector is which character. 
 
 \end{itemize}
 
-\begin{figure}[h!]
-\center
-\includegraphics[scale=0.5]{neighbourhoods.png}
-\caption{Tested neighborhoods}
-\end{figure}
+To our knowledge, LBP has yet not been used in this manner before. Therefore,
+it will be the first thing to implement, to see if it lives up to the
+expectations. When the proof of concept is there, it can be used in a final
+program.
 
-We have tried the neighborhoods as showed in figure 3. We chose these
-neighborhoods to prevent having to use interpolation, which would add a
-computational step, thus making the code execute slower. In the next section we
-will describe what the best neighborhood was.
+Later we will show that taking a histogram over the entire image (basically
+working with just one cell) gives us the best results.
 
-Take an example where the full square can be evaluated, there are cases where
-the neighbors are out of bounds. The first to be checked is the pixel in the
-left bottom corner in the square 3 x 3, with coordinate $(x - 1, y - 1)$ with
-$g_c$ as center pixel that has coordinates $(x, y)$. If the gray-scale value of
-the neighbor in the left corner is greater than the gray-scale value of the
-center pixel than return true. Bit-shift the first bit with 7. The outcome is
-now 1000000. The second neighbor will be bit-shifted with 6, and so on. Until we
-are at 0. The result is a binary pattern of the local point just evaluated. Now
-only the edge pixels are a problem, but a simple check if the location of the
-neighbor is still in the image can resolve this. We simply return false if it
-is.
+\subsection{Matching the database}
 
+Given the LBP of a character, a Support Vector Machine can be used to classify
+the character to a character in a learning set. The SVM uses a concatenation
+of each cell in an image as a feature vector (in the case we check the entire
+image no concatenation has to be done of course. The SVM can be trained with a
+subset of the given dataset called the ''Learning set''. Once trained, the
+entire classifier can be saved as a Pickle object\footnote{See
+\url{http://docs.python.org/library/pickle.html}} for later usage.
+
+\section{Implementation}
 
-\subsubsection{Creating histograms and the feature vector}
+In this section we will describe our implementations in more detail, explaining
+choices we made.
 
-After all the Local Binary Patterns are created for every pixel. This pattern is
-divided in to cells. The feature vector is the vector of concatenated
+\subsection{Character retrieval}
+
+In order to retrieve the characters from the entire image, we need to
+perform a perspective transformation. However, to do this, we need to know the 
+coordinates of the four corners of each character. For our dataset, this is
+stored in XML files. So, the first step is to read these XML files.
+
+\paragraph*{XML reader}
+
+The XML reader will return a 'license plate' object when given an XML file. The
+licence plate holds a list of, up to six, NormalizedImage characters and from
+which country the plate is from. The reader is currently assuming the XML file
+and image name are corresponding, since this was the case for the given
+dataset. This can easily be adjusted if required. 
+
+To parse the XML file, the minidom module is used. So the XML file can be
+treated as a tree, where one can search for certain nodes. In each XML
+file it is possible that multiple versions exist, so the first thing the reader
+will do is retrieve the current and most up-to-date version of the plate. The
+reader will only get results from this version.
+
+Now we are only interested in the individual characters so we can skip the
+location of the entire license plate. Each character has 
+a single character value, indicating what someone thought what the letter or
+digit was and four coordinates to create a bounding box. To make things not to
+complicated a Character class and Point class are used. They
+act pretty much as associative lists, but it gives extra freedom on using the
+data. If less then four points have been set the character will not be saved.
+
+When four points have been gathered the data from the actual image is being
+requested. For each corner a small margin is added (around 3 pixels) so that no
+features will be lost and minimum amounts of new features will be introduced by
+noise in the margin. 
+
+In the next section you can read more about the perspective transformation that
+is being done. After the transformation the character can be saved: Converted
+to grayscale, but nothing further. This was used to create a learning set. If
+it does not need to be saved as an actual image it will be converted to a
+NormalizedImage. When these actions have been completed for each character the
+license plate is usable in the rest of the code.
+
+\paragraph*{Perspective transformation}
+Once we retrieved the corner points of the character, we feed those to a
+module that extracts the (warped) character from the original image, and
+creates a new image where the character is cut out, and is transformed to a
+rectangle.
+
+\subsection{Noise reduction}
+
+The image contains a lot of noise, both from camera errors due to dark noise 
+etc., as from dirt on the license plate. In this case, noise therefore means 
+any unwanted difference in color from the surrounding pixels.
+
+\paragraph*{Camera noise and small amounts of dirt}
+The dirt on the license plate can be of different sizes. We can reduce the 
+smaller amounts of dirt in the same way as we reduce normal noise, by applying
+a Gaussian blur to the image. This is the next step in our program.\\
+\\
+The Gaussian filter we use comes from the \texttt{scipy.ndimage} module. We use
+this function instead of our own function, because the standard functions are
+most likely more optimized then our own implementation, and speed is an
+important factor in this application.
+
+\paragraph*{Larger amounts of dirt}
+Larger amounts of dirt are not going to be resolved by using a Gaussian filter.
+We rely on one of the characteristics of the Local Binary Pattern, only looking
+at the difference between two pixels, to take care of these problems.\\
+Because there will probably always be a difference between the characters and
+the dirt, and the fact that the characters are very black, the shape of the
+characters will still be conserved in the LBP, even if there is dirt
+surrounding the character.
+
+\subsection{Creating Local Binary Patterns and feature vector}
+Every pixel is a center pixel and it is also a value to evaluate but not at the 
+same time. Every pixel is evaluated as shown in the explanation
+of the LBP algorithm. There are several neighbourhoods we can evaluate. We have
+tried the following neighbourhoods:
+
+\begin{figure}[H]
+\center
+\includegraphics[scale=0.5]{neighbourhoods.png}
+\caption{Tested neighbourhoods}
+\end{figure}
+
+We chose these neighbourhoods to prevent having to use interpolation, which
+would add a computational step, thus making the code execute slower. In the
+next section we will describe what the best neighbourhood was.
+
+Take an example where the full square can be evaluated, so none of the
+neighbours are out of bounds. The first to be checked is the pixel in the left 
+bottom corner in the square 3 x 3, with coordinate $(x - 1, y - 1)$ with $g_c$ 
+as center pixel that has coordinates $(x, y)$. If the grayscale value of the
+neighbour in the left corner is greater than the grayscale
+value of the center pixel than return true. Bit-shift the first bit with 7. The
+outcome is now 1000000. The second neighbour will be bit-shifted with 6, and so 
+on. Until we are at 0. The result is a binary pattern of the local point just
+evaluated.
+Now only the edge pixels are a problem, but a simple check if the location of
+the neighbour is still in the image can resolve this. We simply state that the
+pixel has a lower value then the center pixel if it is outside the image
+bounds.
+
+\paragraph*{Histogram and Feature Vector}
+After all the Local Binary Patterns are created for every pixel, this pattern
+is divided into cells. The feature vector is the vector of concatenated
 histograms. These histograms are created for cells. These cells are created by
 dividing the \textbf{pattern} in to cells and create a histogram of that. So
-multiple cells are related to one histogram. All the histograms are concatenated
-and fed to the SVM that will be discussed in the next section, Classification.
-We did however find out that the use of several cells was not increasing our
-performance, so we only have one histogram to feed to the SVM.
+multiple cells are related to one histogram. All the histograms are
+concatenated and fed to the SVM that will be discussed in the next section,
+Classification. We did however find out that the use of several cells was not
+increasing our performance, so we only have one histogram to feed to the SVM.
 
+\subsection{Classification}
 
-\subsection{Matching the database}
-
-Given the LBP of a character, a Support Vector Machine can be used to classify
-the character to a character in a learning set. The SVM uses a concatenation of
-each cell in an image as a feature vector (in the case we check the entire image
-no concatenation has to be done of course. The SVM can be trained with a subset
-of the given dataset called the Learning set. Once trained, the entire
-classifier can be saved as a Pickle object\footnote{See
-\url{http://docs.python.org/library/pickle.html}} for later usage.
 
 
-\section{Determining optimal parameters}
+\section{Finding parameters}
 
 Now that we have a functioning system, we need to tune it to work properly for
-license plates. This means we need to find the parameters. Throughout the
+license plates. This means we need to find the parameters. Throughout the 
 program we have a number of parameters for which no standard choice is
-available. These parameters are:
-
+available. These parameters are:\\
+\\
 \begin{tabular}{l|l}
-  Parameter       & Description\\
-  \hline
-  $\sigma$        & The size of the Gaussian blur.\\
-  \emph{cell size}  & The size of a cell for which a histogram of LBP's
-                        will be generated.\\
-  \emph{Neighborhood}& The neighborhood to use for creating the LBP.\\
-  $\gamma$      & Parameter for the Radial kernel used in the SVM.\\
-  $c$         & The soft margin of the SVM. Allows how much training
-              errors are accepted.\\
-\end{tabular}
-
-For each of these parameters, we will describe how we searched for a good value,
-and what value we decided on.
-
+	Parameter 			& Description\\
+	\hline
+	$\sigma$  			& The size of the Gaussian blur.\\
+	\emph{cell size}	& The size of a cell for which a histogram of LBP's
+	                      will be generated.\\
+	\emph{Neighbourhood}& The neighbourhood to use for creating the LBP.\\
+	$\gamma$			& Parameter for the Radial kernel used in the SVM.\\
+	$c$					& The soft margin of the SVM. Allows how much training
+						  errors are accepted.\\
+\end{tabular}\\
+\\
+For each of these parameters, we will describe how we searched for a good
+value, and what value we decided on.
 
 \subsection{Parameter $\sigma$}
 
 The first parameter to decide on, is the $\sigma$ used in the Gaussian blur. To
-find this parameter, we tested a few values, by checking visually what value
-removed most noise out of the image, while keeping the edges sharp enough to
-work with. It turned out the best value is $\sigma = 0.5$.
-
+find this parameter, we tested a few values, by trying them and checking the
+results. It turned out that the best value was $\sigma = 1.1$.
 
 \subsection{Parameter \emph{cell size}}
 
@@ -319,31 +350,30 @@ The cell size of the Local Binary Patterns determines over what region a
 histogram is made. The trade-off here is that a bigger cell size makes the
 classification less affected by relative movement of a character compared to
 those in the learning set, since the important structure will be more likely to
-remain in the same cell. However, if the cell size is too big, there will not be
-enough cells to properly describe the different areas of the character, and the
-feature vectors will not have enough elements.
-
+remain in the same cell. However, if the cell size is too big, there will not
+be enough cells to properly describe the different areas of the character, and
+the feature vectors will not have enough elements.\\
+\\
 In order to find this parameter, we used a trial-and-error technique on a few
 cell sizes. During this testing, we discovered that a lot better score was
-reached when we take the histogram over the entire image, so with a single cell.
-Therefore, we decided to work without cells.
-
-The reason that using one cell works best is probably because the size of a
-single character on a license plate in the provided dataset is very small. That
-means that when dividing it into cells, these cells become simply too small to
-have a really representative histogram. Therefore, the concatenated histograms
-are then a list of only very small numbers, which are not significant enough to
-allow for reliable classification.
+reached when we take the histogram over the entire image, so with a single
+cell. Therefore, we decided to work without cells.\\
+\\
+A reason we can think of why using one cell works best is that the size of a
+single character on a license plate in the provided dataset is very small.
+That means that when dividing it into cells, these cells become simply too
+small to have a really representative histogram. Therefore, the
+concatenated histograms are then a list of only very small numbers, which
+are not significant enough to allow for reliable classification.
 
+\subsection{Parameter \emph{Neighbourhood}}
 
-\subsection{Parameter \emph{Neighborhood}}
+The neighbourhood to use can only be determined through testing. We did a test
+with each of these neighbourhoods, and we found that the best results were
+reached with the following neighbourhood, which we will call the
+()-neighbourhood.
 
-The neighborhood to use can only be determined through testing. We did a test
-with each of these neighborhoods, and we found that the best results were
-reached with the following neighborhood, which we will call the ()-neighborhood.
-
-
-\subsection{Parameter $\gamma$ \& $c$}
+\subsection{Parameters $\gamma$ \& $c$}
 
 The parameters $\gamma$ and $c$ are used for the SVM. $c$ is a standard
 parameter for each type of SVM, called the 'soft margin'. This indicates how
@@ -351,23 +381,24 @@ exact each element in the learning set should be taken. A large soft margin
 means that an element in the learning set that accidentally has a completely
 different feature vector than expected, due to noise for example, is not taken
 into account. If the soft margin is very small, then almost all vectors will be
-taken into account, unless they differ extreme amounts.
-
+taken into account, unless they differ extreme amounts.\\
 $\gamma$ is a variable that determines the size of the radial kernel, and as
-such determines how steep the difference between two classes can be.
-
+such determines how steep the difference between two classes can be.\\
+\\
 Since these parameters both influence the SVM, we need to find the best
-combination of values. To do this, we perform a so-called grid-search. A grid-
-search takes exponentially growing sequences for each parameter, and checks for
-each combination of values what the score is. The combination with the highest
-score is then used as our parameters, and the entire SVM will be trained using
-those parameters.
-
-We found that the best values for these parameters are $c = ?$ and $\gamma = ?$.
-
+combination of values. To do this, we perform a so-called grid-search. A
+grid-search takes exponentially growing sequences for each parameter, and
+checks for each combination of values what the score is. The combination with
+the highest score is then used as our parameters, and the entire SVM will be
+trained using those parameters.\\
+\\
+We found that the best values for these parameters are $c = ?$ and
+$\gamma = ?$.
 
 \section{Results}
 
+The goal was to find out two things with this research: The speed of the
+classification and the accuracy. In this section we will show our findings.
 
 \subsection{Speed}
 
@@ -375,64 +406,64 @@ Recognizing license plates is something that has to be done fast, since there
 can be a lot of cars passing a camera in a short time, especially on a highway.
 Therefore, we measured how well our program performed in terms of speed. We
 measure the time used to classify a license plate, not the training of the
-dataset, since that can be done off-line, and speed is not a primary necessity
-there.
-
+dataset, since that can be done offline, and speed is not a primary necessity
+there.\\
+\\
 The speed of a classification turned out to be ???.
 
-
 \subsection{Accuracy}
 
 Of course, it is vital that the recognition of a license plate is correct,
 almost correct is not good enough here. Therefore, we have to get the highest
-accuracy score we possibly can.
-
-According to Wikipedia \footnote{
+accuracy score we possibly can.\\
+\\ According to Wikipedia
+\footnote{
 \url{http://en.wikipedia.org/wiki/Automatic_number_plate_recognition}},
 commercial license plate recognition software score about $90\%$ to $94\%$,
-under optimal conditions and with modern equipment.
-
-Our program scores an average of ???.
-
+under optimal conditions and with modern equipment. Our program scores an
+average of ???.
 
 \section{Conclusion}
 
-It turns out that using Local Binary Patterns is a promising technique for
-License Plate Recognition. It seems to be relatively insensitive by dirty
-license plates and different fonts on these plates.
-
-The performance speed wise is ???
-
+In the end it turns out that using Local Binary Patterns is a promising
+technique for License Plate Recognition. It seems to be relatively unsensitive
+for the amount of dirt on license plates and different fonts on these plates.\\
+\\
+The performance speedwise is ???
 
 \section{Reflection}
 
+\subsection{Difficulties}
+
+During the implementation and testing of the program, we did encounter a
+number of difficulties. In this section we will state what these difficulties
+were and whether we were able to find a proper solution for them.
 
-\subsection{Dataset}
+\subsubsection*{Dataset}
 
+We did experience a number of problems with the provided dataset. A number of
+these are problems to be expected in a real world problem, but which make
+development harder. Others are more elemental problems.\\
 The first problem was that the dataset contains a lot of license plates which
 are problematic to read, due to excessive amounts of dirt on them. Of course,
-this is something you would encounter in the real situation, but it made it hard
-for us to see whether there was a coding error or just a bad example.
-
-Another problem was that there were license plates of several countries in the
-dataset. Each of these countries has it own font, which also makes it hard to
-identify these plates, unless there are a lot of these plates in the learning
-set.
-
+this is something you would encounter in the real situation, but it made it
+hard for us to see whether there was a coding error or just a bad example.\\
+Another problem was that there were license plates of several countries in
+the dataset. Each of these countries has it own font, which also makes it
+hard to identify these plates, unless there are a lot of these plates in the
+learning set.\\
 A problem that is more elemental is that some of the characters in the dataset
 are not properly classified. This is of course very problematic, both for
 training the SVM as for checking the performance. This meant we had to check
 each character whether its description was correct.
 
-
-\subsection{SVM}
+\subsubsection*{SVM}
 
 We also had trouble with the SVM for Python. The standard Python SVM, libsvm,
 had a poor documentation. There was no explanation what so ever on which
 parameter had to be what. This made it a lot harder for us to see what went
 wrong in the program.
 
-
 \subsection{Workload distribution}
 
 The first two weeks were team based. Basically the LBP algorithm could be
@@ -441,21 +472,21 @@ Some additional 'basics' where created in similar fashion. This ensured that
 every team member was up-to-date and could start figuring out which part of the
 implementation was most suited to be done by one individually or in a pair.
 
-Gijs created the basic classes we could use and helped the rest everyone by
-keeping track of what required to be finished and whom was working on what.
+\subsubsection*{Who did what}
+Gijs created the basic classes we could use and helped the rest everyone by 
+keeping track of what required to be finished and whom was working on what. 
 Tadde\"us and Jayke were mostly working on the SVM and all kinds of tests
 whether the histograms were matching and alike. Fabi\"en created the functions
-to read and parse the given xml files with information about the license plates.
-Upon completion all kinds of learning and data sets could be created.
+to read and parse the given xml files with information about the license
+plates. Upon completion all kinds of learning and data sets could be created.
+Richard helped out wherever anyone needed a helping hand, and was always
+available when someone had to talk or ask something.
 
-%Richard je moet even toevoegen wat je hebt gedaan :P:P maar miss is dit hele
-%ding wel overbodig Ik dacht dat Rein het zei tijdens gesprek van ik wil weten
-%hoe het ging enzo.
+\subsubsection*{How it went}
 
 Sometimes one cannot hear the alarm bell and wake up properly. This however was
 not a big problem as no one was afraid of staying at Science Park a bit longer
-to help out. Further communication usually went through e-mails and replies were
-instantaneous! A crew to remember.
-
+to help out. Further communication usually went through e-mails and replies
+were instantaneous! A crew to remember.
 
 \end{document}