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@ -261,7 +261,7 @@ simplification.
\end{figure} \end{figure}
Same rivers, unprocessed but in higher scales (1:\numprint{50000} and Same rivers, unprocessed but in higher scales (1:\numprint{50000} and
1:\numprint{250000}), are depicted in figure~\ref{fig:salvis-50-250}. Some 1:\numprint{250000}), are depicted in Figure~\ref{fig:salvis-50-250}. Some
river features are so compact that a reasonably thin line depicting the river river features are so compact that a reasonably thin line depicting the river
is touching itself, creating a thicker line. We can assume that some is touching itself, creating a thicker line. We can assume that some
simplification for scale 1:\numprint{50000} and especially for simplification for scale 1:\numprint{50000} and especially for
@ -286,7 +286,7 @@ Figure~\ref{fig:salvis-generalized-50k} illustrates the same river bend, but
simplified using {\DP} and {\VW} algorithms. The resulting lines are jagged, simplified using {\DP} and {\VW} algorithms. The resulting lines are jagged,
and thus the resulting line looks unlike a real river. To smoothen the jaggedness, and thus the resulting line looks unlike a real river. To smoothen the jaggedness,
traditionally, Chaikin's\cite{chaikin1974algorithm} is applied after traditionally, Chaikin's\cite{chaikin1974algorithm} is applied after
generalization, illustrated in figure~\ref{fig:salvis-generalized-chaikin-50k}. generalization, illustrated in Figure~\ref{fig:salvis-generalized-chaikin-50k}.
\begin{figure}[ht!] \begin{figure}[ht!]
\centering \centering
@ -334,7 +334,7 @@ generalization, illustrated in figure~\ref{fig:salvis-generalized-chaikin-50k}.
\end{figure} \end{figure}
The resulting simplified and smoothened example The resulting simplified and smoothened example
(figure~\onpage{fig:salvis-generalized-chaikin-50k}) yields a more (Figure~\onpage{fig:salvis-generalized-chaikin-50k}) yields a more
aesthetically pleasing result; however, it obscures natural river features. aesthetically pleasing result; however, it obscures natural river features.
Given the absence of rocks, the only natural features that influence the river Given the absence of rocks, the only natural features that influence the river
@ -740,7 +740,7 @@ results have been manually calculated. The test suite executes parts of the
algorithm against a predefined set of geometries, and asserts that the output algorithm against a predefined set of geometries, and asserts that the output
matches the resulting hand-calculated geometries. matches the resulting hand-calculated geometries.
The full set of test geometries is visualized in figure~\ref{fig:test-figures}. The full set of test geometries is visualized in Figure~\ref{fig:test-figures}.
\begin{figure}[ht] \begin{figure}[ht]
\centering \centering
@ -762,7 +762,7 @@ the implementation:
a subtle bug, created a self-crossing bend in Visinčia. The offending a subtle bug, created a self-crossing bend in Visinčia. The offending
bend was copied to the automated test suite, which helped fix the bug. bend was copied to the automated test suite, which helped fix the bug.
Now the test suite contains the same bend (a hook-like bend on the Now the test suite contains the same bend (a hook-like bend on the
right-hand side of figure~\ref{fig:test-figures}) and code to verify right-hand side of Figure~\ref{fig:test-figures}) and code to verify
that it was correctly exaggerated. that it was correctly exaggerated.
\item During algorithm development, automated tests run about once a \item During algorithm development, automated tests run about once a
@ -883,7 +883,7 @@ purpose of each column in \textsc{wm\_debug} is described below:
sub-stage name, e.g., \textsc{bbends-polygon} creates polygon sub-stage name, e.g., \textsc{bbends-polygon} creates polygon
geometries after polygons have been detected; this particular example geometries after polygons have been detected; this particular example
is used to generate colored polygons in is used to generate colored polygons in
figure~\ref{fig:fig8-definition-of-a-bend}. Figure~\ref{fig:fig8-definition-of-a-bend}.
\item[\normalfont\textsc{name}] is the name of the geometry, which comes from \item[\normalfont\textsc{name}] is the name of the geometry, which comes from
parameter~\textsc{dbgname}. parameter~\textsc{dbgname}.
@ -891,7 +891,7 @@ purpose of each column in \textsc{wm\_debug} is described below:
\item[\normalfont\textsc{gen}] is the top-level iteration number. In other \item[\normalfont\textsc{gen}] is the top-level iteration number. In other
words, the number of times the execution flow passes through words, the number of times the execution flow passes through
\textsc{detect bends} phase as depicted in \textsc{detect bends} phase as depicted in
figure~\onpage{fig:flow-chart}. Figure~\onpage{fig:flow-chart}.
\item[\normalfont\textsc{nbend}] is the bend's index in its \textsc{line}. \item[\normalfont\textsc{nbend}] is the bend's index in its \textsc{line}.
@ -947,7 +947,7 @@ of 45 cm (1.5 feet), is 1.5 mm, as analyzed in \titlecite{mappingunits}.
In our case, our target is line bend, rather than a symbol. Assume 1.5 mm is a In our case, our target is line bend, rather than a symbol. Assume 1.5 mm is a
diameter of the bend. A semi-circle of 1.5 mm diameter is depicted in diameter of the bend. A semi-circle of 1.5 mm diameter is depicted in
figure~\ref{fig:half-circle}. A bend of this size or larger, when adjusted to Figure~\ref{fig:half-circle}. A bend of this size or larger, when adjusted to
scale, will not be simplified. scale, will not be simplified.
\begin{figure}[ht] \begin{figure}[ht]
@ -1056,7 +1056,7 @@ on the number of iterations.
To prove that the algorithm implementation is correct for multiple vertices, To prove that the algorithm implementation is correct for multiple vertices,
additional example was created and illustrated in additional example was created and illustrated in
figure~\ref{fig:inflection-1-gentle-inflection}: the rule re-assigns two Figure~\ref{fig:inflection-1-gentle-inflection}: the rule re-assigns two
vertices to the next bend. vertices to the next bend.
\begin{figure}[ht] \begin{figure}[ht]
@ -1124,7 +1124,7 @@ Looking at the {\WM} paper alone, it may seem like self-crossing may happen
only with the neighboring bend. This would mean an efficient $O(n)$ only with the neighboring bend. This would mean an efficient $O(n)$
implementation\footnote{where $n$ is the number of bends in a line. See implementation\footnote{where $n$ is the number of bends in a line. See
explanation of \textsc{algorithmic complexity} in section~\ref{sec:vocab}.}. explanation of \textsc{algorithmic complexity} in section~\ref{sec:vocab}.}.
However, as one can see in figure~\ref{fig:selfcrossing-1-non-neighbor}, it may However, as one can see in Figure~\ref{fig:selfcrossing-1-non-neighbor}, it may
not be the case: any other bend in the line may be crossing it. not be the case: any other bend in the line may be crossing it.
If one translates the requirements to code in a straightforward way, it would If one translates the requirements to code in a straightforward way, it would
@ -1399,30 +1399,30 @@ Our generalized results are viewed from the following angles:
\label{fig:salvis-wm-50k} \label{fig:salvis-wm-50k}
\end{figure} \end{figure}
As one can see in figure~\ref{fig:salvis-wm-50k}, the illustrations deliver As one can see in Figure~\ref{fig:salvis-wm-50k}, the illustrations deliver
what was promised by the algorithm, but with a few caveats. Left side of the what was promised by the algorithm, but with a few caveats. Left side of the
figure looks reasonably well simplified: long bends remain slightly curved, figure looks reasonably well simplified: long bends remain slightly curved,
small bends are removed or slightly exaggerated. small bends are removed or slightly exaggerated.
Figure's~\ref{fig:salvis-wm-50k} left part is clipped to Figure's~\ref{fig:salvis-wm-50k} left part is clipped to
figure~\ref{fig:salvis-wm-50k-nw}. As one can see, some bends were well Figure~\ref{fig:salvis-wm-50k-nw}. As one can see, some bends were well
exaggerated, and some bends were eliminated. exaggerated, and some bends were eliminated.
\begin{figure}[h!] \begin{figure}[h!]
\centering \centering
\includegraphics[width=\textwidth]{salvis-wm-50k-nw} \includegraphics[width=\textwidth]{salvis-wm-50k-nw}
\caption{Left part of figure~\ref{fig:salvis-wm-50k}.} \caption{Left part of Figure~\ref{fig:salvis-wm-50k}.}
\label{fig:salvis-wm-50k-nw} \label{fig:salvis-wm-50k-nw}
\end{figure} \end{figure}
Top--right side (clipped in figure~\ref{fig:salvis-wm-50k-ne}) some jagged Top--right side (clipped in Figure~\ref{fig:salvis-wm-50k-ne}) some jagged
and sharp bends appear. These will become more pronounced in even larger-scale and sharp bends appear. These will become more pronounced in even larger-scale
simplification in the next section. simplification in the next section.
\begin{figure}[h!] \begin{figure}[h!]
\centering \centering
\includegraphics[width=\textwidth]{salvis-wm-50k-ne} \includegraphics[width=\textwidth]{salvis-wm-50k-ne}
\caption{Top--right part of figure~\ref{fig:salvis-wm-50k}.} \caption{Top--right part of Figure~\ref{fig:salvis-wm-50k}.}
\label{fig:salvis-wm-50k-ne} \label{fig:salvis-wm-50k-ne}
\end{figure} \end{figure}
@ -1432,7 +1432,7 @@ sharp edges for others.
\subsubsection{Large-scale (1:\numprint{250000})} \subsubsection{Large-scale (1:\numprint{250000})}
\label{sec:analyzed-large-scale} \label{sec:analyzed-large-scale}
As visible in figure~\ref{fig:salvis-wm-250k-10x}, for large-scale map, some of the As visible in Figure~\ref{fig:salvis-wm-250k-10x}, for large-scale map, some of the
resulting bends look significantly exaggerated. Why is that? resulting bends look significantly exaggerated. Why is that?
Figure~\ref{fig:salvis-wm-250k-overlaid-zoom} zooms in the large-scale Figure~\ref{fig:salvis-wm-250k-overlaid-zoom} zooms in the large-scale
simplification and overlays the original. simplification and overlays the original.
@ -1527,11 +1527,11 @@ all three shapes: GDR50LT, {\WM}--simplified GDB10LT, and the original GDB10LT.
\begin{figure}[h!] \begin{figure}[h!]
\centering \centering
\includegraphics[width=\textwidth]{salvis-wm-gdr50-ne} \includegraphics[width=\textwidth]{salvis-wm-gdr50-ne}
\caption{Top--right side of figure~\ref{fig:salvis-wm-gdr50}.} \caption{Top--right side of Figure~\ref{fig:salvis-wm-gdr50}.}
\label{fig:salvis-wm-gdr50-ne} \label{fig:salvis-wm-gdr50-ne}
\end{figure} \end{figure}
Although figures are almost identical, figure~\ref{fig:salvis-wm-gdr50-ne} Although figures are almost identical, Figure~\ref{fig:salvis-wm-gdr50-ne}
illustrates two small bends that have been removed in GDR50LT, but have been illustrates two small bends that have been removed in GDR50LT, but have been
exaggerated by our implementation. exaggerated by our implementation.
@ -1688,8 +1688,8 @@ researched and extended.
\section{Acknowledgments} \section{Acknowledgments}
\label{sec:acknowledgments} \label{sec:acknowledgments}
I would like to thank my thesis supervisor, Andrius Balčiūnas, for his help in I would like to thank my thesis supervisor, Dr. Andrius Balčiūnas, for his help
formulating the requirements and providing early editorial feedback for the in formulating the requirements and providing early editorial feedback for the
thesis. thesis.
I am grateful to Tomas Straupis, who handed me the {\WM}\cite{wang1998line} I am grateful to Tomas Straupis, who handed me the {\WM}\cite{wang1998line}