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@ -76,7 +76,7 @@
\textbf{\MYAUTHOR} \\[8ex] \textbf{\MYAUTHOR} \\[8ex]
\normalsize \normalsize
A thesis presented for the degree of Master in Cartography \\[8ex] A Thesis Presented for the Degree of Master in Cartography \\[8ex]
\LARGE \LARGE
\textbf{\textsc{\MYTITLENOCAPS}} \textbf{\textsc{\MYTITLENOCAPS}}
@ -93,7 +93,7 @@
\begin{abstract} \begin{abstract}
\label{sec:abstract} \label{sec:abstract}
Currently available line simplification algorithms are rooted in mathematics Currently available line simplification algorithms are rooted in mathematics
and geometry, and are unfit for bendy map features like rivers and and geometry, and are unfit for bendy map features like rivers and
coastlines. {\WnM} observed how cartographers simplify these natural coastlines. {\WnM} observed how cartographers simplify these natural
features and created an algorithm. We implemented this algorithm and features and created an algorithm. We implemented this algorithm and
@ -126,17 +126,17 @@ Textwidth in cm: {\printinunitsof{cm}\prntlen{\textwidth}}
When creating small-scale maps, often the detail of the data source is greater When creating small-scale maps, often the detail of the data source is greater
than desired for the map. While many features can be removed or simplified, it than desired for the map. While many features can be removed or simplified, it
is more tricky with natural features that have many bends, like coastlines, is more tricky with natural features that have many bends, like coastlines,
rivers or forest boundaries. rivers, or forest boundaries.
To create a small-scale map from a large-scale data source, features need to be To create a small-scale map from a large-scale data source, features need to be
simplified, i.e., detail should be reduced. While performing the simplified, i.e., detail should be reduced. While performing the
simplification, it is important to retain the "defining" shape of the original simplification, it is important to retain the "defining" shape of the original
feature. Otherwise, if the simplified feature looks too different than the feature. Otherwise, if the simplified feature looks too different from the
original, the result will look unrealistic. Simplification problem for some original, the result will look unrealistic. Simplification problem for some
objects can often be solved by non-geometric means: objects can often be solved by non-geometric means:
\begin{itemize} \begin{itemize}
\item Towns and cities can be filtered by number of inhabitants. \item Towns and cities can be filtered by the number of inhabitants.
\item Roads can be eliminated by the road length, number of lanes, or \item Roads can be eliminated by the road length, number of lanes, or
classification of the road (local, regional, international). classification of the road (local, regional, international).
\end{itemize} \end{itemize}
@ -151,7 +151,7 @@ viewed as a task of finding a delicate balance between two competing goals:
\begin{itemize} \begin{itemize}
\item Reduce detail by removing or simplifying "less important" features. \item Reduce detail by removing or simplifying "less important" features.
\item Retain enough detail, so the original is still recognize-able. \item Retain enough detail, so the original is still recognizable.
\end{itemize} \end{itemize}
Given the discussed complexities with natural features, a fine line between Given the discussed complexities with natural features, a fine line between
@ -160,11 +160,11 @@ straight line) needs to be found. Therein lies the complexity of simplification
algorithms: all have different trade-offs. algorithms: all have different trade-offs.
The purpose of the thesis is to implement a cartographic line generalization The purpose of the thesis is to implement a cartographic line generalization
algorithm on the basis of {\WM} algorithm using open-source software. Tasks: algorithm on the basis of {\WM} algorithm, using open-source software. Tasks:
\begin{itemize} \begin{itemize}
\item Evaluate existing line simplification algorithms. \item Evaluate existing line simplification algorithms.
\item Identify main river generalization problems using classical line \item Identify main river generalization problems, using classical line
simplification algorithms. simplification algorithms.
\item Define the method of the {\WM} technical implementation. \item Define the method of the {\WM} technical implementation.
\item Realize {\WM} algorithm technically, explaining the geometric \item Realize {\WM} algorithm technically, explaining the geometric
@ -174,24 +174,23 @@ algorithm on the basis of {\WM} algorithm using open-source software. Tasks:
\end{itemize} \end{itemize}
Scientific relevance of this work --- the simplification processes (steps) Scientific relevance of this work --- the simplification processes (steps)
described by the {\WM} algorithm are analyzed in detail, practically described by the {\WM} algorithm --- are analyzed in detail, practically
implemented and the implementation --- described. That expands the knowledge of implemented, and the implementation is described. That expands the knowledge of
cartographic theory about the generalization of natural objects' boundaries cartographic theory about the generalization of natural objects' boundaries
after their natural defining properties. after their natural defining properties.
In the original {\WM} article introducing the algorithm, the steps are not In the original {\WM} article introducing the algorithm, the steps are not
detailed in a way that can be put into practice for specific data; steps are detailed in a way that can be put into practice for specific data; the steps are
specified in this work. Practically this work makes it possible to use open specified in this work. Practically, this work makes it possible to use open-source software to perform cartographic line generalization. The developed
source software to perform cartographic line generalization. The developed
specialized cartographic line simplification algorithm can be applied by specialized cartographic line simplification algorithm can be applied by
cartographers to implement automatic data generalization solutions. Given the cartographers to implement automatic data generalization solutions. Given the
open-source nature of this work, the algorithm implementation can be modified open-source nature of this work, the algorithm implementation can be modified
freely. freely.
\section{Literature Review and Problematic} \section{Literature Review And Problematic}
\label{sec:literature-review-problematic} \label{sec:literature-review-problematic}
\subsection{Available algorithms} \subsection{Available Algorithms}
This section reviews the classical line simplification algorithms, which, This section reviews the classical line simplification algorithms, which,
besides being around for a long time, offer easily accessible implementations, besides being around for a long time, offer easily accessible implementations,
@ -203,11 +202,11 @@ implementation.
{\DP}\cite{douglas1973algorithms} and {\VW}\cite{visvalingam1993line} are {\DP}\cite{douglas1973algorithms} and {\VW}\cite{visvalingam1993line} are
"classical" line simplification computer graphics algorithms. They are "classical" line simplification computer graphics algorithms. They are
relatively simple to implement, require few runtime resources. Both of them relatively simple to implement and require few runtime resources. Both of them
accept a single parameter, based on desired scale of the map, which makes them accept a single parameter based on desired scale of the map, which makes them
straightforward to adjust for different scales. straightforward to adjust for different scales.
Both algorithms available in PostGIS, a free-software GIS suite: Both algorithms are available in PostGIS, a free-software GIS suite:
\begin{itemize} \begin{itemize}
\item {\DP} via \item {\DP} via
\href{https://postgis.net/docs/ST_Simplify.html}{PostGIS \textsc{st\_simplify}}. \href{https://postgis.net/docs/ST_Simplify.html}{PostGIS \textsc{st\_simplify}}.
@ -222,9 +221,9 @@ algorithm\cite{chaikin1974algorithm} via
\href{https://postgis.net/docs/ST_ChaikinSmoothing.html}{PostGIS \href{https://postgis.net/docs/ST_ChaikinSmoothing.html}{PostGIS
\textsc{st\_chaikinsmoothing}}. \textsc{st\_chaikinsmoothing}}.
To use in generalization examples, we will use two rivers: Šalčia and Visinčia. In generalization examples, we will use two rivers: Šalčia and Visinčia.
These rivers were chosen, because they have both large and small bends, and These rivers were chosen because they have both large and small bends, and
thus convenient to analyze for both small and large scale generalization. thus are convenient to analyze for both small- and large-scale generalization.
Figure~\onpage{fig:salvis-25} illustrates the original two rivers without any Figure~\onpage{fig:salvis-25} illustrates the original two rivers without any
simplification. simplification.
@ -252,8 +251,8 @@ simplification.
\label{fig:salvis-50-250} \label{fig:salvis-50-250}
\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
@ -276,7 +275,7 @@ simplification for scale 1:\numprint{50000} and especially for
Figure~\ref{fig:salvis-generalized-50k} illustrates the same river bend, but 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,
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}.
@ -327,12 +326,12 @@ generalization, illustrated in figure~\ref{fig:salvis-generalized-chaikin-50k}.
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
direction are topographic: direction are topographic:
\begin{itemize} \begin{itemize}
\item Relatively straight river (completely straight or with small-angled \item Relatively straight river (completely straight or with small-angled
bends over a relatively long distance) implies greater slope, more bends over a relatively long distance) implies greater slope, more
water, and/or faster flow. water, and/or faster flow.
@ -341,7 +340,6 @@ direction are topographic:
and/or less water. and/or less water.
\end{itemize} \end{itemize}
Both {\VW} and {\DP} have a tendency to remove the small bends altogether, Both {\VW} and {\DP} have a tendency to remove the small bends altogether,
removing a valuable characterization of the river. removing a valuable characterization of the river.
@ -355,11 +353,13 @@ example, classical algorithms would remove these bends altogether. A
cartographer would retain a few of those distinctive bends, but would increase cartographer would retain a few of those distinctive bends, but would increase
the distance between the bends, remove some of the bends, or both. the distance between the bends, remove some of the bends, or both.
For the reasons discussed in this section, the "classical" {\DP} and {\VW} are % TODO: figues shouldn't split the sentence.
not well suited for natural river generalization, and a more robust line
generalization algorithm is worthwhile for to look for.
\subsubsection{Modern approaches} For the reasons discussed in this section, the "classical" {\DP} and {\VW} are
not well-suited for natural river generalization, and a more robust line
generalization algorithm is worthwhile to look for.
\subsubsection{Modern Approaches}
Due to their simplicity and ubiquity, {\DP} and {\VW} have been established as Due to their simplicity and ubiquity, {\DP} and {\VW} have been established as
go-to algorithms for line generalization. During recent years, alternatives go-to algorithms for line generalization. During recent years, alternatives
@ -380,8 +380,7 @@ have emerged. These modern replacements fall into roughly two categories:
Authors of most of the aforementioned articles have implemented the Authors of most of the aforementioned articles have implemented the
generalization algorithm, at least to generate the illustrations in the generalization algorithm, at least to generate the illustrations in the
articles. However, code is not available for evaluation with a desired data articles. However, code is not available for evaluation with a desired dataset, much less for use as a basis for creating new maps. To the author's knowledge,
set, much less for use as a basis for creating new maps. To author's knowledge,
{\WM}\cite{wang1998line} is available in a commercial product, but requires a {\WM}\cite{wang1998line} is available in a commercial product, but requires a
purchase of the commercial product suite, without a way to license the purchase of the commercial product suite, without a way to license the
standalone algorithm. standalone algorithm.
@ -393,13 +392,13 @@ especially suitable for generalization of natural linear features:
\begin{figure}[b] \begin{figure}[b]
\centering \centering
\includegraphics[width=.8\textwidth]{wang125} \includegraphics[width=.8\textwidth]{wang125}
\caption{figure 12.5 in \cite{wang1998line}: example of cartographic line \caption{Figure 12.5 in \cite{wang1998line}: example of cartographic line
generalization.} generalization.}
\label{fig:wang125} \label{fig:wang125}
\end{figure} \end{figure}
\begin{itemize} \begin{itemize}
\item Small bends are not always removed, but either combined (for example, \item Small bends are not always removed, but either combined (e.g.,
3 bends into 2), exaggerated, or removed, depending on the neighboring 3 bends into 2), exaggerated, or removed, depending on the neighboring
bends. bends.
\item Long and gentle bends are not straightened, but kept as-is. \item Long and gentle bends are not straightened, but kept as-is.
@ -413,10 +412,7 @@ frequent small bends.
Figure~\ref{fig:wang125}, sub-figure labeled "proposed method" (from the Figure~\ref{fig:wang125}, sub-figure labeled "proposed method" (from the
original \titlecite{wang1998line}) illustrates the {\WM} algorithm. original \titlecite{wang1998line}) illustrates the {\WM} algorithm.
\subsection{Problematic with generalization of rivers} \subsection{Problematic with Generalization of Rivers}
% DONE subscection: andriub: Į šį skyrių turi būti perkeltas tekstas iš From Simplification to Generalization ir mano pakomentuota dalis iš Modern approaches skyriaus.
% DONE: [Skyriaus pradžioje pateikiama bendra informacija: Upių generalizavimo problemą galima skaidyti į dvi dalis: egzistuojantys algoritmai skirti geometrijos supaprastinimui, tačiau neturi kartografinės logikos; egzistuojantys sprendimai nėra laisvai prieinami. Atitinkamai tuomet seka tekstas iš From Simplification to Generalization skyriaus, o toliau - dalis iš Modern approaches skyriaus.
This section introduces the reader to simplification and generalization, and This section introduces the reader to simplification and generalization, and
discusses two main problems with current-day automatic cartographic line discusses two main problems with current-day automatic cartographic line
@ -433,9 +429,9 @@ generalization:
\subsubsection{Simplification versus Generalization} \subsubsection{Simplification versus Generalization}
It is important to note the distinction between simplification, line It is important to note the distinction between simplification, line
generalization and cartographic generalization. generalization, and cartographic generalization.
Simplification reduces object's detail in isolation, not taking object's Simplification reduces an object's detail in isolation, not taking the object's
natural properties or surrounding objects into account. For example, if a natural properties or surrounding objects into account. For example, if a
river is simplified, it may have an approximate shape of the original river, river is simplified, it may have an approximate shape of the original river,
but lose some shapes that define it. For example: but lose some shapes that define it. For example:
@ -450,13 +446,13 @@ but lose some shapes that define it. For example:
\item Low-angle river bend river over a long distance differs significantly \item Low-angle river bend river over a long distance differs significantly
from a completely straight canal. Non-cartographic line simplification from a completely straight canal. Non-cartographic line simplification
may replace a that bend with a straight line, making the river more may replace that bend with a straight line, making the river more
similar to a canal than a river. similar to a canal than a river.
\end{itemize} \end{itemize}
In other words, simplification processes the line ignoring its geographic In other words, simplification processes the line, ignoring its geographic
features. It is works well when the features are human-made (e.g., roads, features. It works well when the features are human-made (e.g., roads,
administrative boundaries, buildings). There is a number of freely available administrative boundaries, buildings). There is a number of freely available
non-cartographic line simplification algorithms, which this paper will review. non-cartographic line simplification algorithms, which this paper will review.
@ -473,7 +469,7 @@ line generalization deals with a single feature class, takes into account its
geographic properties, but ignores other features. This paper examines {\WM}'s geographic properties, but ignores other features. This paper examines {\WM}'s
\titlecite{wang1998line}, a cartographic line generalization algorithm. \titlecite{wang1998line}, a cartographic line generalization algorithm.
\subsubsection{Availability of generalization algorithms} \subsubsection{Availability of Generalization Algorithms}
Lack of robust openly available generalization algorithm implementations poses Lack of robust openly available generalization algorithm implementations poses
a problem for map creation with free software: there is no high-quality a problem for map creation with free software: there is no high-quality
@ -481,31 +477,31 @@ simplification algorithm to create down-scaled maps, so any cartographic work,
which uses line generalization as part of its processing, will be of sub-par which uses line generalization as part of its processing, will be of sub-par
quality. We believe that availability of high-quality open-source tools is an quality. We believe that availability of high-quality open-source tools is an
important foundation for future cartographic experimentation and development, important foundation for future cartographic experimentation and development,
thus it it benefits the cartographic society as a whole. thus it benefits the cartographic society as a whole.
{\WM}'s commercial availability signals something about the value of the {\WM}'s commercial availability signals something about the value of the
algorithm: at least the authors of the commercial software suite deemed it algorithm: at least the authors of the commercial software suite deemed it
worthwhile to include it. However, not everyone has access to the commercial worthwhile to include it. However, not everyone has access to the commercial
software suite, access to funds to buy the commercial suite, or access to the software suite, access to funds to buy the commercial suite, or access to the
operating system required to run the commercial suite. PostGIS, in contrast, is operating system required to run the commercial suite. PostGIS, in contrast, is
free on itself, and runs on free platforms. Therefore, algorithm free itself, and runs on free platforms. Therefore, algorithm
implementations that run on PostGIS or other free platforms are useful to a implementations that run on PostGIS or other free platforms are useful to a
wider cartographic society than proprietary ones. wider cartographic society than proprietary ones.
\subsubsection{Unfitness of line simplification algorithms} \subsubsection{Unfitness of Line Simplification Algorithms}
Section~\ref{sec:dp-vw-chaikin} illustrates the current gaps with Line Section~\ref{sec:dp-vw-chaikin} illustrates the current gaps with line
Simplification algorithms for real rivers. To sum up, we highlight the simplification algorithms for real rivers. To sum up, we highlight the
following cartographic problems from our examples: following cartographic problems from our examples:
\begin{description} \begin{description}
\item[Long bends] should remain as long bends, instead of become fully \item[Long bends] should remain as long bends, instead of becoming fully
straight lines. straight lines.
\item[Many small bends] should not be removed. To retain river's character, \item[Many small bends] should not be removed. To retain a river's character,
the algorithm should retain some small bends, and, when they are too the algorithm should retain some small bends, and, when they are too
small to be visible, should be combined or exaggerated. small to be visible, they should be combined or exaggerated.
\end{description} \end{description}
@ -514,7 +510,7 @@ cartographic generalization, which takes topology and other feature classes
into account, is out of scope. into account, is out of scope.
Figure~\onpage{fig:wang125} illustrates {\WM} algorithm from their original Figure~\onpage{fig:wang125} illustrates {\WM} algorithm from their original
paper. Note how the long bends retain curvy, and how some small bends got paper. Note how the long bends retain curvy, and how some small bends get
exaggerated. exaggerated.
\section{Methodology} \section{Methodology}
@ -525,18 +521,18 @@ desired for a practical implementation: it is not straightforward to implement
the algorithm from the paper alone. the algorithm from the paper alone.
Explanations in this document are meant to expand, rather than substitute, the Explanations in this document are meant to expand, rather than substitute, the
original description in {\WM}. Therefore familiarity with the original paper is original description in {\WM}. Therefore, familiarity with the original paper is
assumed, and, for some sections, having the original close-by is necessary to assumed, and, for some sections, having the original close-by is necessary to
meaningfully follow this document. meaningfully follow this document.
This paper describes {\WM} in detail that is more useful for anyone who wishes This paper describes {\WM} in detail that is more useful for anyone who wishes
to follow the algorithm implementation more closely: each section is expanded to follow the algorithm implementation more closely: each section is expanded
with additional commentary, and illustrations for non-obvious steps. Corner with additional commentary, and illustrations added for non-obvious steps. Corner
cases are discussed too. cases are discussed, too.
Assume Euclidean geometry throughout this document, unless noted otherwise. Assume Euclidean geometry throughout this document, unless noted otherwise.
\subsection{Main geometry elements used by algorithm} \subsection{Main Geometry Elements Used by Algorithm}
\label{sec:vocab} \label{sec:vocab}
This section defines and explains the geometry elements that are used This section defines and explains the geometry elements that are used
@ -549,37 +545,37 @@ throughout this paper and the implementation.
\item[\normalfont\textsc{line segment}] or \textsc{segment} joins two \item[\normalfont\textsc{line segment}] or \textsc{segment} joins two
vertices by a straight line. A segment can be expressed by two vertices by a straight line. A segment can be expressed by two
coordinate pairs: $(x_1, y_1)$ and $(x_2, y_2)$. Line Segment and coordinate pairs: $(x_1, y_1)$ and $(x_2, y_2)$. Line segment and
Segment are used interchangeably. segment are used interchangeably.
\item[\normalfont\textsc{line}] or \textsc{linestring}, represents a single \item[\normalfont\textsc{line}] or \textsc{linestring} represents a single
linear feature. For example, a river or a coastline. linear feature. For example, a river or a coastline.
Geometrically, A line is a series of connected line segments, or, Geometrically, a line is a series of connected line segments, or,
equivalently, a series of connected vertices. Each vertex connects to equivalently, a series of connected vertices. Each vertex connects to
two other vertices, except those vertices at either ends of the line: two other vertices, with the exception of the vertices at either ends of the line:
these two connect to a single other vertex. these two connect to a single other vertex.
\item[\normalfont\textsc{multiline}] or \textsc{multilinestring} is a \item[\normalfont\textsc{multiline}] or \textsc{multilinestring} is a
collection of linear features. Throughout this implementation this is collection of linear features. Throughout this implementation, this is
used rarely (normally, a river is a single line), but can be valid used rarely (normally, a river is a single line) but can be valid
when, for example, a river has an island. when, for example, a river has an island.
\item[\normalfont\textsc{bend}] is a subset of a line that humans perceive \item[\normalfont\textsc{bend}] is a subset of a line that humans perceive
as a curve. The geometric definition is complex and is discussed in as a curve. The geometric definition is complex and is discussed in
section~\ref{sec:definition-of-a-bend}. section~\ref{sec:definition-of-a-bend}.
\item[\normalfont\textsc{baseline}] is a line between bend's first and last \item[\normalfont\textsc{baseline}] is a line between the bend's first and last
vertices. vertices.
\item[\normalfont\textsc{sum of inner angles}] is a measure of how "curved" \item[\normalfont\textsc{sum of inner angles}] is a measure of how "curved"
the bend is. Assume first and last bend vertices are vectors. Then sum the bend is. Assume that first and last bend vertices are vectors. Then sum
of inner angles will be the angular difference of those two vectors. of inner angles will be the angular difference of those two vectors.
\item[\normalfont\textsc{algorithmic complexity}] measured in \textsc{big o \item[\normalfont\textsc{algorithmic complexity}] measured in \textsc{big o
notation}, is a relative measure that helps explain how notation}, is a relative measure that helps explain how
long\footnote{the upper bound, i.e., the worst case.} will the long\footnote{the upper bound, i.e., the worst case.} the
algorithm run depending on it's input. It is widely used in computing algorithm will run depending on its input. It is widely used in computing
science when discussing the efficiency of a given algorithm. science when discussing the efficiency of a given algorithm.
For example, given $n$ objects and time complexity of $O(log(n))$, the For example, given $n$ objects and time complexity of $O(log(n))$, the
@ -596,7 +592,7 @@ throughout this paper and the implementation.
\end{description} \end{description}
\subsection{Algorithm implementation process} \subsection{Algorithm Implementation Process}
\tikzset{ \tikzset{
startstop/.style={trapezium,text centered,minimum height=2em, startstop/.style={trapezium,text centered,minimum height=2em,
@ -659,35 +655,34 @@ We have taken a different approach: process each step fully for the line,
before moving to the next step. This way provides the following advantages: before moving to the next step. This way provides the following advantages:
\begin{itemize} \begin{itemize}
\item For \textsc{eliminate self-crossing} stage, when it finds a bend with the right
\item \textsc{eliminate self-crossing}, when finds a bend with the right
sum of inflection angles, it checks the whole line for self-crossings. sum of inflection angles, it checks the whole line for self-crossings.
This is impossible with streaming, because it requires having the full This is impossible with streaming because it requires having the full
line in memory. It could be optimized by, for example, looking for a line in memory. It could be optimized by, for example, looking for a
fixed number of neighboring bends (say, 10), but that would complicate fixed number of neighboring bends (say, 10), but that would complicate
the implementation. the implementation.
\item \textsc{fix gentle inflections} is iterating the same line twice from \item \textsc{fix gentle inflections} is iterating the same line twice from
opposite directions. That could be re-written to streaming fashion, but opposite directions. That could be re-written to streaming fashion, but
that complicates the implementation too. it complicates the implementation, too.
\end{itemize} \end{itemize}
On the other hand, comparing to the {\WM} prototype flow chart, our On the other hand, comparing to the {\WM} prototype flow chart, our
implementation uses more memory (because it needs to have the full line before implementation uses more memory (because it needs to have the full line before
processing), and some steps are unnecessarily repeated, like re-computing the processing), and some steps are unnecessarily repeated, like re-computing the
bend's attributes during repeated iterations. bend's attributes during repeated iterations.
\subsection{Technical implementation} \subsection{Technical Implementation}
\label{sec:technical-implementation} \label{sec:technical-implementation}
Technical algorithm realization was created in \titlecite{postgis311}. PostGIS Technical algorithm realization was created in \titlecite{postgis311}. PostGIS
is a PostgreSQL extension for working with spatial data. is a PostgreSQL extension for working with spatial data.
PostgreSQL is an open-source relational database, widely used in industry and PostgreSQL is an open-source relational database, widely used in industry and
academia. PostgreSQL can be interfaced from nearly any programming language, academia. PostgreSQL can be interfaced from nearly any programming language;
therefore solutions written in PostgreSQL (and their extensions) are usable in therefore, solutions written in PostgreSQL (and their extensions) are usable in
many environments. On top of that, PostGIS has implements a rich set of many environments. On top of that, PostGIS implements a rich set of
functions\cite{postgisref} for working with geometric and geographic objects. functions\cite{postgisref} for working with geometric and geographic objects.
Due to its wide applicability and rich library of spatial functions, PostGIS is Due to its wide applicability and rich library of spatial functions, PostGIS is
@ -722,15 +717,15 @@ This function accepts the following parameters:
\end{description} \end{description}
The function \textsc{st\_simplifywm} calls into helper functions, which detect, The function \textsc{st\_simplifywm} calls into helper functions, which detect,
transform or remove bends. These helper functions are also defined in the transform, or remove bends. These helper functions are also defined in the
implementation and are part of the algorithm technical realization. All implementation and are part of the algorithm technical realization. All
supporting functions use spatial manipulation functions provided by PostGIS. supporting functions use spatial manipulation functions provided by PostGIS.
\subsection{Automated tests} \subsection{Automated Tests}
\label{sec:automated-tests} \label{sec:automated-tests}
As part of the algorithm realization, an automated test suite has been As part of the algorithm realization, an automated test suite has been
developed. Shapes to test each function have been hand-crafted and expected developed. Shapes to test each function have been hand-crafted, and expected
results have been manually calculated. The test suite executes parts of the 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.
@ -744,31 +739,31 @@ The full set of test geometries is visualized in figure~\ref{fig:test-figures}.
\label{fig:test-figures} \label{fig:test-figures}
\end{figure} \end{figure}
Test suite can be executed with a single command, and completes in about a Test suite can be executed with a single command and completes in about a
second. Having an easily accessible test suite boosts confidence that no second. Having an easily accessible test suite boosts confidence that no
unexpected bugs have snug in while modifying the algorithm. unexpected bugs have snug in while modifying the algorithm.
We will explain two instances on when automated tests were very useful during We will explain two instances when automated tests were very useful during
the implementation: the implementation:
\begin{itemize} \begin{itemize}
\item Created a function \textsc{wm\_exaggeration}, which exaggerates bends \item Created a function \textsc{wm\_exaggeration}, which exaggerates bends
following the rules. It worked well over simple geometries, but, due to following the rules. It worked well over simple geometries but, due to
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-looking 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
minute. They quickly find logical and syntax errors. In contrast, minute. They quickly find logical and syntax errors. In contrast,
running the algorithm with real rivers takes a few minutes, which is running the algorithm with real rivers takes a few minutes, which
increases the feedback loop, and takes longer to fix the "simple" increases the feedback loop, and takes longer to fix the "simple"
errors. errors.
\end{itemize} \end{itemize}
Whenever I find and fix a bug, I aim to create an automated test case for it, Whenever we find and fix a bug, we aim to create an automated test case for it,
so the same bug is not re-introduced by whoever works next on the same piece of so the same bug is not re-introduced by whoever works next on the same piece of
code. code.
@ -782,39 +777,39 @@ already-working code.
It is widely believed that the ability to reproduce the results of a published It is widely believed that the ability to reproduce the results of a published
study is important to the scientific community. In practice, however, it is study is important to the scientific community. In practice, however, it is
often hard to impossible: research methodologies, as well as algorithms often hard or impossible: research methodologies, as well as algorithms
themselves, are explained in prose, which, due to the nature of the non-machine themselves, are explained in prose, which, due to the nature of the non-machine
language, lends itself to inexact interpretations. language, lends itself to inexact interpretations.
This article, besides explaining the algorithm in prose, includes the program This article, besides explaining the algorithm in prose, includes the program
of the algorithm in a way that can be executed on reader's workstation. On top of the algorithm in a way that can be executed on reader's workstation. On top
of it, all the illustrations in this paper are generated using that algorithm, of it, all the illustrations in this paper are generated using that algorithm
from a predefined list of test geometries (see from a predefined list of test geometries (see
section~\ref{sec:automated-tests}). section~\ref{sec:automated-tests}).
Besides embedded in this document, this article itself, and code for this This article and accompanying code are accessible on GitHub as of 05/21/2021
article are accessible on github as of 2021-05-21\cite{wmsql}. \cite{wmsql}.
Instructions how to re-generate all the visualizations are in Instructions how to re-generate all the visualizations are in
appendix~\ref{sec:code-regenerate}. The visualization code serves as a good appendix~\ref{sec:code-regenerate}. The visualization code serves as a good
example reference for anyone willing to start using the algorithm. example reference for anyone willing to start using the algorithm.
\section{Algorithm implementation} \section{Algorithm Implementation}
Like alluded in section~\ref{sec:introduction}, {\WM} paper skims over Like alluded in section~\ref{sec:introduction}, {\WM} paper skims over
certain details, which are important to implement the algorithm. This section certain details which are important to implement the algorithm. This section
goes through each algorithm stage, illustrating the intermediate steps and goes through each algorithm stage, illustrating the intermediate steps and
explaining the author's desiderata for a more detailed description. explaining the author's desiderata for a more detailed description.
Illustrations of the following sections are extracted from the automated test Illustrations of the following sections are extracted from the automated test
cases, which were written during the algorithm implementation (as discussed in cases which were written during the algorithm implementation (as discussed in
section~\ref{sec:automated-tests}). section~\ref{sec:automated-tests}).
\subsection{Debugging} \subsection{Debugging}
\label{sec:debugging} \label{sec:debugging}
This implementation includes debugging facilities, in a form of a table This implementation includes debugging facilities in a form of a table
\textsc{wm\_debug}. Table's schema is written in \textsc{wm\_debug}. The table's schema is written in
listing~\ref{lst:wm-debug-sql}. listing~\ref{lst:wm-debug-sql}.
\begin{listing}[h] \begin{listing}[h]
@ -835,20 +830,20 @@ create table wm_debug(
\end{listing} \end{listing}
When debug mode is active, implementation steps will store their results, which When debug mode is active, implementation steps will store their results, which
can be useful to manually inspect results of intermediate actions. Besides can be useful to manually inspect the results of intermediate actions. Besides
manual inspection, most of the figure illustrations in this article are manual inspection, most of the figure illustrations in this article are
visualized from the \textsc{wm\_debug} table. Debugging mode can be activated visualized from the \textsc{wm\_debug} table. Debugging mode can be activated
by passing a non-empty \textsc{dbgname} string to the function by passing a non-empty \textsc{dbgname} string to the function
\textsc{st\_simplifywm} (this function was described in \textsc{st\_simplifywm} (this function was described in
section~\ref{sec:technical-implementation}). By convention, \textsc{dbgname} is section~\ref{sec:technical-implementation}). By convention, \textsc{dbgname} is
the name of the geometry that's being simplified, e.g., \textsc{šalčia}. the name of the geometry that is being simplified, e.g., \textsc{šalčia}. The
Purpose of each column in \textsc{wm\_debug} is described below: purpose of each column in \textsc{wm\_debug} is described below:
\begin{description} \begin{description}
\item[\normalfont\textsc{id}] is a unique identifier for each feature. \item[\normalfont\textsc{id}] is a unique identifier for each feature.
Generated automatically by PostgreSQL. Useful when it is necessary to Generated automatically by PostgreSQL. Useful when it is necessary to
copy one or more features to a separate table for unit tests, like copy one or more features to a separate table for unit tests, as
described in section~\ref{sec:automated-tests}. described in section~\ref{sec:automated-tests}.
\item[\normalfont\textsc{stage}] is the stage of the algorithm. As of \item[\normalfont\textsc{stage}] is the stage of the algorithm. As of
@ -874,17 +869,17 @@ Purpose of each column in \textsc{wm\_debug} is described below:
\end{description} \end{description}
Some of these have sub-stages, which are encoded by a dash and a Some of these have sub-stages which are encoded by a dash and a
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, comes from \item[\normalfont\textsc{name}] is the name of the geometry, which comes from
parameter~\textsc{dbgname}. parameter~\textsc{dbgname}.
\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, 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}.
@ -895,53 +890,53 @@ Purpose of each column in \textsc{wm\_debug} is described below:
\item[\normalfont\textsc{props}] is a free-form JSON object to store \item[\normalfont\textsc{props}] is a free-form JSON object to store
miscellaneous values. For example, \textsc{ebendattrs} phase stores a miscellaneous values. For example, \textsc{ebendattrs} phase stores a
boolean property \textsc{isolated}, which signifies whether the bend is boolean property \textsc{isolated}, which signifies whether the bend is
isolated or not (explained in section~\ref{sec:isolated-bend}. isolated or not (explained in section~\ref{sec:isolated-bend}).
\end{description} \end{description}
When debug mode is turned off (that is, \textsc{dbgname} is left unspecified), When debug mode is turned off (that is, \textsc{dbgname} is left unspecified),
\textsc{wm\_debug} is empty and the algorithm runs slightly faster. \textsc{wm\_debug} is empty and the algorithm runs slightly faster.
\subsection{Merging pieces of the river into one} \subsection{Merging Pieces of the River into One}
Example river geometries were sourced from OpenStreetMap\cite{openstreetmap} Example river geometries were sourced from OpenStreetMap\cite{openstreetmap}
and NŽT\cite{nzt}. Rivers in both data sources are stored in shorter line and NŽT\cite{nzt}. Rivers in both data sources are stored in shorter line
segments, and multiple segments (usually hundreds or thousands for significant segments, and multiple segments (usually hundreds or thousands for significant
rivers) define one full river. While it is convenient to store and edit, these rivers) define one full river. While it is convenient to store and edit, these
segments are not explicitly related to each other. This poses a problem for segments are not explicitly related to each other. This poses a problem for
simplification algorithms, which manipulate on full linear features at a time: simplification algorithms which manipulate on full linear features at a time:
full river geometries, but not their parts. full river geometries, but not their parts.
Since these rivers do not have an explicit relationship to connect them Since these rivers do not have an explicit relationship to connect them
together, they were connected using heuristics: if two line segments share a together, they were connected using heuristics: if two line segments share a
name and are within 500 meters from each other, then they form a single river. name and are within 500 meters from each other, then they form a single river.
For all line simplification algorithms, all rivers need to be combined, and For all line simplification algorithms, all rivers need to be combined and
this way proved to be reasonably effective. Source code for this operation can this way proved to be reasonably effective. Source code for this operation can
be found in listings~\onpage{lst:aggregate-rivers.sql}. be found in listing~\onpage{lst:aggregate-rivers.sql}.
\subsection{Bend scaling and dimensions} \subsection{Bend Scaling And Dimensions}
\label{sec:bend-scaling-and-dimensions} \label{sec:bend-scaling-and-dimensions}
{\WM} accepts a single input parameter: the diameter of a half-circle. If the {\WM} accepts a single input parameter: the diameter of a half-circle. If the
bend's adjusted size (explained in detail in section~\ref{sec:shape-of-a-bend}) bend's adjusted size (explained in detail in section~\ref{sec:shape-of-a-bend})
is greater than the area of the half-circle, then the bend will be left is greater than the area of the half-circle, then the bend will be left
untouched. If the bend's adjusted size is smaller than the area of the provided untouched. If the bend's adjusted size is smaller than the area of the provided
half-circle, the bend will be simplified: either exaggerated, combined or half-circle, the bend will be simplified: either exaggerated, combined, or
eliminated. eliminated.
The extent of line simplification, as well as the half-circle's diameter, The extent of line simplification, as well as the half-circle's diameter,
depends on the desired target scale. Simplification should be more aggressive depends on the desired target scale. Simplification should be more aggressive
for smaller target scales, and less aggressive for larger scales. This section for smaller target scales and less aggressive for larger scales. This section
goes through the process of finding the correct variable to {\WM} algorithm. goes through the process of finding the correct variable to {\WM} algorithm.
What is the minimal, but still eligible figure that can should be displayed on What is the minimal, but still eligible, figure that should be displayed on
the map? the map?
According to \titlecite{cartoucheMinimalDimensions}, the map is typically held According to \titlecite{cartoucheMinimalDimensions}, the map is typically held
at a distance of 30cm. Recommended minimum symbol size given viewing distance at a distance of 30 cm. Recommended minimum symbol size, given viewing distance
of 45cm (1.5 feet) is 1.5mm, as analyzed in \titlecite{mappingunits}. 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 1.5mm 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 1.5mm 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.
@ -999,7 +994,7 @@ are necessary when writing code to detect the bends:
segments) are also the first vertex of the next bend. segments) are also the first vertex of the next bend.
\end{itemize} \end{itemize}
Figure~\ref{fig:fig8-definition-of-a-bend} illustrates article's figure 8, Figure~\ref{fig:fig8-definition-of-a-bend} illustrates the article's figure 8,
but with bends colored as polygons: each color is a distinctive bend. but with bends colored as polygons: each color is a distinctive bend.
\begin{figure}[ht] \begin{figure}[ht]
@ -1012,7 +1007,7 @@ but with bends colored as polygons: each color is a distinctive bend.
\label{fig:fig8-definition-of-a-bend} \label{fig:fig8-definition-of-a-bend}
\end{figure} \end{figure}
\subsection{Gentle Inflection at End of a Bend} \subsection{Gentle Inflection at the End of a Bend}
The gist of the section is in the original article: The gist of the section is in the original article:
@ -1021,7 +1016,7 @@ The gist of the section is in the original article:
would not recognize this as the bend point of a bend would not recognize this as the bend point of a bend
\end{displaycquote} \end{displaycquote}
Figure~\ref{fig:fig5-gentle-inflection} visualizes original paper's figure 5, Figure~\ref{fig:fig5-gentle-inflection} visualizes the original paper's figure 5,
when a single vertex is moved outwards the end of the bend. when a single vertex is moved outwards the end of the bend.
\begin{figure}[ht] \begin{figure}[ht]
@ -1040,15 +1035,17 @@ when a single vertex is moved outwards the end of the bend.
\label{fig:fig5-gentle-inflection} \label{fig:fig5-gentle-inflection}
\end{figure} \end{figure}
The illustration for this section was clear, but insufficient: it does not % TODO: figure should not split the text.
The illustration for this section was clear but insufficient: it does not
specify how many vertices should be included when calculating the end-of-bend specify how many vertices should be included when calculating the end-of-bend
inflection. The iterative approach was chosen --- as long as the angle is inflection. The iterative approach was chosen: as long as the angle is
"right" and the baseline is becoming shorter, the algorithm should keep "right" and the baseline is becoming shorter, the algorithm should keep
re-assigning vertices to different bends; practically not having an upper bound re-assigning vertices to different bends. There is no upper bound
on the number of iterations. 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.
@ -1073,14 +1070,14 @@ fail to match some bends that should be adjusted. Current implementation works
as follows: as follows:
\begin{enumerate} \begin{enumerate}
\item Run the algorithm from beginning to the end. \item Run the algorithm from the beginning to the end.
\item \label{rev1} Reverse the line and each bend. \item \label{rev1} Reverse the line and each bend.
\item Run the algorithm again. \item Run the algorithm again.
\item \label{rev2} Reverse the line and each bend. \item \label{rev2} Reverse the line and each bend.
\item Return result. \item Return result.
\end{enumerate} \end{enumerate}
Reversing the line and its bends is straightforward to implement, but costly: Reversing the line and its bends is straightforward to implement but costly:
the two reversal steps cost additional time and memory. The algorithm could be the two reversal steps cost additional time and memory. The algorithm could be
made more optimal with a similar version of the algorithm, but the one which made more optimal with a similar version of the algorithm, but the one which
goes backwards. In this case, steps \ref{rev1} and \ref{rev2} could be spared, goes backwards. In this case, steps \ref{rev1} and \ref{rev2} could be spared,
@ -1088,10 +1085,10 @@ that way saving memory and computation time.
The "quite small angle" was arbitrarily chosen to \smallAngle. The "quite small angle" was arbitrarily chosen to \smallAngle.
\subsection{Self-line Crossing When Cutting a Bend} \subsection{Self-Line Crossing When Cutting a Bend}
When bend's baseline crosses another bend, it is called self-crossing. When a bend's baseline crosses another bend, it is called self-crossing.
Self-crossing is undesirable for the upcoming bend manipulation operators, thus Self-crossing is undesirable for the upcoming bend manipulation operators; therefore,
should be removed. There are a few rules on when and how they should be removed should be removed. There are a few rules on when and how they should be removed
--- this section explains them in higher detail, discusses their time --- this section explains them in higher detail, discusses their time
complexity and applied optimizations. Figure~\ref{fig:fig6-selfcrossing} is complexity and applied optimizations. Figure~\ref{fig:fig6-selfcrossing} is
@ -1100,17 +1097,19 @@ copied from the original article.
\begin{figure}[ht] \begin{figure}[ht]
\centering \centering
\includegraphics[width=.5\textwidth]{fig6-selfcrossing} \includegraphics[width=.5\textwidth]{fig6-selfcrossing}
\caption{Originally figure 6: bend's baseline (orange) is crossing a neighboring bend.} \caption{Originally figure 6: the bend's baseline (orange) is crossing a neighboring bend.}
\label{fig:fig6-selfcrossing} \label{fig:fig6-selfcrossing}
\end{figure} \end{figure}
\begin{figure}[ht] \begin{figure}[ht]
\centering \centering
\includegraphics[width=.5\textwidth]{selfcrossing-1} \includegraphics[width=.5\textwidth]{selfcrossing-1}
\caption{Bends baseline (orange) is crossing a non-neighboring bend.} \caption{The bend's baseline (orange) is crossing a non-neighboring bend.}
\label{fig:selfcrossing-1-non-neighbor} \label{fig:selfcrossing-1-non-neighbor}
\end{figure} \end{figure}
% TODO: figure should not split the text.
Looking at the {\WM} paper alone, it may seem like self-crossing may happen 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
@ -1121,15 +1120,15 @@ 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
be quite computationally expensive: naively implemented, complexity of checking be quite computationally expensive: naively implemented, complexity of checking
every bend with every bend is $O(n^2)$. In other words, the time it takes to every bend with every bend is $O(n^2)$. In other words, the time it takes to
run the algorithm grows quadratically with the with the number of vertices. run the algorithm grows quadratically with the number of vertices.
It is possible to optimize this step and skip checking a large number of bends. It is possible to optimize this step and skip checking a large number of bends.
Only bends whose sum of inner angles is larger than $180^\circ$ can ever Only bends, the inner angles' sum of which is larger than $180^\circ$, can ever
self-cross. That way, only a fraction of bends need to be checked. The self-cross. That way, only a fraction of bends need to be checked. The
worst-case complexity is still $O(n^2)$, when all bends' inner angles are worst-case complexity is still $O(n^2)$, when all bends' inner angles are
larger than $180^\circ$. Having this optimization, the algorithmic complexity larger than $180^\circ$. Having this optimization, the algorithmic complexity
(as a result, the time it takes to execute the algorithm) is drops by the (as a result, the time it takes to execute the algorithm) drops by the
fraction of bends whose sum of inner angles is smaller than $180^\circ$. fraction of bends, the inner angles' sum of which is smaller than $180^\circ$.
\subsection{Attributes of a Single Bend} \subsection{Attributes of a Single Bend}
@ -1142,12 +1141,12 @@ compactness index is calculated as follows:
\item Construct a polygon by joining first and last vertices of the bend. \item Construct a polygon by joining first and last vertices of the bend.
\item Calculate area of the polygon $A_{p}$. \item Calculate the area of the polygon $A_{p}$.
\item Calculate perimeter $P$ of the polygon. The same value is the \item Calculate perimeter $P$ of the polygon. The same value is the
circumference of the circle: $C = P$. circumference of the circle: $C = P$.
\item Given circle's circumference $C$, circle's area $A_{c}$ is: \item Given the circle's circumference $C$, the circle's area $A_{c}$ is:
\[ \[
A_c = \frac{C^2}{4\pi} A_c = \frac{C^2}{4\pi}
@ -1164,20 +1163,20 @@ compactness index is calculated as follows:
\end{enumerate} \end{enumerate}
Once this operation is complete, each bend will have a list of properties, Once this operation is complete, each bend will have a list of properties
which will be used by other modifying operators. which will be used by other modifying operators.
\subsection{Shape of a Bend} \subsection{Shape of a Bend}
\label{sec:shape-of-a-bend} \label{sec:shape-of-a-bend}
This section introduces \textsc{adjusted size} $A_{adj}$, which trivially This section introduces \textsc{adjusted size} $A_{adj}$ which trivially
derives from \textsc{compactness index} $c$ and "polygonized" bend's area $A_{p}$: derives from \textsc{compactness index} $c$ and "polygonized" bend's area $A_{p}$:
\[ \[
A_{adj} = \frac{0.75 A_{p}}{c} A_{adj} = \frac{0.75 A_{p}}{c}
\] \]
Adjusted size is necessary later to compare bends with each other, or decide if Adjusted size is necessary later to compare bends with each other, or to decide if
the bend is within the simplification threshold. the bend is within the simplification threshold.
Sometimes, when working with {\WM}, it is useful to convert between Sometimes, when working with {\WM}, it is useful to convert between
@ -1201,12 +1200,12 @@ Bend itself and its "isolation" can be described by \textsc{average curvature},
which is \textcquote{wang1998line}{geometrically defined as the ratio of which is \textcquote{wang1998line}{geometrically defined as the ratio of
inflection over the length of a curve.} inflection over the length of a curve.}
Two conditions must be true to claim that a bend is isolated: Two conditions must be followed to claim that a bend is isolated:
\begin{enumerate} \begin{enumerate}
\item \textsc{average curvature} of neighboring bends, should be larger \item \textsc{average curvature} of neighboring bends should be larger
than the "candidate" bend's curvature. The article did not offer a than the "candidate" bend's curvature. The article did not offer a
value, this implementation arbitrarily chose $\isolationThreshold$. value; this implementation arbitrarily chose $\isolationThreshold$.
\item Bends on both sides of the "candidate" bend should be longer than a \item Bends on both sides of the "candidate" bend should be longer than a
certain value. This implementation does not (yet) define such a certain value. This implementation does not (yet) define such a
@ -1214,7 +1213,7 @@ Two conditions must be true to claim that a bend is isolated:
\end{enumerate} \end{enumerate}
\subsection{The Context of a Bend: Isolated and Similar Bends} \subsection{The Context of a Bend: Isolated And Similar Bends}
To find out whether two bends are similar, they are compared by 3 components: To find out whether two bends are similar, they are compared by 3 components:
@ -1256,7 +1255,7 @@ beyond repeating the elimination steps in an illustrated example.
\includegraphics[width=\textwidth]{fig8-elimination-gen3} \includegraphics[width=\textwidth]{fig8-elimination-gen3}
\caption{Iteration 2 (result)} \caption{Iteration 2 (result)}
\end{subfigure} \end{subfigure}
\caption{Originally figure 8: bend elimination through iterations.} \caption{Originally figure 8: the bend elimination through iterations.}
\label{fig:elimination-through-iterations} \label{fig:elimination-through-iterations}
\end{figure} \end{figure}
@ -1267,12 +1266,12 @@ Combination operator was not implemented in this version.
\subsection{Exaggeration Operator} \subsection{Exaggeration Operator}
\label{sec:exaggeration-operator} \label{sec:exaggeration-operator}
Exaggeration operator finds bends of which \textsc{adjusted size} is smaller Exaggeration operator finds bends, of which \textsc{adjusted size} is smaller
than the \textsc{diameter of the half-circle}. Once a target bend is found, it than the \textsc{diameter of the half-circle}. Once a target bend is found, it
will be exaggerated it in increments until either becomes true: will be exaggerated in increments until either becomes true:
\begin{itemize} \begin{itemize}
\item \textsc{adjusted size} of the exaggerated bend is larger than area of \item \textsc{adjusted size} of the exaggerated bend is larger than the area of
the half-circle. the half-circle.
\item The exaggerated bend starts intersecting with a neighboring bend. \item The exaggerated bend starts intersecting with a neighboring bend.
@ -1308,7 +1307,7 @@ implementation. A single exaggeration increment is done as follows:
is linearly interpolated between $[s,1]$, using the same rules as for is linearly interpolated between $[s,1]$, using the same rules as for
the first half. the first half.
First version of the algorithm used simple linear interpolation based The first version of the algorithm used simple linear interpolation based
on the point's position in the line. The current version applies a few on the point's position in the line. The current version applies a few
coefficients, which were derived empirically, by observing the coefficients, which were derived empirically, by observing the
resulting bend. resulting bend.
@ -1331,7 +1330,7 @@ the algorithm.
\section{Results} \section{Results}
\subsection{Generalization results of Analyzed Rivers} \subsection{Generalization Results of Analyzed Rivers}
Figures~\ref{fig:salvis-wm-50k} and~\ref{fig:salvis-wm-250k} visualize Figures~\ref{fig:salvis-wm-50k} and~\ref{fig:salvis-wm-250k} visualize
the generalization result for Šalčia and Visinčia using {\WM} with the the generalization result for Šalčia and Visinčia using {\WM} with the
@ -1343,7 +1342,8 @@ table~\ref{table:scale-halfcirlce-diameter}:
\item 1:\numprint{250000}: 220. \item 1:\numprint{250000}: 220.
\end{itemize} \end{itemize}
\subsubsection{Mid-scale (1:\numprint{50000})} The original feature is orange. As can be seen, some isolated bends are
exaggerated, and some small bends are removed.
\begin{figure}[ht] \begin{figure}[ht]
\centering \centering
@ -1417,11 +1417,11 @@ leaving no space to exaggerate for those between the two. For the same reason,
the 1:\numprint{50000} figure~\ref{fig:salvis-wm-50k} has many smaller bends the 1:\numprint{50000} figure~\ref{fig:salvis-wm-50k} has many smaller bends
at approximately the same location. at approximately the same location.
\subsection{Generalization result comparison with national spatial data sets} \subsection{Comparison of Generalization Result with National Spatial Datasets}
% TODO: GDR50LT and GDR250LT % TODO: GDR50LT and GDR250LT
\subsection{Testing results online} \subsection{Testing Results Online}
\label{sec:testing-results-online} \label{sec:testing-results-online}
An on-line tool\cite{openmapwm} has been developed to test incoming parameters An on-line tool\cite{openmapwm} has been developed to test incoming parameters
@ -1451,18 +1451,18 @@ produces poorly simplified results for some geometries.
\label{sec:conclusions} \label{sec:conclusions}
Classical and modern algorithms line simplification algorithms were evaluated, Classical and modern algorithms line simplification algorithms were evaluated,
main problems with them --- identified. A method for {\WM} technical main problems with them identified. A method for {\WM} technical
implementation was defined, and the algorithm --- implemented. Each geometric implementation was defined, and the algorithm implemented. Each geometric
transformation was described and visualized. The implemented algorithm was transformation was described and visualized. The implemented algorithm was
applied for different shapes and compared to national (Lithuanian) datasets. applied for different shapes and compared to national (Lithuanian) datasets.
About 1000 lines of Procedural SQL were written for the algorithm and tests, About 1,000 lines of Procedural SQL were written for the algorithm and tests,
and a few hundred lines of supporting scripts in Make, Python, Awk, Bash. and a few hundred lines of supporting scripts in Make, Python, Awk, Bash.
Helped by its permissive license and early interest, the algorithm code has With the help of its permissive license and early interest, the algorithm code has
already been used to create a prototype on-line service to evaluate the already been used to create a prototype on-line service to evaluate the
algorithm robustness. algorithm robustness.
\section{Related Work and future suggestions} \section{Related Work And Future Suggestions}
\label{sec:related_work} \label{sec:related_work}
% TODO: write after section~\ref{sec:conclusions} is complete. % TODO: write after section~\ref{sec:conclusions} is complete.
@ -1470,12 +1470,12 @@ algorithm robustness.
\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, Andrius Balčiūnas, for his help in
formulating the requirements and providing early editorial feedback for the 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}
paper in a warm pre-COVID summer evening. I got intrigued. He was also an early paper on a warm pre-COVID summer evening. I got intrigued. He was also an early
beta-tester of my implementation, and helped me understand where the initial beta-tester of my implementation, and helped me understand where the initial
algorithm descriptions were ambiguous. algorithm descriptions were ambiguous.
@ -1485,17 +1485,17 @@ Many thanks to NŽT for providing the datasets with a very permissive license.
\begin{appendices} \begin{appendices}
\section{Code listings} \section{Code Listings}
This section contains code listings of the {\WM} algorithm. This section contains code listings of the {\WM} algorithm.
\subsection{Re-generating this paper} \subsection{Re-Generating This Paper}
\label{sec:code-regenerate} \label{sec:code-regenerate}
Like explained in section~\ref{sec:reproducing-the-paper}, illustrations in Like explained in section~\ref{sec:reproducing-the-paper}, illustrations in
this paper are generated from a small list of sample geometries. To observe this paper are generated from a small list of sample geometries. To observe
the source geometries or regenerate this paper, run this script (assuming the source geometries or regenerate this paper, run this script (assuming
name of this document is \textsc{mj-msc-full.pdf}). the name of this document is \textsc{mj-msc-full.pdf}).
Listing~\ref{lst:extract-and-generate} will extract the source files from Listing~\ref{lst:extract-and-generate} will extract the source files from
the \textsc{mj-msc-full.pdf} to a temporary directory, run the top-level the \textsc{mj-msc-full.pdf} to a temporary directory, run the top-level