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Motiejus Jakštys 2021-05-19 22:57:48 +03:00 committed by Motiejus Jakštys
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Makefile
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@ -132,7 +132,7 @@ salvis-50k_WIDTHDIV = 2
salvis-250k_1SELECT = wm_rivers where name='Šalčia' OR name='Visinčia'
salvis-250k_WIDTHDIV = 10
.faux_test-rivers: tests-rivers.sql wm.sql .faux_db
.faux_test-rivers: tests-rivers.sql wm.sql Makefile .faux_db
./db -v scaledwidth=$(SCALEDWIDTH) -f $<
touch $@

168
mj-msc.tex
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@ -19,11 +19,12 @@
\usepackage{float}
\usepackage{tikz}
\usepackage{fancyvrb}
%\usepackage{charter}
\iffalse
\iftrue
% requires minted
\usepackage{minted}
\newcommand{\inputcode}[2]{\inputminted[fontsize=\small}{#1}{#2}
\newcommand{\inputcode}[2]{\inputminted[fontsize=\small]{#1}{#2}}
\else
% does not require minted
\usepackage{verbatim}
@ -98,18 +99,20 @@ Textwidth in cm: {\printinunitsof{cm}\prntlen{\textwidth}}
\fi
When creating small-scale maps, often the detail of the data source is greater
than desired for the map. This becomes especially acute for natural features
that have many bends, like coastlines, rivers and forest boundaries.
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,
rivers and forest boundaries.
To create a small-scale map from a large-scale data source, these features need
to be generalized: detail should be reduced. However, while doing so, it is
important to preserve the "defining" shape of the original feature, otherwise
the result will look unrealistic.
To create a small-scale map from a large-scale data source, features need to be
generalized: detail should be reduced. While performing the generalization, it
is important to retain the "defining" shape of the original feature. Otherwise,
if the generalized feature looks too different than the original, the result
will look unrealistic.
For example, if a river is nearly straight, it should be nearly straight after
generalization, otherwise a too straightened river will look like a canal.
Conversely, if the river is highly wiggly, the number of bends should be
reduced, but not removed.
reduced, but not removed altogether.
Generalization problem for other objects can often be solved by other
non-geometric means:
@ -121,8 +124,8 @@ non-geometric means:
classification of the road (local, regional, international).
\end{itemize}
Natural line generalization problem can be viewed as having two competing
goals:
To sum up, natural line generalization problem can be viewed as a task of
finding a delicate balance between two competing goals:
\begin{itemize}
\item Reduce detail by removing or simplifying "less important" features.
@ -130,16 +133,21 @@ goals:
\end{itemize}
Given the discussed complexities, a fine line between under-generalization
(leaving object as-is) and over-generalization (making a straight line) must be
found. Therein lies the complexity of generalization algorithms: all have
(leaving object as-is) and over-generalization (making a straight line) needs
to be found. Therein lies the complexity of generalization algorithms: all have
different trade-offs.
\section{Literature review and problematic}
\label{sec:literature-review}
A number of cartographic line generalization algorithms have been researched.
The "classical" ones are {\DP} and {\VW} in combination with Chaikin's. There
are also modern ones.
The "classical" ones are {\DP}\cite{douglas1973algorithms} and
{\VW}\cite{visvalingam1993line} in combination with
Chaikin's\cite{chaikin1974algorithm}.
This section reviews the classical ones, which, besides being around for a long
time, offer easily accessible implementations, as well as more modern ones,
which only theorize, but do not provide an implementation.
\subsection{Available algorithms}
@ -149,25 +157,28 @@ are also modern ones.
"classical" line generalization computer graphics algorithms. They are
relatively simple to implement, require few runtime resources. Both of them
accept only a single parameter, based on desired scale of the map, which makes
them very simple to adjust for different scales.
them straightforward to adjust for different scales.
Both algorithms are part of PostGIS, a free-software GIS suite:
\begin{itemize}
\item {\DP} via
\href{https://postgis.net/docs/ST_Simplify.html}{PostGIS Simplify}.
\href{https://postgis.net/docs/ST_Simplify.html}{PostGIS \texttt{ST\_Simplify}}.
\item {\VW} via
\href{https://postgis.net/docs/ST_SimplifyVW.html}{PostGIS SimplifyVW}.
\href{https://postgis.net/docs/ST_SimplifyVW.html}{PostGIS \texttt{SimplifyVW}}.
\end{itemize}
It may be worthwhile to post-process those through a widely available Chaikin's
line smoothing algorithm\cite{chaikin1974algorithm} via
\href{https://postgis.net/docs/ST_ChaikinSmoothing.html}{PostGIS
ChaikinSmoothing}.
\texttt{ST\_ChaikinSmoothing}}.
To use in generalization examples, we will use two rivers: Žeimena and Šalčia
(they flow into one). Figure~\onpage{fig:salvis-25} illustrates the original
two rivers without any processing (yet).
To use in generalization examples, we will use two rivers: Šalčia and Visinčia.
Figure~\ref{fig:salvis-25} illustrates the original two rivers without any
processing.
These rivers were chosen, because they have both large and small bends, and
thus convenient to analyze for both small and large scale generalization.
\begin{figure}[h]
\centering
@ -177,9 +188,10 @@ two rivers without any processing (yet).
\end{figure}
Same rivers, unprocessed, but with higher density (scales 1:50000 and 1:250000)
are depicted in figure~\onpage{fig:salvis-50-250}. Some river features are so
compact that a reasonably thin line depicting them is overlapping with itself.
As can be seen in the article example, generalization is worthy.
are depicted in figure~\ref{fig:salvis-50-250}. Some river features are so
compact that a reasonably thin line depicting the river is overlapping with
itself, creating a thicker line in print. As a result, generalization for this
river for a smaller scale is worthy.
\begin{figure}[h]
\centering
@ -197,8 +209,6 @@ As can be seen in the article example, generalization is worthy.
\label{fig:salvis-50-250}
\end{figure}
\subsubsection{Modern approaches}
Due to their simplicity and ubiquity, {\DP} and {\VW} have been established as
@ -219,10 +229,12 @@ have emerged. These modern replacements fall into roughly two categories:
\end{itemize}
Authors of most of the aforementioned articles have implemented the
generalization algorithm, at least to generate the visuals in the articles.
However, I wasn't able to find code for any of those to evaluate with my
desired data set, or use as a basis for my own maps. {\WM} \cite{wang1998line}
is available in a commercial product.
generalization algorithm, at least to generate the illustrations in the
articles. However, code is not available for evaluation with a desired data
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
purchase of the commercial product suite, without a way to license the
standalone algorithm.
Lack of robust openly available generalization algorithm implementations poses
a problem for map creation with free software: there is not a similar
@ -233,6 +245,15 @@ open-source tools is an important foundation for future cartographic
experimentation and development, thus it it benefits the cartographic society
as a whole.
{\WM}'s commercial availability signals something about the value of the
algorithm: at least the authors of the commercial software suite deemed it
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
operating system required to run the commercial suite. PostGIS, in contrast, is
free on itself, and runs on free platforms. Therefore, algorithm
implementations that run on PostGIS or other free platforms are useful to a
wider cartographic society than proprietary ones.
\subsection{Problematic with generalization of rivers}
\section{Methodology}
@ -244,14 +265,15 @@ the algorithm from the paper alone.
Explanations in this document are meant to expand, rather than substitute, the
original description in {\WM}. Therefore familiarity with the original paper is
assumed, and, for some sections, having it close-by is necessary to
assumed, and, for some sections, having the original close-by is necessary to
meaningfully follow this document.
In this paper we describe {\WM} in a detail that is more useful for algorithm:
each section will be expanded, with more elaborate and exact illustrations for
every step of the algorithm.
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
with additional commentary, and richer illustrations for non-obvious steps. In
many cases, corner cases are discussed and clarified.
Algorithms discussed in this paper assume Euclidean geometry.
Assume Euclidean geometry throughout this document, unless noted otherwise.
\subsection{Vocabulary and terminology}
@ -267,8 +289,8 @@ This section defines vocabulary and terms as defined in the rest of the paper.
$(x_2, y_2)$. Line Segment and Segment are used interchangeably
throughout the paper.
\item[Line] represents a single linear feature in the real world. For
example, a river or a coastline. {\tt LINESTRING} in GIS terms.
\item[Line] (or \textsc{linestring}) represents a single linear feature in
the real world. For example, a river or a coastline.
Geometrically, A line is a series of connected line segments, or,
equivalently, a series of connected vertices. Each vertex connects to
@ -300,7 +322,7 @@ and the implementation.
Radians & $\nicefrac{\pi}{6}$ & $\nicefrac{\pi}{4}$ & $\nicefrac{\pi}{2}$ & $\pi$ & $2\pi$ \\
\hline
\end{tabular}
\caption{Popular degree and radian values}
\caption{Some angular degree and radian values mentioned in this article.}
\label{table:radians}
\end{table}
@ -314,8 +336,7 @@ algorithm against a predefined set of geometries, and asserts that the output
matches the resulting hand-calculated geometry.
The full set of test geometries is visualized in
figure~\onpage{fig:test-figures}. The figure includes arrows depicting line
direction.
figure~\ref{fig:test-figures}.
\begin{figure}[h]
\centering
@ -330,7 +351,7 @@ unexpected bugs have snug in while modifying the algorithm.
\section{Description of the implementation}
Like alluded in section~\onpage{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
goes through each algorithm stage, illustrating the intermediate steps and
explaining the author's desiderata for a more detailed description.
@ -339,15 +360,23 @@ Illustrations of the following sections are extracted from the automated test
cases, which were written during the algorithm implementation (as discussed in
section~\onpage{sec:automated-tests}).
Lines in illustrations are black, and bends are heavily colored after
converting them to polygons. Bends are converted to polygons (for illustration
purposes) using the following algorithm:
Illustrated lines are black. Bends themselves are linear features.
Discriminating between bends in illustrations might be tricky, because
sometimes a single \textsc{line segment} can belong to two bends.
Given that, there is another way to highlight bends in a schematic drawing: by
converting them to polygons and by altering their background colors. It works
as follows:
\begin{itemize}
\item Join the first and last vertices of the bend, creating a polygon.
\item Color the polygons using distinct colors.
\end{itemize}
This type of illustration works quite well, since polygons created from bends
are almost never overlapping, and discriminating different backgrounds is
easier than discriminating different line shapes or colors.
\subsection{Definition of a Bend}
\label{sec:definition-of-a-bend}
@ -369,20 +398,20 @@ are necessary when writing code to detect the bends:
segments belong to 1 or 2 bends.
\item First and last segments of each bend (except for the two end-line
segments) is also the first vertex of the next bend.
segments) are also the first vertex of the next bend.
\end{itemize}
Properties above may be apparent when looking at illustrations at this article
or reading here, but they are nowhere as such when looking at the original
article.
Figure~\ref{fig:fig8-definition-of-a-bend} illustrates article's Figure 8,
Figure~\ref{fig:fig8-definition-of-a-bend} illustrates article's figure 8,
but with bends colored as polygons: each color is a distinctive bend.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{fig8-definition-of-a-bend}
\caption{Originally Figure 8: detected bends are highlighted}
\caption{Originally figure 8: detected bends are highlighted}
\label{fig:fig8-definition-of-a-bend}
\end{figure}
@ -395,7 +424,7 @@ The gist of the section is in the original article:
would not recognize this as the bend point of a bend
\end{displaycquote}
Figure~\ref{fig:fig5-gentle-inflection} visualizes original paper's Figure 5,
Figure~\ref{fig:fig5-gentle-inflection} visualizes original paper's figure 5,
when a single vertex is moved outwards the end of the bend.
\begin{figure}[h]
@ -409,13 +438,13 @@ when a single vertex is moved outwards the end of the bend.
\includegraphics[width=\textwidth]{fig5-gentle-inflection-after}
\caption{After applying the inflection rule}
\end{subfigure}
\caption{Originally Figure 5: gentle inflections at the ends of the bend}
\caption{Originally figure 5: gentle inflections at the ends of the bend}
\label{fig:fig5-gentle-inflection}
\end{figure}
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
inflection. We chose the iterative approach --- as long as the angle is "right"
inflection. The iterative approach was chosen --- as long as the angle is "right"
and the distance is decreasing, the algorithm should keep re-assigning vertices
to different bends; practically not having an upper bound on the number of
iterations.
@ -423,7 +452,7 @@ iterations.
To prove that the algorithm implementation is correct for multiple vertices,
additional example was created, and illustrated in
figure~\ref{fig:inflection-1-gentle-inflection}: the rule re-assigns two
vertices to the next bend instead of one.
vertices to the next bend.
\begin{figure}[h]
\centering
@ -436,14 +465,15 @@ vertices to the next bend instead of one.
\includegraphics[width=\textwidth]{inflection-1-gentle-inflection-after}
\caption{After applying the inflection rule}
\end{subfigure}
\caption{Gentle inflection at the end of the bend when multiple vertices is moved}
\caption{Gentle inflection at the end of the bend when multiple vertices
are moved}
\label{fig:inflection-1-gentle-inflection}
\end{figure}
To find and fix the gentle bends' inflections requires to run the algorithm in
both directions; if implemented as documented, the steps will fail to match
some bends that should be mutated. This implementation does it in the following
way:
Note that to find and fix the gentle bends' inflections, the algorithm should
run twice, both ways. Otherwise, if it is executed only one way, the steps will
fail to match some bends that should be adjusted. Current implementation works
as follows:
\begin{enumerate}
\item Run the algorithm from beginning to the end.
@ -453,17 +483,18 @@ way:
\item Return result.
\end{enumerate}
The current implementation is the most straightforward, but not optimal:
reversing of lines and bends could be avoided by walking backwards the lines.
In this case, steps \ref{rev1} and \ref{rev2} could be spared, thus saving
memory and computation time.
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
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,
that way saving memory and computation time.
The "quite small angle" was arbitrarily chosen to $\smallAngle$.
\subsection{Self-line Crossing When Cutting a Bend}
When bend's baseline crosses another bend, it is called self-crossing. This is
undesirable in the upcoming operators, and self-crossings should be removed
When bend's baseline crosses another bend, it is called self-crossing.
Self-crossing is undesirable in the upcoming operators, thus should be removed
following the rules of the article.
\begin{figure}[h]
@ -477,14 +508,15 @@ following the rules of the article.
\includegraphics[width=\textwidth]{fig6-selfcrossing-after}
\caption{Self-crossing removed following the algorithm}
\end{subfigure}
\caption{Originally Figure 6: simple case of self-line crossing}
\caption{Originally figure 6: simple case of self-line crossing}
\label{fig:fig6-selfcrossing}
\end{figure}
The original description does not go into detail which bends may self-cross, and which <TBD>
The self-line-crossing may happen not by the neighboring bend, but by any other
bend in the line. For example, the baseline of the bend $(A, B)$ may cross
different bends in between, as depicted in
figure~\onpage{fig:selfcrossing-1-non-neighbor}.
bend in the line. For example, the baseline of the bend may cross different
bends in between, as depicted in figure~\ref{fig:selfcrossing-1-non-neighbor}.
\begin{figure}[h]
\centering
@ -632,7 +664,7 @@ We strongly believe in the ability to reproduce the results is critical for any
This was tested on Linux Debian 11 with upstream packages only.
\subsection{Algorithm code listings}
\inputcode{postgresql}{wm.sql}
%\inputcode{postgresql}{wm.sql}
\end{appendices}
\end{document}