\documentclass[a4paper]{article} \usepackage[T1]{fontenc} \usepackage[american]{babel} \usepackage[utf8]{inputenc} \usepackage{fvextra} \usepackage[autostyle,english=american]{csquotes} \MakeOuterQuote{"} \usepackage[ maxbibnames=99, style=numeric, sorting=none, alldates=iso, seconds=true ]{biblatex} \addbibresource{bib.bib} \usepackage[ pdfusetitle, pdfkeywords={Line Generalization,Line Simplification,Wang--Mueller}, pdfborderstyle={/S/U/W 0} % /S/U/W 1 to enable reasonable decorations ]{hyperref} \usepackage{enumitem} \usepackage[toc,page,title]{appendix} \usepackage{caption} \usepackage{subcaption} \usepackage{dcolumn} \usepackage{gensymb} \usepackage{units} \usepackage{varwidth} \usepackage{tabularx} \usepackage{float} \usepackage{numprint} \usepackage{tikz} \usetikzlibrary{shapes.geometric,arrows,positioning} \usepackage{fancyvrb} \usepackage{layouts} \usepackage{minted} %\usepackage{charter} %\usepackage{setspace} %\doublespacing \input{version.inc} \input{vars.inc} \newcommand{\onpage}[1]{\ref{#1} on page~\pageref{#1}} \newcommand{\titlecite}[1]{\citetitle{#1}\cite{#1}} \newcommand{\titleciteauthor}[1]{\citetitle{#1} by \citeauthor{#1}\cite{#1}} \newcommand{\DP}{Douglas \& Peucker} \newcommand{\VW}{Visvalingam--Whyatt} \newcommand{\WM}{Wang--M{\"u}ller} \newcommand{\WnM}{Wang and M{\"u}ller} \newcommand{\WirM}{Wang ir M{\"u}ller} % {\WM} algoritmo realizacija kartografinei upių generalizacijai \newcommand{\MYTITLE}{{\WM} algorithm realization for cartographic line generalization} \newcommand{\MYTITLENOCAPS}{wang--m{\"u}ller algorithm realization for cartographic line generalization} \newcommand{\MYAUTHOR}{Motiejus Jakštys} \newcommand{\inputcode}[2]{\inputminted[fontsize=\small]{#1}{#2}} \newenvironment{longlisting}{\captionsetup{type=listing}}{} \title{\MYTITLE} \author{\MYAUTHOR} \date{\VCDescribe} \begin{document} \begin{titlepage} \begin{center} \includegraphics[width=0.2\textwidth]{vu.pdf} \\[4ex] \large \textbf{\textsc{ vilnius university \\ faculty of chemistry and geosciences \\ department of cartography and geoinformatics }} \\[8ex] \textbf{\MYAUTHOR} \\[8ex] \normalsize A Thesis Presented for the Degree of Master in Cartography \\[8ex] \LARGE \textbf{\textsc{\MYTITLENOCAPS}} \vfill \normalsize Supervisor Dr. Andrius Balčiūnas \\[16ex] \VCDescribe \end{center} \end{titlepage} \begin{abstract} \label{sec:abstract} Currently available line simplification algorithms are rooted in mathematics and geometry, and are unfit for bendy map features like rivers and coastlines. {\WnM} observed how cartographers simplify these natural features and created an algorithm. We implemented this algorithm and documented it in great detail. Our implementation makes {\WM} algorithm freely available in PostGIS, and this paper explains it. \vfill Šiuo metu esami linijų supaprastinimo algoritmai yra kilę iš matematikos ir geometrijos, bet nėra tinkami lankstiems geografiniams objektams, tokiems kaip upės ir pakrantės, atvaizduoti. {\WirM} ištyrė, kaip kartografai atlieka upių generalizaciją, ir sukūrė algoritmą. Mes realizavome šį algoritmą ir išsamiai jį dokumentavome. Mūsų {\WM} realizacija ir dokumentacija yra nemokamos ir laisvai prieinamos, naudojant PostGIS platformą. \end{abstract} \clearpage \tableofcontents \listoftables \listoflistings \newpage \section{Introduction} \label{sec:introduction} \iffalse NOTICE: this value should be copied to layer2img.py:TEXTWIDTH, so dimensions of inline images are reasonable. 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. While many features can be removed or simplified, it is more tricky with natural features that have many bends, like coastlines, rivers, or forest boundaries. 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 simplification, it is important to retain the "defining" shape of the original feature. Otherwise, if the simplified feature looks too different from the original, the result will look unrealistic. Simplification problem for some objects can often be solved by non-geometric means: \begin{itemize} \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 classification of the road (local, regional, international). \end{itemize} However, things are not as simple for natural features like rivers or coastlines. If a river is nearly straight, it should remain such after simplification. An overly straightened river will look like a canal, and the other way around --- too curvy would not reflect the natural shape. Conversely, if the river originally is highly wiggly, the number of bends should be reduced, but not removed altogether. Natural line simplification 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. \item Retain enough detail, so the original is still recognizable. \end{itemize} Given the discussed complexities with natural features, a fine line between under-simplification (leaving an object as-is) and over-simplification (making a straight line) needs to be found. Therein lies the complexity of simplification algorithms: all have different trade-offs. The purpose of the thesis is to implement a cartographic line generalization algorithm on the basis of {\WM} algorithm, using open-source software. Tasks: \begin{itemize} \item Evaluate existing line simplification algorithms. \item Identify main river generalization problems, using classical line simplification algorithms. \item Define the method of the {\WM} technical implementation. \item Realize {\WM} algorithm technically, explaining the geometric transformations in detail. \item Apply the created algorithm for different datasets and compare the results with national datasets. \end{itemize} Scientific relevance of this work --- the simplification processes (steps) described by the {\WM} algorithm --- are analyzed in detail, practically implemented, and the implementation is described. That expands the knowledge of cartographic theory about the generalization of natural objects' boundaries after their natural defining properties. 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; the steps are specified in this work. Practically, this work makes it possible to use open-source software to perform cartographic line generalization. The developed specialized cartographic line simplification algorithm can be applied by cartographers to implement automatic data generalization solutions. Given the open-source nature of this work, the algorithm implementation can be modified freely. \section{Literature Review And Problematic} \label{sec:literature-review-problematic} \subsection{Available Algorithms} This section reviews the classical line simplification algorithms, 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. \subsubsection{{\DP}, {\VW} and Chaikin's} \label{sec:dp-vwchaikin} {\DP}\cite{douglas1973algorithms} and {\VW}\cite{visvalingam1993line} are "classical" line simplification computer graphics algorithms. They are 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 straightforward to adjust for different scales. Both algorithms are available in PostGIS, a free-software GIS suite: \begin{itemize} \item {\DP} via \href{https://postgis.net/docs/ST_Simplify.html}{PostGIS \textsc{st\_simplify}}. \item {\VW} via \href{https://postgis.net/docs/ST_SimplifyVW.html}{PostGIS \textsc{st\_simplifyvw}}. \end{itemize} It may be worthwhile to post-process those through Chaikin's line smoothing algorithm\cite{chaikin1974algorithm} via \href{https://postgis.net/docs/ST_ChaikinSmoothing.html}{PostGIS \textsc{st\_chaikinsmoothing}}. In generalization examples, we will use two rivers: Šalčia and Visinčia. These rivers were chosen because they have both large and small bends, and thus are convenient to analyze for both small- and large-scale generalization. Figure~\onpage{fig:salvis-25} illustrates the original two rivers without any simplification. \begin{figure}[ht] \centering \includegraphics[width=\textwidth]{salvis-25k} \caption{Example rivers for visual tests (1:{\numprint{25000}}).} \label{fig:salvis-25} \end{figure} \begin{figure}[ht] \centering \begin{subfigure}[b]{.49\textwidth} \includegraphics[width=\textwidth]{salvis-2x50k} \caption{Example scaled 1:\numprint{50000}.} \label{fig:salvis-2x50k} \end{subfigure} \hfill \begin{subfigure}[b]{.49\textwidth} \centering \includegraphics[width=.2\textwidth]{salvis-250k-10x} \caption{Example scaled 1:\numprint{250000}.} \end{subfigure} \caption{Down-scaled original river.} \label{fig:salvis-50-250} \end{figure} Same rivers, unprocessed but in higher scales (1:\numprint{50000} and 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 is touching itself, creating a thicker line. We can assume that some simplification for scale 1:\numprint{50000} and especially for 1:\numprint{250000} is worthwhile. \begin{figure}[ht] \centering \begin{subfigure}[b]{.49\textwidth} \includegraphics[width=\textwidth]{salvis-dp64-2x50k} \caption{Using {\DP}.} \end{subfigure} \hfill \begin{subfigure}[b]{.49\textwidth} \includegraphics[width=\textwidth]{salvis-vw64-2x50k} \caption{Using {\VW}.} \end{subfigure} \caption{Simplified using classical algorithms (1:\numprint{50000}).} \label{fig:salvis-generalized-1x50k} \end{figure} Figure~\ref{fig:salvis-generalized-1x50k} illustrates the same river bend, but 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, traditionally, Chaikin's\cite{chaikin1974algorithm} is applied after generalization, illustrated in Figure~\ref{fig:salvis-generalized-chaikin-1x50k}. \begin{figure}[ht!] \centering \begin{subfigure}[b]{.49\textwidth} \includegraphics[width=\textwidth]{salvis-dpchaikin64-2x50k} \caption{{\DP} and Chaikin's.} \label{fig:salvis-dpchaikin64-2x50k} \end{subfigure} \hfill \begin{subfigure}[b]{.49\textwidth} \includegraphics[width=\textwidth]{salvis-vwchaikin64-2x50k} \caption{{\VW} and Chaikin's.} \label{fig:salvis-vwchaikin64-2x50k} \end{subfigure} \caption{Simplified and smoothened river (1:\numprint{50000}).} \label{fig:salvis-generalized-chaikin-1x50k} \end{figure} \begin{figure}[ht!] \centering \begin{subfigure}[b]{.49\textwidth} \includegraphics[width=\textwidth]{salvis-overlaid-dpchaikin64-2x50k} \caption{Original (fig.~\ref{fig:salvis-2x50k}) and simplified (fig.~\ref{fig:salvis-dpchaikin64-2x50k}).} \end{subfigure} \hfill \begin{subfigure}[b]{.49\textwidth} \includegraphics[width=\textwidth]{salvis-overlaid-vwchaikin64-2x50k} \caption{Original (fig.~\ref{fig:salvis-2x50k}) and simplified (fig.~\ref{fig:salvis-vwchaikin64-2x50k}.)} \end{subfigure} \caption{Zoomed-in simplified and smoothened river and original.} \label{fig:salvis-overlaid-generalized-chaikin-1x50k} \end{figure} \begin{figure}[b] \centering \includegraphics[width=\textwidth]{amalgamate1} \caption{Narrow bends amalgamating into thick unintelligible blobs.} \label{fig:pixel-amalgamation} \end{figure} The resulting simplified and smoothened example (Figure~\onpage{fig:salvis-generalized-chaikin-1x50k}) yields a more aesthetically pleasing result; however, it obscures natural river features. Given the absence of rocks, the only natural features that influence the river direction are topographic: \begin{itemize} \item Relatively straight river (completely straight or with small-angled bends over a relatively long distance) implies greater slope, more water, and/or faster flow. \item Bendy river, on the contrary, implies slower flow, slighter slope, and/or less water. \end{itemize} Both {\VW} and {\DP} have a tendency to remove the small bends altogether, removing a valuable characterization of the river. Sometimes low-water rivers in slender slopes have many bends next to each other. In low resolutions (either in small-DPI screens or paper, or when the river is sufficiently zoomed out, or both), the small bends will amalgamate to a unintelligible blob. Figure~\ref{fig:pixel-amalgamation} illustrates a real-world example where a bendy river, normally 1 or 2 pixels wide, creates a wide area, of which the shapes of the bend become unintelligible. In this example, classical algorithms would remove these bends altogether. A cartographer would retain a few of those distinctive bends, but would increase the distance between the bends, remove some of the bends, or both. % TODO: figues shouldn't split the sentence. 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. \clearpage \subsubsection{Modern Approaches} Due to their simplicity and ubiquity, {\DP} and {\VW} have been established as go-to algorithms for line generalization. During recent years, alternatives have emerged. These modern replacements fall into roughly two categories: \begin{itemize} \item Cartographic knowledge was encoded to an algorithm (bottom-up approach). One among these are \titlecite{wang1998line}, also known as {\WM}'s algorithm. \item Mathematical shape transformation which yields a more cartographic result. E.g., \titlecite{jiang2003line}, \titlecite{dyken2009simultaneous}, \titlecite{mustafa2006dynamic}, \titlecite{nollenburg2008morphing}, \titlecite{devangleserrorbends}. \end{itemize} Authors of most of the aforementioned articles have implemented the generalization algorithm, at least to generate the illustrations in the 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, {\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. {\WM} algorithm was created by encoding professional cartographers' knowledge into a computer algorithm. It has a few main properties which make it especially suitable for generalization of natural linear features: \begin{figure}[h!] \centering \includegraphics[width=.8\textwidth]{wang125} \caption{Figure 12.5 in \cite{wang1998line}: example of cartographic line generalization.} \label{fig:wang125} \end{figure} \begin{itemize} \item Small bends are not always removed, but either combined (e.g., 3 bends into 2), exaggerated, or removed, depending on the neighboring bends. \item Long and gentle bends are not straightened, but kept as-is. \end{itemize} As a result of these properties, {\WM} algorithm retains the defining properties of the natural features: high-current rivers keep their appearance as such, instead of becoming canals; low-stream bendy rivers retain their frequent small bends. Figure~\ref{fig:wang125}, sub-figure labeled "proposed method" (from the original \titlecite{wang1998line}) illustrates the {\WM} algorithm. \subsection{Problematic with Generalization of Rivers} This section introduces the reader to simplification and generalization, and discusses two main problems with current-day automatic cartographic line generalization: \begin{itemize} \item Currently available line simplification algorithms were created to simplify geometries, but do not encode cartographic knowledge. \item Existing cartographic line generalization algorithms are not freely accessible. \end{itemize} \subsubsection{Simplification versus Generalization} It is important to note the distinction between simplification, line generalization, and cartographic generalization. Simplification reduces an object's detail in isolation, not taking the object's natural properties or surrounding objects into account. For example, if a river is simplified, it may have an approximate shape of the original river, but lose some shapes that define it. For example: \begin{itemize} \item Low-water rivers in slender slopes have many small bends next to each other. A non-cartographic line simplification may remove all of them, thus losing an important river's characteristic feature: after such simplification, it will be hard to tell that the original river was low-water in a slender slope. \item Low-angle river bend river over a long distance differs significantly from a completely straight canal. Non-cartographic line simplification may replace that bend with a straight line, making the river more similar to a canal than a river. \end{itemize} In other words, simplification processes the line, ignoring its geographic features. It works well when the features are human-made (e.g., roads, administrative boundaries, buildings). There is a number of freely available non-cartographic line simplification algorithms, which this paper will review. Contrary to line simplification, cartographic generalization does not focus into a single feature class (e.g., rivers), but the whole map. For example, line simplification may change river bends in a way that bridges (and roads to the bridges) become misplaced. While line simplification is limited to a single feature class, cartographic generalization is not. Fully automatic cartographic generalization is not yet a solved problem. % . Cartographic line generalization falls in between the two: it does more than line simplification, and less than cartographic generalization. Cartographic line generalization deals with a single feature class, takes into account its geographic properties, but ignores other features. This paper examines {\WM}'s \titlecite{wang1998line}, a cartographic line generalization algorithm. \subsubsection{Availability of Generalization Algorithms} Lack of robust openly available generalization algorithm implementations poses a problem for map creation with free software: there is no high-quality 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 quality. We believe that the availability of high-quality open-source tools is an important foundation for future cartographic experimentation and development, thus 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 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. \subsubsection{Unfitness of Line Simplification Algorithms} Section~\ref{sec:dp-vwchaikin} illustrates the current gaps with line simplification algorithms for real rivers. To sum up, we highlight the following cartographic problems from our examples: \begin{description} \item[Long bends] should remain as long bends, instead of becoming fully straight lines. \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 small to be visible, they should be combined or exaggerated. \end{description} We are limiting the problem to cartographic line generalization. That is, full cartographic generalization, which takes topology and other feature classes into account, is out of scope. Figure~\onpage{fig:wang125} illustrates {\WM} algorithm from their original paper. Note how the long bends retain curvy, and how some small bends get exaggerated. \section{Methodology} \label{sec:methodology} The original {\WM}'s algorithm \cite{wang1998line} leaves something to be desired for a practical implementation: it is not straightforward to implement 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 the original close-by is necessary to meaningfully follow this document. 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 illustrations added for non-obvious steps. Corner cases are discussed, too. \subsection{Main Geometry Elements Used by Algorithm} \label{sec:vocab} This section defines and explains the geometry elements that are used throughout this paper and the implementation. Assume Euclidean geometry throughout this document, unless noted otherwise. \begin{description} \item[\normalfont\textsc{vertex}] is a point on a plane, can be expressed by a pair of $(x,y)$ coordinates. \item[\normalfont\textsc{line segment}] or \textsc{segment} joins 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 segment are used interchangeably. \item[\normalfont\textsc{line}] or \textsc{linestring} represents a single linear feature. 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 two other vertices, with the exception of the vertices at either ends of the line: these two connect to a single other vertex. \item[\normalfont\textsc{multiline}] or \textsc{multilinestring} is a collection of linear features. Throughout this implementation, this is used rarely (normally, a river is a single line) but can be valid when, for example, a river has an island. \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 section~\ref{sec:definition-of-a-bend}. \item[\normalfont\textsc{baseline}] is a line between the bend's first and last vertices. \item[\normalfont\textsc{sum of inner angles}] is a measure of how "curved" 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. \item[\normalfont\textsc{algorithmic complexity}] measured in \textsc{big o notation}, is a relative measure that helps explain how long\footnote{the upper bound, i.e., the worst case.} the algorithm will run depending on its input. It is widely used in computing science when discussing the efficiency of a given algorithm. For example, given $n$ objects and time complexity of $O(log(n))$, the time it takes to execute the algorithm is logarithmic to $n$. Conversely, if complexity is $O(n^2)$, then the time it takes to execute the algorithm grows quadratically with input. Importantly, if the input size doubles, the time it takes to run the algorithm quadruples. \textsc{big o notation} was first suggested by Bachmann\cite{bachmann1894analytische} and Landau\cite{landau1911} in late \textsc{xix} century, and clarified and popularized for computing science by Donald Knuth\cite{knuth1976big} in the 1970s. \end{description} \clearpage \subsection{Algorithm Implementation Process} \label{sec:algorithm-implementation-process} \tikzset{ startstop/.style={trapezium,text centered,minimum height=2em, trapezium left angle=70,trapezium right angle=110,draw=black,fill=red!20}, proc/.style={rectangle,minimum height=2em,text centered,draw=black, fill=orange!20}, decision/.style={diamond,minimum height=2em,text centered,aspect=3, draw=black,fill=green!20}, arrow/.style={thick,->,>=stealth}, } \begin{figure}[!ht] \centering \begin{tikzpicture}[node distance=1.5cm,auto] \node (start) [startstop] {Read \textsc{linestring}}; \node (detect) [proc,below of=start] {Detect bends}; \node (inflections) [proc,below of=detect] {Fix gentle inflections}; \node (selfcrossing) [proc,below of=inflections] {Eliminate self-crossing}; \node (mutated1) [decision,below of=selfcrossing] {Mutated?}; \node (bendattrs) [proc,below of=mutated1] {Compute bend attributes}; \node (exaggeration) [proc,below of=bendattrs] {Exaggeration}; \node (mutated2) [decision,below of=exaggeration] {Mutated?}; \node (elimination) [proc,below of=mutated2] {Elimination}; \node (mutated3) [decision,below of=elimination] {Mutated?}; \node (stop) [startstop,below of=mutated3] {Stop}; \coordinate [right of=mutated1,node distance=5cm] (mutated1y) {}; \coordinate [right of=mutated2,node distance=5cm] (mutated2y) {}; \coordinate [right of=mutated3,node distance=5cm] (mutated3y) {}; \draw [arrow] (start) -- (detect); \draw [arrow] (detect) -- (inflections); \draw [arrow] (inflections) -- (selfcrossing); \draw [arrow] (selfcrossing) -- (mutated1); \draw [arrow] (mutated1) -| node [near start] {Yes} (mutated1y) |- (detect); \draw [arrow] (mutated1) -- node[anchor=west] {No} (bendattrs); \draw [arrow] (bendattrs) -- (exaggeration); \draw [arrow] (exaggeration) -- (mutated2); \draw [arrow] (mutated2) -| node [near start] {Yes} (mutated2y) |- (detect); \draw [arrow] (mutated2) -- node[anchor=west] {No} (elimination); \draw [arrow] (mutated3) -| node [near start] {Yes} (mutated3y) |- (detect); \draw [arrow] (mutated3) -- node[anchor=west] {No} (stop); \draw [arrow] (elimination) -- (mutated3); \end{tikzpicture} \caption{Flow chart of the implementation workflow.} \label{fig:flow-chart} \end{figure} Figure~\ref{fig:flow-chart} visualizes the algorithm steps for each line. \textsc{multilinestring} features are split to \textsc{linestring} features and executed in order. Judging from {\WM} prototype flow chart (depicted in figure 11 of the original paper), their approach is iterative for the line: it will process the line in sequence, doing all steps, before moving on to the next step. We will call this approach "streaming", because it does not require to have the full line to process it. 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: \begin{itemize} \item For \textsc{eliminate self-crossing} stage, when it finds a bend with the right sum of inflection angles, it checks the whole line for self-crossings. This is impossible with streaming because it requires having the full line in memory. It could be optimized by, for example, looking for a fixed number of neighboring bends (say, 10), but that would complicate the implementation. \item \textsc{fix gentle inflections} is iterating the same line twice from opposite directions. That could be re-written to streaming fashion, but it complicates the implementation, too. \end{itemize} 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 processing), and some steps are unnecessarily repeated, like re-computing the bend's attributes during repeated iterations. \subsection{Technical Implementation} \label{sec:technical-implementation} Technical algorithm realization was created in \titlecite{postgis311}. PostGIS is a PostgreSQL extension for working with spatial data. PostgreSQL is an open-source relational database, widely used in industry and academia. PostgreSQL can be interfaced from nearly any programming language; therefore, solutions written in PostgreSQL (and their extensions) are usable in many environments. On top of that, PostGIS implements a rich set of functions\cite{postgisref} for working with geometric and geographic objects. Due to its wide applicability and rich library of spatial functions, PostGIS is the implementation language of the {\WM} algorithm. The implementation exposes the entrypoint function \textsc{st\_simplifywm}, in listing~\ref{lst:st-simplifywm}. \begin{listing} \begin{minted}[fontsize=\small]{sql} create function ST_SimplifyWM( geom geometry, dhalfcircle float, intersect_patience integer default 10, dbgname text default null ) returns geometry \end{minted} \caption{Function \textsc{st\_simplifywm}.} \label{lst:st-simplifywm} \end{listing} This function accepts the following parameters: \begin{description} \item[\normalfont\textsc{geom}] is the input geometry. Either \textsc{linestring} or \textsc{multilinestring}. \item[\normalfont\textsc{dhalfcircle}] is the diameter of the half-circle. Explained in section~\ref{sec:bend-scaling-and-dimensions}. \item[\normalfont\textsc{intersect\_patience}] is an optional parameter to exaggeration operator, explained in section~\ref{sec:exaggeration-operator}. \item[\normalfont\textsc{dbgname}] is an optional human-readable name of the figure. Explained in section~\ref{sec:debugging}. \end{description} The function \textsc{st\_simplifywm} calls into helper functions, which detect, transform, or remove bends. These helper functions are also defined in the implementation and are part of the algorithm technical realization. All supporting functions use spatial manipulation functions provided by PostGIS. \subsection{Automated Tests} \label{sec:automated-tests} As part of the algorithm realization, an automated test suite has been developed. Shapes to test each function have been hand-crafted, and expected results have been manually calculated. The test suite executes parts of the algorithm against a predefined set of geometries, and asserts that the output matches the resulting hand-calculated geometries. The full set of test geometries is visualized in Figure~\ref{fig:test-figures}. \begin{figure}[ht] \centering \includegraphics[width=\textwidth]{test-figures} \caption{Geometries for automated test cases.} \label{fig:test-figures} \end{figure} 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 unexpected bugs have snug in while modifying the algorithm. We will explain two instances when automated tests were very useful during the implementation: \begin{itemize} \item Created a function \textsc{wm\_exaggeration}, which exaggerates bends following the rules. It worked well over simple geometries but, due to 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. 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 that it was correctly exaggerated. \item During algorithm development, automated tests run about once a minute. They quickly find logical and syntax errors. In contrast, running the algorithm with real rivers takes a few minutes, which increases the feedback loop, and takes longer to fix the "simple" errors. \end{itemize} 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 code. Besides testing for specific cases, an automated test suite ensures future stability and longevity of the implementation itself: when new contributors start changing code, they have higher assurance they have not broken an already-working code. \subsection{Reproducibility} \label{sec:reproducing-the-paper} 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 often hard or impossible: research methodologies, as well as algorithms themselves, are explained in prose, which, due to the nature of the non-machine language, lends itself to inexact interpretations. 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 it, all the illustrations in this paper are generated using that algorithm from a predefined list of test geometries (see section~\ref{sec:automated-tests}). This article and accompanying code are accessible on GitHub as of 2021-05-19 \cite{wmsql}. Instructions how to re-generate all the visualizations are in appendix~\ref{sec:code-regenerate}. The visualization code serves as a good example reference for anyone willing to start using the algorithm. \section{Algorithm Implementation} As 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. Illustrations of the following sections are extracted from the automated test cases which were written during the algorithm implementation (as discussed in section~\ref{sec:automated-tests}). \subsection{Debugging} \label{sec:debugging} This implementation includes debugging facilities in a form of a table \textsc{wm\_debug}. The table's schema is written in listing~\ref{lst:wm-debug-sql}. When debug mode is active, implementation steps will store their results, which can be useful to manually inspect the results of intermediate actions. Besides manual inspection, most of the figure illustrations in this article are visualized from the \textsc{wm\_debug} table. Debugging mode can be activated by passing a non-empty \textsc{dbgname} string to the function \textsc{st\_simplifywm} (this function was described in section~\ref{sec:technical-implementation}). By convention, \textsc{dbgname} is 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: \begin{description} \item[\normalfont\textsc{id}] is a unique identifier for each feature. Generated automatically by PostgreSQL. Useful when it is necessary to copy one or more features to a separate table for unit tests, as described in section~\ref{sec:automated-tests}. \item[\normalfont\textsc{stage}] is the stage of the algorithm. As of writing, there are a few: \begin{description} \item[\normalfont\textsc{afigures}] at the beginning of the loop. \item[\normalfont\textsc{bbends}] after bends are detected. \item[\normalfont\textsc{cinflections}] after gentle inflections are fixed. \item[\normalfont\textsc{dcrossings}] after self-crossings are eliminated. \item[\normalfont\textsc{ebendattrs}] after bend attributes are calculated. \item[\normalfont\textsc{gexaggeration}] after bends have been exaggerated. \item[\normalfont\textsc{helimination}] after bends have been eliminated. \end{description} Some of these have sub-stages which are encoded by a dash and a sub-stage name, e.g., \textsc{bbends-polygon} creates polygon geometries after polygons have been detected; this particular example is used to generate colored polygons in Figure~\ref{fig:fig8-definition-of-a-bend}. \item[\normalfont\textsc{name}] is the name of the geometry, which comes from parameter~\textsc{dbgname}. \item[\normalfont\textsc{gen}] is the top-level iteration number. In other words, the number of times the execution flow passes through \textsc{detect bends} phase as depicted in Figure~\onpage{fig:flow-chart}. \item[\normalfont\textsc{nbend}] is the bend's index in its \textsc{line}. \item[\normalfont\textsc{way}] is the geometry column. \item[\normalfont\textsc{props}] is a free-form JSON object to store miscellaneous values. For example, \textsc{ebendattrs} phase stores a boolean property \textsc{isolated}, which signifies whether the bend is isolated or not (explained in section~\ref{sec:isolated-bend}). \end{description} When debug mode is turned off (that is, \textsc{dbgname} is left unspecified), \textsc{wm\_debug} is empty and the algorithm runs slightly faster. \begin{listing}[h!] \begin{minted}[fontsize=\small]{sql} drop table if exists wm_debug; create table wm_debug( id serial, stage text not null, name text not null, gen bigint not null, nbend bigint, way geometry, props jsonb ); \end{minted} \caption{\textsc{wm\_debug} table definition} \label{lst:wm-debug-sql} \end{listing} \subsection{Merging Pieces of a River into One} Example river geometries were sourced from OpenStreetMap\cite{openstreetmap} and NŽT\cite{nzt}. Rivers in both data sources are stored in shorter line segments, and multiple segments (usually hundreds or thousands for significant 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 simplification algorithms which manipulate on full linear features at a time: full river geometries, but not their parts. Since these rivers do not have an explicit relationship to connect them 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. 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 be found in listing~\onpage{lst:aggregate-rivers.sql}. \subsection{Bend Scaling And Dimensions} \label{sec:bend-scaling-and-dimensions} {\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}) 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 half-circle, the bend will be simplified: either exaggerated, combined, or eliminated. The extent of line simplification, as well as the half-circle's diameter, depends on the desired target scale. Simplification should be more aggressive for smaller target scales and less aggressive for larger scales. This section goes through the process of finding the correct variable to {\WM} algorithm. What is the minimal, but still eligible, figure that should be displayed on the map? According to \titlecite{cartoucheMinimalDimensions}, the map is typically held at a distance of 30 cm. Recommended minimum symbol size, given viewing distance 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 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 scale, will not be simplified. \begin{figure}[ht] \centering \begin{tikzpicture}[x=1mm,y=1mm] \draw[] (-10, 0) -- (-.75,0) arc (225:-45:.75) -- (10, 0); \end{tikzpicture} \caption{Smallest feature that will be not simplified (to scale).} \label{fig:half-circle} \end{figure} \begin{table}[h!] \centering \begin{tabular}{ c D{.}{.}{1} } Scale & \multicolumn{1}{c}{$D(m)$} \\ \hline 1:\numprint{10000} & 15 \\ 1:\numprint{15000} & 22.5 \\ 1:\numprint{25000} & 37.5 \\ 1:\numprint{50000} & 75 \\ 1:\numprint{250000} & 220 \\ \end{tabular} \caption{{\WM} half-circle diameter $D$ for popular scales.} \label{table:scale-halfcirlce-diameter} \end{table} {\WM} algorithm does not have a notion of scale, but it does have a notion of distance: it accepts a single parameter $D$, the half-circle's diameter. Assuming measurement units in projected coordinate system are meters (for example, \titlecite{epsg3857}), some popular scales are highlighted in table~\ref{table:scale-halfcirlce-diameter}. \subsection{Definition of a Bend} \label{sec:definition-of-a-bend} The original article describes a bend as follows: \begin{displaycquote}{wang1998line} A bend can be defined as that part of a line which contains a number of subsequent vertices, with the inflection angles on all vertices included in the bend being either positive or negative and the inflection of the bend's two end vertices being in opposite signs. \end{displaycquote} \begin{figure}[h!] \centering \includegraphics[width=\textwidth]{fig8-definition-of-a-bend} \caption{Similar to figure 8 in \cite{wang1998line}: detected bends are highlighted.} \label{fig:fig8-definition-of-a-bend} \end{figure} Here are some non-obvious characteristics that are necessary when writing code to detect the bends: \begin{itemize} \item End segments of each line should also belong to bends. That way, all segments belong to 1 or 2 bends. \item First and last segments of each bend (except for the two end-line segments) are also the first vertex of the next bend. \end{itemize} \subsection{Gentle Inflection at the End of a Bend} \begin{displaycquote}{wang1998line} But if the inflection that marks the end of a bend is quite small, people would not recognize this as the bend point of a bend \end{displaycquote} 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. \begin{figure}[ht] \centering \begin{subfigure}[b]{.49\textwidth} \includegraphics[width=\textwidth]{fig5-gentle-inflection-before} \caption{Before applying the inflection rule.} \end{subfigure} \hfill \begin{subfigure}[b]{.49\textwidth} \includegraphics[width=\textwidth]{fig5-gentle-inflection-after} \caption{After applying the inflection rule.} \end{subfigure} \caption{Figure 5 in \cite{wang1998line}: gentle inflections at the ends of the bend.} \label{fig:fig5-gentle-inflection} \end{figure} \begin{figure}[h!] \centering \begin{subfigure}[b]{.49\textwidth} \includegraphics[width=\textwidth]{inflection-1-gentle-inflection-before} \caption{Before applying the inflection rule.} \end{subfigure} \hfill \begin{subfigure}[b]{.49\textwidth} \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 with multiple vertices.} \label{fig:inflection-1-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. The iterative approach was chosen: as long as the angle is "right" and the baseline is becoming shorter, the algorithm should keep re-assigning vertices to different bends. There is no upper bound on the number of 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. 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 the beginning to the end. \item \label{rev1} Reverse the line and each bend. \item Run the algorithm again. \item \label{rev2} Reverse the line and each bend. \item Return result. \end{enumerate} 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 a bend's baseline crosses another bend, it is called self-crossing. 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 --- this section explains them in higher detail, discusses their time complexity and applied optimizations. Figure~\ref{fig:fig6-selfcrossing} is copied from the original article. \begin{figure}[ht] \centering \includegraphics[width=.5\textwidth]{fig6-selfcrossing} \caption{Originally figure 6: the bend's baseline (orange) is crossing a neighboring bend.} \label{fig:fig6-selfcrossing} \end{figure} \begin{figure}[ht] \centering \includegraphics[width=.5\textwidth]{selfcrossing-1} \caption{The bend's baseline (orange) is crossing a non-neighboring bend.} \label{fig:selfcrossing-1-non-neighbor} \end{figure} % TODO: figure should not split the text. Looking at the {\WM} paper alone, it may seem like self-crossing may happen only with the neighboring bend. This would mean an efficient $O(n)$ implementation\footnote{where $n$ is the number of bends in a line. See 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 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 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 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. 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 worst-case complexity is still $O(n^2)$, when all bends' inner angles are larger than $180^\circ$. Having this optimization, the algorithmic complexity (as a result, the time it takes to execute the algorithm) drops by the fraction of bends, the inner angles' sum of which is smaller than $180^\circ$. \subsection{Attributes of a Single Bend} \textsc{compactness index} is "the ratio of the area of the polygon over the circle whose circumference length is the same as the length of the circumference of the polygon" \cite{wang1998line}. Given a bend, its compactness index is calculated as follows: \begin{enumerate} \item Construct a polygon by joining first and last vertices of the bend. \item Calculate the area of the polygon $A_{p}$. \item Calculate perimeter $P$ of the polygon. The same value is the circumference of the circle: $C = P$. \item Given the circle's circumference $C$, the circle's area $A_{c}$ is: \[ A_c = \frac{C^2}{4\pi} \] \item Compactness index $c$ is the area of the polygon $A_p$ divided by the area of the circle $A_c$: \[ c = \frac{A_p}{A_c} = \frac{A_p}{ \frac{C^2}{4\pi} } = \frac{4\pi A_p}{C^2} \] \end{enumerate} Once this operation is complete, each bend will have a list of properties which will be used by other modifying operators. \subsection{Shape of a Bend} \label{sec:shape-of-a-bend} This section introduces \textsc{adjusted size} $A_{adj}$ which trivially derives from \textsc{compactness index} $c$ and "polygonized" bend's area $A_{p}$: \[ A_{adj} = \frac{0.75 A_{p}}{c} \] Adjusted size is necessary later to compare bends with each other, or to decide if the bend is within the simplification threshold. Sometimes, when working with {\WM}, it is useful to convert between half-circle's diameter $D$ and adjusted size $A_{adj}$. These easily derive from circle's area formula $A = 2\pi \frac{D}{2}^2$: \[ D = 2\sqrt{\frac{2 A_{adj}}{\pi}} \] In reverse, adjusted size $A_{adj}$ from half-circle's diameter: \[ A_{adj} = \frac{\pi D^2}{8} \] \subsection{Isolated Bend} \label{sec:isolated-bend} Bend itself and its "isolation" can be described by \textsc{average curvature}, which is \textcquote{wang1998line}{geometrically defined as the ratio of inflection over the length of a curve.} Two conditions must be followed to claim that a bend is isolated: \begin{enumerate} \item \textsc{average curvature} of neighboring bends should be larger than the "candidate" bend's curvature. The article did not offer a value; this implementation arbitrarily chose $\isolationThreshold$. \item Bends on both sides of the "candidate" bend should be longer than a certain value. This implementation does not (yet) define such a constraint and will only follow the average curvature constraint above. \end{enumerate} We believe unclear criteria for \textsc{isolated bend} is one of the main causes for jagged lines in section~\ref{sec:results}, and is a suggested further area of research in section~\ref{sec:future-suggestions}. \subsection{The Context of a Bend: Isolated And Similar Bends} To find out whether two bends are similar, they are compared by 3 components: \begin{enumerate} \item \textsc{adjusted size} $A_{adj}$. \item \textsc{compactness index} $c$. \item \textsc{baseline length} $l$. \end{enumerate} Components 1, 2 and 3 represent a point in a 3-dimensional space, and Euclidean distance $d(p,q)$ between those is calculated to differentiate bends $p$ and $q$: \[ d(p,q) = \sqrt{(A_{adj(p)}-A_{adj(q)})^2 + (c_p-c_q)^2 + (l_p-l_q)^2} \] The smaller the distance $d$, the more similar the bends are. \clearpage \subsection{Elimination Operator} Figure~\ref{fig:elimination-through-iterations} illustrates steps of figure 8 from the original paper. There is not much to add to the original description beyond repeating the elimination steps in an illustrated example. \begin{figure}[ht] \centering \begin{subfigure}[b]{\textwidth} \includegraphics[width=\textwidth]{fig8-elimination-gen1} \caption{Original} \end{subfigure} \begin{subfigure}[b]{\textwidth} \includegraphics[width=\textwidth]{fig8-elimination-gen2} \caption{Iteration 1} \end{subfigure} \begin{subfigure}[b]{\textwidth} \includegraphics[width=\textwidth]{fig8-elimination-gen3} \caption{Iteration 2 (result)} \end{subfigure} \caption{Originally figure 8: the bend elimination through iterations.} \label{fig:elimination-through-iterations} \end{figure} \subsection{Combination Operator} \label{sec:combination-operator} Combination operator was not implemented in this version. \subsection{Exaggeration Operator} \label{sec:exaggeration-operator} 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 will be exaggerated in increments until either becomes true: \begin{itemize} \item \textsc{adjusted size} of the exaggerated bend is larger than the area of the half-circle. \item The exaggerated bend starts intersecting with a neighboring bend. Then exaggeration aborts, and the bend remains as if it were one step before the intersection. \end{itemize} Exaggeration operator uses a hardcoded parameter \textsc{exaggeration step} $s \in (1,2]$. It was arbitrarily picked to {\exaggerationEnthusiasm} for this implementation. A single exaggeration increment is done as follows: \begin{enumerate} \item Find a candidate bend. \item Find the bend's baseline. \item Find \textsc{midpoint}, the center of the bend's baseline. \item Find \textsc{midbend}, the center of the bend. Distance from one baseline vertex to \textsc{midbend} should be the same as from \textsc{midbend} to the other baseline vertex. \item Mark each bend's vertex with a number between $[1,s]$. The number is derived with elements linearly between the start vertex and \textsc{midbend}, with values somewhat proportional to the azimuth between these lines: \begin{itemize} \item \textsc{midbend} and the point. \item \textsc{midpoint} and the point. \end{itemize} The other half of the bend, from \textsc{midbend} to the final vertex, is linearly interpolated between $[s,1]$, using the same rules as for the first half. 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 coefficients, which were derived empirically, by observing the resulting bend. \item Each point (except the beginning and end vertices of the bend) will be placed farther away from the baseline. The length of misplacement is the marked value in the previous step. \end{enumerate} \begin{figure}[ht] \centering \includegraphics[width=.5\textwidth]{isolated-1-exaggerated} \caption{Example isolated exaggerated bend.} \label{fig:isolated-1-exaggerated} \end{figure} The technical implementation of the algorithm contains two implementations of exaggeration operator: \textsc{wm\_exaggerate\_bend} is the original one. It uses simple linear interpolation. It is fast, but simple. It tends to leave jagged bends. \textsc{wm\_exaggerate\_bend2} is a more computationally expensive function, which leaves better-looking exaggerated bends. Both functions are interchangeable and can be found in listing~\ref{lst:wm.sql}. Figure~\ref{fig:isolated-1-exaggerated} illustrates an exaggerated bend using \textsc{wm\_exaggerate\_bend2}. \section{Results} \label{sec:results} This section visualizes the results, discusses robustness and issues of the generalization, and suggests specific improvements. One of our goals is to compare the generalized lines with the official generalized dataset\cite{nzt}. Therefore, we have selected the target scales that the official sources offer, too: 1:\numprint{50000} and 1:\numprint{250000}. The \textsc{dhalfcircle} values for the subset are as follow: \begin{table}[ht] \centering \begin{tabular}{ c D{.}{.}{1} } Scale & \multicolumn{1}{c}{$D(m)$} \\ \hline 1:\numprint{50000} & 75 \\ 1:\numprint{250000} & 220 \\ \end{tabular} \end{table} Our generalized results are viewed from the following angles: \begin{itemize} \item Compare to the non-simplified originals. \item Compare to the official datasets. \item Compare to {\DP} and {\VW}. \end{itemize} \subsection{Generalization Results of Analyzed Rivers} \label{sec:generalization-results-of-analyzed-rivers} \subsubsection{Medium-Scale (1:\numprint{50000})} \label{sec:analyzed-medium-scale} \begin{figure}[h!] \centering \includegraphics[width=\textwidth]{salvis-wm75--grpk10-1x50k} \caption{2x zoomed-in {\WM} for 1:\numprint{50000}.} \label{fig:salvis-wm75--grpk10-1x50k} \end{figure} \begin{figure}[h!] \centering \includegraphics[width=\textwidth]{salvis-wm75--grpk10-1x50ktr} \caption{Top--right part of Figure~\ref{fig:salvis-wm75--grpk10-1x50k}.} \label{fig:salvis-wm75--grpk10-1x50ktr} \end{figure} \begin{figure}[h!] \centering \includegraphics[width=\textwidth]{salvis-wm75--grpk10-1x50ktl} \caption{Left part of Figure~\ref{fig:salvis-wm75--grpk10-1x50k}.} \label{fig:salvis-wm75--grpk10-1x50ktl} \end{figure} As one can see in Figure~\ref{fig:salvis-wm75--grpk10-1x50k}, the illustrations deliver 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, small bends are removed or slightly exaggerated. Figure's~\ref{fig:salvis-wm75--grpk10-1x50k} left part is clipped to Figure~\ref{fig:salvis-wm75--grpk10-1x50ktl}. As one can see, some bends were well exaggerated, and some bends were eliminated. Top--right side (clipped in Figure~\ref{fig:salvis-wm75--grpk10-1x50ktl}), some jagged and sharp bends appear. These will become more pronounced in even larger-scale simplification in the next section. To sum up, mid-scale simplification works well for some geometries, but creates sharp edges for others. \clearpage \subsubsection{Large-Scale (1:\numprint{250000})} \label{sec:analyzed-large-scale} As visible in Figure~\ref{fig:salvis-wm220-10x}, for large-scale map, some of the resulting bends look significantly exaggerated. Why is that? Figure~\ref{fig:salvis-wm220-overlaid-zoom} zooms in the large-scale simplification and overlays the original. \begin{figure}[ht] \centering \begin{subfigure}[b]{.49\textwidth} \centering \includegraphics[width=.2\textwidth]{salvis-250k-10x} \caption{Original.} \end{subfigure} \hfill \begin{subfigure}[b]{.49\textwidth} \centering \includegraphics[width=.2\textwidth]{salvis-wm220-10x} \caption{Simplified.} \end{subfigure} \caption{GRPK10 simplified with {\WM} for 1:\numprint{250000}.} \label{fig:salvis-wm220-10x} \end{figure} \begin{figure}[ht] \centering \includegraphics[width=.8\textwidth]{salvis-wm-overlaid-250k-zoom} \caption{10x zoomed-in {\WM} for 1:\numprint{250000}.} \label{fig:salvis-wm220-overlaid-zoom} \end{figure} A conglomeration of bends is visible, especially in top--right side of the illustration. We assume this was caused by two bends significantly exaggerated, leaving no space to exaggerate those between the two. \subsubsection{Discussion} For mid-size scales of 1:\numprint{50000}, the implemented algorithm works well for certain geometries, and poorly for others. This test surfaced two areas for future research and improvement: \begin{itemize} \item Exaggeration is sometimes creating sharp edges, especially when the exaggerated bend is quite small. When sharp edges are created, exaggeration could interpolate more points in the bend, and exaggerate using the interpolated points. \item In larger scales, when bends do not have space to exaggerate, they should be combined or eliminated instead. \end{itemize} \subsection{Comparison with National Spatial Datasets} There are a few datasets used in this comparison: GRPK10, GRPK50 and GRPK250. They are vector datasets which include rivers. They can be downloaded for free from \cite{nzt}. Here are the meanings of the codenames: \begin{description} \item[GRPK10] is a dataset of highest detail. Suited for maps of scale 1:\numprint{10000}. \item[GRPK50] is suited for maps of scale 1:\numprint{50000}. \item[GRPK250] offers the least detail, and is suited for maps of scale 1:\numprint{250000}. \end{description} During the analysis, we ran {\WM} on GRPK10 for 2 destination scales: 1:\numprint{50000} and 1:\numprint{250000}.\footnote{parameter calculation is detailed in section~\ref{sec:bend-scaling-and-dimensions}.} This section compares the resulting {\WM}--generalized rivers to GRPK50 and GRPK250. \subsubsection{Medium-Scale (1:\numprint{50000})} For our research location, the national dataset GRPK10 is almost equivalent to GRPK50, with a few nuances. Figure~\ref{fig:salvis-wm75-grpk50-grpk10-1x50k} illustrates all three shapes: GRPK50, {\WM}--simplified GRPK10, and the original GRPK10. \begin{figure}[h!] \centering \includegraphics[width=\textwidth]{salvis-wm75-grpk50-grpk10-1x50k} \caption{GRPK10, GRPK50 and {\WM} for 1:\numprint{50000}.} \label{fig:salvis-wm75-grpk50-grpk10-1x50k} \end{figure} \begin{figure}[h!] \centering \includegraphics[width=\textwidth]{salvis-wm75-grpk50-grpk10-1x50ktl} \caption{Left side of Figure~\ref{fig:salvis-wm75-grpk50-grpk10-1x50k}.} \label{fig:salvis-wm75-grpk50-grpk10-1x50ktl} \end{figure} \begin{figure}[h!] \centering \includegraphics[width=\textwidth]{salvis-wm75-grpk50-grpk10-1x50ktr} \caption{Top--right side of Figure~\ref{fig:salvis-wm75-grpk50-grpk10-1x50k}.} \label{fig:salvis-wm75-grpk50-grpk10-1x50ktr} \end{figure} Although figures are almost identical, Figure~\ref{fig:salvis-wm75-grpk50-grpk10-1x50ktr} illustrates two small bends that have been removed in GRPK50, but have been exaggerated by our implementation. \clearpage \subsubsection{Large-Scale (1:\numprint{250000})} \label{sec:national-large-scale} Figure~\ref{fig:salvis-wm220} illustrates the original GRPK250 and the {\WM}--simplified version. As section~\ref{sec:analyzed-large-scale} explains, the algorithm tries to exaggerate many bends to a great size. However, GRPK250 takes the opposite approach --- only the very basic shapes of the largest bends are retained. Time and customers will tell, which approach is more appropriate, after the current {\WM} implementation receives some time and attention, as desired in section~\ref{sec:future-suggestions}. \begin{figure}[h!] \centering \begin{subfigure}[b]{.49\textwidth} \includegraphics[width=\textwidth]{salvis-grpk250-2x} \caption{GRPK250.} \end{subfigure} \hfill \begin{subfigure}[b]{.49\textwidth} \centering \includegraphics[width=\textwidth]{salvis-wm220} \caption{{\WM}-simplified GRPK10.} \end{subfigure} \caption{GRPK250 and {\WM}--simplified GRPK10.} \label{fig:salvis-wm220} \end{figure} \subsection{Comparison with {\DP} and {\VW}} It is time to visually compare our implementation with the classical algorithms: {\DP}, {\VW} and Chaikin. Since we have established that more work is needed for small-scale maps (1:\numprint{250000}), we will limit the comparison in this section to 1:\numprint{50000}. \begin{figure}[h!] \includegraphics[width=\textwidth]{salvis-wm75-dp64-grpk10-1x50k} \caption{} \label{fig:salvis-wm75-dp64-grpk10-1x50k} \end{figure} \begin{figure}[h!] \includegraphics[width=\textwidth]{salvis-wm75-dpchaikin64-grpk10-1x50k} \caption{} \label{fig:salvis-wm75-dpchaikin64-grpk10-1x50k} \end{figure} \clearpage \begin{figure}[h!] \includegraphics[width=\textwidth]{salvis-wm75-vw64-grpk10-1x50k} \caption{} \label{fig:salvis-wm75-vw64-grpk10-1x50k} \end{figure} \begin{figure}[h!] \includegraphics[width=\textwidth]{salvis-wm75-vwchaikin64-grpk10-1x50k} \caption{} \label{fig:salvis-wm75-vwchaikin64-grpk10-1x50k} \end{figure} \clearpage \subsection{Testing Results Online} \label{sec:testing-results-online} An on-line tool\cite{openmapwm} has been developed to test incoming parameters to {\WM} algorithm. A user should select a river of interest, enter the \textsc{dhalfcircle} parameter and click "Submit". The simplified line feature will be overlaid on top of the map. Figure~\ref{fig:openmap-wm-good} illustrates the end result that looks reasonably well. Figure~\ref{fig:openmap-wm-bad} illustrates that the algorithm produces poorly simplified results for some geometries. \begin{figure}[ht] \centering \includegraphics[width=\textwidth]{openmap-wm-good.png} \caption{Example on-line test tool for {\WM} algorithm.} \label{fig:openmap-wm-good} \end{figure} \begin{figure}[ht] \centering \includegraphics[width=.5\textwidth]{openmap-wm-bad.png} \caption{Another example from the on-line test tool.} \label{fig:openmap-wm-bad} \end{figure} \section{Conclusions} \label{sec:conclusions} Classical and modern line simplification algorithms were evaluated, main problems with them identified. A method for {\WM} technical implementation was defined, and the algorithm implemented. Each geometric transformation was described and visualized. The implemented algorithm was applied for different shapes and compared to national (Lithuanian) datasets. 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. 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 algorithm robustness. \section{Future Suggestions} \label{sec:future-suggestions} These are the areas for possible future work with this, published, implementation: \begin{itemize} \item Implement bend combination operator (section~\ref{sec:combination-operator}). \item Fine-tune parameters for bend exaggeration. Section~\ref{sec:generalization-results-of-analyzed-rivers} contains a exaggerated bends that became sharp and includes some future ideas. \item What are the exaggeration limits when working with large scales? Section~\ref{sec:national-large-scale} discusses examples that some limits are necessary. \item Research when bends should be marked as \textsc{isolated}. As is seen from examples, the current criteria are not robust enough. \item Once the points above yield a satisfactory result, the efficiency of the algorithm could be improved to work on the lines in "streaming" fashion (more details in section~\ref{sec:algorithm-implementation-process}). \end{itemize} That sums up what could be improved without changing the algorithm in a significant way. Other than that, further area of research is working towards graduating the algorithm from "isolated cartographic generalization" to "full cartographic generalization". The current operators of {\WM} algorithm have a few venues to preserve the surrounding topology. This could be further researched and extended. \section{Acknowledgments} \label{sec:acknowledgments} I would like to thank my thesis supervisor, Dr. Andrius Balčiūnas, for his help in formulating the requirements and providing early editorial feedback for the thesis. I am grateful to Tomas Straupis, who handed me the {\WM}\cite{wang1998line} 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 algorithm descriptions were ambiguous. Many thanks to NŽT\cite{nzt} for providing the datasets with a very permissive license. \printbibliography \begin{appendices} \section{Code Listings} This section contains a subset of files for the {\WM} algorithm. As a reminder, full listings, including supporting programs, can be found on GitHub\cite{wmsql}. \subsection{Re-Generating This Paper} \label{sec:code-regenerate} As explained in section~\ref{sec:reproducing-the-paper}, illustrations in 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 name of this document is \textsc{mj-msc-full.pdf}). 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 \textsc{make} command, and display the generated document. Source code for the algorithm, as well as other supporting files, can be found in the temporary directory. \begin{longlisting} \inputcode{bash}{extract-and-generate} \caption{\textsc{extract-and-generate}} \label{lst:extract-and-generate} \end{longlisting} \subsection{Function \textsc{st\_simplifywm}} \begin{longlisting} \inputcode{postgresql}{wm.sql} \caption{\textsc{wm.sql}} \label{lst:wm.sql} \end{longlisting} \subsection{Function \textsc{aggregate\_rivers}} \begin{longlisting} \inputcode{postgresql}{aggregate-rivers.sql} \caption{\textsc{aggregate-rivers.sql}} \label{lst:aggregate-rivers.sql} \end{longlisting} \end{appendices} \end{document}