An Evaluation Model of Level of Detail Consistency of Geographical Features on Digital Maps

International Journal of Geo-Information

Article An Evaluation Model of Level of Detail Consistency of Geographical Features on Digital Maps

Pengcheng Liu 1,2 and Jia Xiao 1,2,*

1 Hubei Province Key Laboratory for Geographical Process Analysis & Simulation, Central China Normal University, Wuhan 430079, China; [email protected] 2 College of Urban and Environmental Science, Central China Normal University, Wuhan 430079, China * Correspondence: [email protected]; Tel.: +86-157-0299-4209

Received: 29 May 2020; Accepted: 24 June 2020; Published: 26 June 2020

Abstract: This paper proposes a method to evaluate the level of detail (LoD) of geographic features on digital maps and assess their LoD consistency. First, the contour of the geometry of the geographic feature is sketched and the hierarchy of its graphical units is constructed. Using the quartile measurement method of statistical analysis, outliers of graphical units are eliminated and the average value of the graphical units below the bottom quartile is used as the statistical LoD parameter for a given data sample. By comparing the LoDs of homogeneous and heterogeneous features, we analyze the diﬀerences between the nominal scale and actual scale to evaluate the LoD consistency of features on a digital map. The validation of this method is demonstrated by experiments conducted on contour lines at a 1:5K scale and artiﬁcial building polygon data at scales of 1:2K and 1:5K. The results show that our proposed method can extract the scale of features on maps and evaluate their LoD consistency.

Keywords: level of detail; graphical unit; geographical feature; digital map

1. Introduction Scale has always been a key study focus in various disciplines. In cartography, scale includes rich meanings. Several terms—semantic resolution, geometric precision, geometric resolution, and feature granularity—are all related to scale to some extent. In the field of geographic information science, the level of detail (LoD) adds another such term. It can be used in the analysis of any type of spatial data, especially that of geospatial vector data. It is obvious that the LoD of cartographic features is directly related to the map scale. Different categories of geographic features on a map should, theoretically, have the same level of detail. A small-scale topographical map is commonly generalized from a relatively large-scale map. However, cartographic generalization can easily cause inconsistency in different cartographic features’ LoDs due to the various generalization models. The consistency of LoD has become a research focus in recent years with the massive application of volunteered geographical information (VGI), e.g., VGI in OpenStreetMap [1–4]. VGI data come from a wide range of sources, e.g., coordinate data collected with the GPS devices, digital results that are vectorized from paper maps, interpretation results of remote sensing images, and so on [5–7]. It is difficult to ensure that the VGI data of various sources are at the same level of detail. Furthermore, the volunteers who provide the VGI data normally have different educational backgrounds and knowledge structures, which makes the inconsistency of LoD inevitable. Thus, the geographic features in the VGI data from different sources and providers generally demonstrate the inconsistent LoD, which reduces the data quality and might result in the misreading of Internet maps. Therefore, it is necessary to develop a unified metric to evaluate the LoD of the geographic features on the map. When we try to develop such a metric for web maps, it will face a lack of metadata as well as scale and resolution.

ISPRS Int. J. Geo-Inf. 2020, 9, 410; doi:10.3390/ijgi9060410 www.mdpi.com/journal/ijgi ISPRS Int. J. Geo-Inf. 2020, 9, 410 2 of 16

ISPRS Int. J. Geo-Inf. 2020, 9, x FOR PEER REVIEW 2 of 16 Consistency evaluation of map LoD can be conducted at both the semantic and representational levels of a geometric graph. Touya et al. [8] divided building polygons in OpenStreetMap into four OpenStreetMap into four hierarchy levels (single building, street, city, and country) according to hierarchy levels (single building, street, city, and country) according to different LoD hierarchies different LoD hierarchies under different scales. As another example, considering features underrepresenting different forest scales. areas As and another single example, buildings, considering these features features bel representingong to different forest semantic areas and levels single: a buildings,forest area thesethat has features few vertices belong and to di overlapsfferent semantic several buildings levels: a forestshould area be considered that has few inconsistent vertices and at overlapsthe semantic several level buildings [9,10]. The should most bedirect considered method inconsistentof measuring at the the LoD semantic of a geometric level [9,10 element]. The most is to directcalculate method the shor of measuringtest distance the between LoD of a adjacent geometric points element belonging is to calculate to thethe feature. shortest For distance example, between Ai [11] adjacentmeasured points the LoD belonging of a point to the cluster feature. using For example,a point-oriented Ai [11] measuredDelaunay thetriangle. LoD of For a point a curve cluster or usingpolygon a point-oriented feature, the length Delaunay of the triangle. shortest For side a curveof the or feature polygon can feature, be used the as length the minimum of the shortest detail sideparameter of the featurefor cartographic can be used representation as the minimum [12 detail]. For parameter the sake forof cartographicclarity and representationreadability, curve [12]. Forsymbols the sake on a of map clarity have and a readability,minimum perceptible curve symbols width. on The a map larger have the a minimumscale, the perceptiblelarger the ground width. Thedistance larger to the which scale, the the line larger width the groundcorresponds. distance When to which the scale the line is less width than corresponds. a certain threshold When the value, scale isneighboring less than a parts certain of thresholdthe feature value, will aggregate. neighboring This parts means of the that feature the curve will aggregate.symbol width This of means curves that at theaggregation curve symbol can widthbe used of curves as a atparameter aggregation for can LoD be usedassessment as a parameter [12,13]. for This LoD assessmentmethod is [12called,13]. Thisincremental method iscurve called detail incremental detection. curve Another detail detection.method involves Another methodrasterizing involves curve rasterizing features curveusing featuresincremental using grid incremental widths. gridWhen widths. the grid When width the gridis greater width than is greater a threshold than a threshold value, the value, part the of partthe ofcurve the curvethat does that doesnot coalesce not coalesce at smaller at smaller grid grid widths widths appears appears to toaggregate. aggregate. The The grid grid size size which which is equal toto thethe threshold threshold value value is usedis used to define to define the LoD the ofLoD a curve of a [curve14]. Figure [14].1 Figure shows the1 shows LoD detection the LoD processesdetection processes employed employed by the aforementioned by the aforementioned methods. Inmethods. Figure1 Inb, Fig theure curve 1b, detailthe curve is clear detail when is clear the symbolwhen the width symbol is 1m; width however, is 1m; the however, symbol the aggregates symbol onceaggregates and twice once when and itstwice width when is 2 its m (Figurewidth is1c) 2 andm (Figure 3.5 m (Figure1c) and1 3.5d), m respectively. (Figure 1d), Figure respectively.1e–g show Figure the same1e-g show curve the rasterized same curve with rasterized di fferent with grid widths.different Results grid widths. obtained Results by the incrementobtained by grid the width increment method grid are consistentwidth method with those are consistent obtained by with the those obtained by the increment symbol width method. The LoD parameter of a feature can be increment symbol width method. The LoD parameter of a feature can be measured by the threshold measured by the threshold width (symbol width or grid width) where parts of the feature width (symbol width or grid width) where parts of the feature conglomerate together. conglomerate together.

Figure 1. Detecting the LoD of a curve using the progressive increment method. (a) A curve with bends. Figure 1. Detecting the LoD of a curve using the progressive increment method. (a) A curve with (b–d) Curve representation when the symbol width is respectively 1m, 2m and 3.5m. (e–g) Rasterizing bends. (b–d) Curve representation when the symbol width is respectively 1m, 2m and 3.5m. (e–g) results of curves, respectively, at 1m, 2m and 3.5m. Rasterizing results of curves, respectively, at 1m, 2m and 3.5m. This paperpaper presents presents a methoda method that that uses uses computational computational geometry geometry to detect to the detect minimum the minimum graphical unitsgraphical of map units features of map and computesfeatures and LoD computes parameters LoD using parameters a mathematical using statistics a mathematical model. This statistics model estimatesmodel. This the model actual mapestimates scale and the itsactual deviation map fromscale map-nominal and its deviation scale andfrom evaluates map-nomi thenal level scale of detail and consistencyevaluates the of maplevel featureof detail representations. consistency of The map method feature obtains representations. the LoD and The feature method representation obtains the by detectingLoD and feature topology representation conflict in curve-oriented by detecting representationstopology conflict on in the curve basis-oriented of the internal representations consistency on principle,the basis whichof the needsinternal to beconsistency visualized severalprincipl timese, which with needs different to parameters.be visualized The several method times presented with indifferent this paper parameters. allows us The to recognizemethod presented the minimal in representationthis paper allows graphical us to unitrecognize and to calculatethe minimal the LoDrepresentation of map features graphical using unit the graphicaland to calculate unit. Compared the LoD withof map previous features methods, using the our graphical algorithm unit. uses mathematicalCompared with models previous while methods, the former our uses algorithm analog uses detection. mathematical models while the former uses analogThe detection. structure of this paper is as follows. In Section 2.1, we present techniques to detect the basic graphicalThe structure units of contour of this linespaper and is artificialas follows. map In features. Section Section2.1, we 2.2 present presents techniques a statistical to methoddetect the to basic graphical units of contour lines and artificial map features. Section 2.2 presents a statistical method to calculate the LoD of each feature and evaluate the LoD consistency of multiple features. In Section 3, we design an experiment to test the proposed method and analyze the experimental results. Section 4 summarizes the paper and prospects for future work.

ISPRS Int. J. Geo-Inf. 2020, 9, 410 3 of 16 calculate the LoD of each feature and evaluate the LoD consistency of multiple features. In Section3, we design an experiment to test the proposed method and analyze the experimental results. Section4 summarizes the paper and prospects for future work.

2. Methods

2.1. Graphical Unit and LoD It is natural to describe the LoD of a geographical feature using its geometric details. For the geographical features represented by different geometries, the representations of their geometric details are quite different. For a polyline, the bends in it could reflect its geometric details. It is obvious that the smaller the bends are, the richer its geometric details will be. Herein, we call the geometry of the geographical feature which could reflect its geometric detail degree the graphical unit. Thus, the bend could be the graphical unit of a curve.

2.1.1. Bend of Curve as Graphical Unit For a digital map, in order to meet the aesthetic requirements of feature representation, cartographic features commonly contain redundant points, so the line segment of a feature cannot be regarded as its graphical unit. According to the above discussion, a bend is regarded as a good graphical unit for the curve feature. However, it is not an easy task to describe and detect the bends of a curve. Scholars have provided various detection bend approaches for a curve [15–18]. Ai [15] used a Delaunay triangle net to build a curve-bending binary tree structure, allowing them to interpret the nested curve relationship and eﬀectively extract a graphical unit, called leaf bend. Herein, we use the method proposed in [15] to detect the bends of a curve. The points on the curve can be set up as a constrained triangle network with the curve as the constraint edge [19,20]. By extracting the left and right triangle network, we can build the tree structure of the left and right bends. The process is shown in (a–h) in Figure2. For the curve shown in Figure2a, the points on the curve are used to build a constrained Delaunay triangle network (Figure2b). Two steps are used to decide whether a triangle is on the left or right side of a curve. Firstly, the order of the three points of a triangle is adjusted according to their location along the curve. Taking the triangle in Figure3 as an example, the adjusted order will be A, B, and P. Secondly, the vector product (0, 0, V) of AP→ and BP→ is calculated as shown in Equation (1); if V > 0, the triangle is on the left side of the curve; if V < 0, it is on the right side. If V = 0, the three points A, B, and P are on the same line. Figure3 shows the processes involved in these operations. (0, 0, V) = AP→ BP→ (1) × where AP→ and BP→ are regarded as space vectors which have three components and the value of the third component is 0, e.g., AP→ = (x, y, 0), and AP→ BP→ refers to the cross product of two space vectors. × ISPRS Int. J. Geo-Inf. 2020, 9, 410 4 of 16 ISPRS Int. J. Geo-Inf. 2020, 9, x FOR PEER REVIEW 4 of 16

Figure 2. The process for identifying the graphical units of a curve. (a) An example curve with the Figure 2. The process for identifying the graphical units of a curve. (a) An example curve with the order number of points on the curve. (b) Delaunay triangulation constructed with the curve points. order number of points on the curve. (b) Delaunay triangulation constructed with the curve points. (c) The triangulation sets on the left side of the forward direction of the curve. (d) The triangulation sets (c) The triangulation sets on the left side of the forward direction of the curve. (d) The triangulation on the right of the forward direction of the curve. (e) The left-bend sets on the left of the forward direction sets on the right of the forward direction of the curve. ( ) The left-bend sets on the left of the forward of the curve. (f) The right-bend sets on the left of the forwarde direction of the curve. (g) The hierarchical structuredirection ofof thethe left-sidecurve. (f curve.) The right (h) The-bend hierarchical sets on the structure left of the of forward the right-side direction curve. of the curve. (g) The hierarchical structure of the left-side curve. (h) The hierarchical structure of the right-side curve.

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Figure 3. Judging the position of Point P, Q relative to the curve L. Figure 3. Judging the position of Point P, Q relative to the curve L. According to Equation (1), the triangles in Figure2b are divided into two sets: left-side triangles and right-sideIf this point triangles. is not the Figure vertex2c,d of represents a triangle, the the subsets next point of left will and be right considered triangles. until To a detect point whether belongs theto a triangletriangle, subsets and this are is onused the as left the orstarting right sidepoint. and For their all the nesting triangle relationship, edges containing we first the detect starting the entrancepoint, select point the of one the first-levelthat is connected curve from with the the starting point pointwith the of the largest curve. number. The point with the largestIf thisnumber point is is called not the the vertex ending of apoint. triangle, Next, the the next part point of the will curve be considered between the until starting a point point belongs and tothe a ending triangle, point and is this taken is used as one as theof the starting first-level point. bends; For all the the ending triangle point edges is the containing starting thepoint starting of the point,next bend. select We the repeat one that this isoperation connected to get with the the next point bend with of the the first largest level, number. and so Theon, until point the with end the of largestthe curve. number For the is calledright bend the ending (Figure point. 2d), the Next, first the-level part bends of the cover curve points between 7–22, the 22 starting–39, 41– point48, 49 and–72, theand ending73–80; pointon the is left, taken the as first one-level of the bend first-level covers bends; points the 0–80 ending (all the point points is the on starting the curve). point For of the nextright bend.bends, We there repeat is also this a operation first-level to bend get thewhich next covers bend ofpoints the first 0–80 level,, as shown and so in on, Figure until the2g. endFor all of thebends curve. under For the the first right bends, bend we (Figure build2 d),up thea bend first-level binary bendstree according cover points to the 7–22, method 22–39, proposed 41–48, 49–72, by Ai and[14].73–80; The bend on the tree left, structures the first-level of the bend left coversand right points sides 0–80 are (all shown the points in Figure on the 2g, curve).h, and Forthe theleft right and bends,right leaf there bends is also are ashown first-level in Figure bend 2 whiche,f. covers points 0–80, as shown in Figure2g. For all bends under the first bends, we build up a bend binary tree according to the method proposed by Ai [14]. 2.1.2. Graphical Unit of Natural Features The bend tree structures of the left and right sides are shown in Figure2g,h, and the left and right leaf bendsHerein, are shown we take in Figure the minimum2e,f. width and depth of the leaf bends as the parameter of the natural features’ graphical unit. The line segment from the starting point to the ending point of the bend 2.1.2.forms Graphicala bend base, Unit the of distance Natural Featuresof it is the width (w) of the bend, and the largest distance from the innerHerein, points weof the take bend the minimumto its base width is the anddepth depth of the of thebend( leafd) bends (Figure as 4 the). The parameter smaller of of the the natural width features’and depth graphical values is unit. regarded The line as segmentthe LoD parameter from the starting of the bend. point toWe the use ending the relationship point of the between bend forms the minimum distinguishable distance (0.3 mm in most cases in this paper) [21] and the LoD parameters a bend base, the distance of it is the width (w) of the bend, and the largest distance from the inner to calculate a scale value to which each bend corresponds (Equation (2)). The calculated scale is points of the bend to its base is the depth of the bend(d) (Figure4). The smaller of the width and depth called the actual scale of the bend. values is regarded as the LoD parameter of the bend. We use the relationship between the minimum distinguishable distance (0.3 mm in most cases in this paper) [21] and the LoD parameters to calculate a scale value to which each bend corresponds (Equation (2)). The calculated scale is called the actual scale of the bend. 1 d = svo (2) MR (l 1000) ∗ where MR is the actual scale denominator (as opposed to the nominal scale denominator); l is the LoD parameter (measured in meters), and dsvo is the minimum distinguishable distance (in millimeters; typically, 0.3). In general, for maps with a scale greater than 1:10,000, it is better to take 0.3mm as the value. For maps with a smaller scale, due to the large amount of data and more complicated degree of Figure 4. The schematic map of the width (w) and depth (d) of a bend. curvature, the value of dsvo can be adjusted to adapt to this trend. Therefore, for maps with a scale less than 1:100,000 and greater than 1:1,000,000, we set the value of dsvo to 0.6mm, and for maps with a scale less than 1:1,000,000, we set the value of dsvo to 1mm. The LoD parameters of all the leaf bends provide 1 dsvo the data for the statistical analysis of the curves’= LoD in Section 2.2. The LoD parameter calculation(2) model based on the structure of a curve is consistentMlR ( *1000) with the incremental LoD detection approach.

where is the actual scale denominator (as opposed to the nominal scale denominator); is the LoD parameter (measured in meters), and is the minimum distinguishable distance (in 𝑀𝑀𝑅𝑅 𝑙𝑙 𝑑𝑑𝑠𝑠𝑠𝑠𝑠𝑠 ISPRS Int. J. Geo-Inf. 2020, 9, x FOR PEER REVIEW 5 of 16

Figure 3. Judging the position of Point P, Q relative to the curve L.

If this point is not the vertex of a triangle, the next point will be considered until a point belongs to a triangle, and this is used as the starting point. For all the triangle edges containing the starting point, select the one that is connected with the point with the largest number. The point with the largest number is called the ending point. Next, the part of the curve between the starting point and the ending point is taken as one of the first-level bends; the ending point is the starting point of the next bend. We repeat this operation to get the next bend of the first level, and so on, until the end of the curve. For the right bend (Figure 2d), the first-level bends cover points 7–22, 22–39, 41–48, 49–72, and 73–80; on the left, the first-level bend covers points 0–80 (all the points on the curve). For the right bends, there is also a first-level bend which covers points 0–80, as shown in Figure 2g. For all bends under the first bends, we build up a bend binary tree according to the method proposed by Ai [14]. The bend tree structures of the left and right sides are shown in Figure 2g,h, and the left and right leaf bends are shown in Figure 2e,f.

2.1.2. Graphical Unit of Natural Features Herein, we take the minimum width and depth of the leaf bends as the parameter of the natural features’ graphical unit. The line segment from the starting point to the ending point of the bend forms a bend base, the distance of it is the width (w) of the bend, and the largest distance from the inner points of the bend to its base is the depth of the bend(d) (Figure 4). The smaller of the width and depth values is regarded as the LoD parameter of the bend. We use the relationship between the

ISPRSminimum Int. J. Geo-Inf.distinguishable2020, 9, 410 distance (0.3 mm in most cases in this paper) [21] and the LoD parameters6 of 16 to calculate a scale value to which each bend corresponds (Equation (2)). The calculated scale is called the actual scale of the bend.

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millimeters; typically, 0.3). In general, for maps with a scale greater than 1:10,000, it is better to take 0.3mm as the value. For maps with a smaller scale, due to the large amount of data and more complicated degree of curvature, the value of can be adjusted to adapt to this trend. Therefore,

for maps with a scale less than 1:100,000 and greater𝑠𝑠𝑠𝑠𝑠𝑠 than 1:1,000,000, we set the value of to 0.6mm, and for maps with a scale less than 1:𝑑𝑑1,000,000, we set the value of to 1mm. The LoD 𝑑𝑑𝑠𝑠𝑠𝑠𝑠𝑠 parameters of all the leaf bends provide the data for the statistical analysis o𝑠𝑠𝑠𝑠𝑠𝑠f the curves’ LoD in Section 2.2. The LoD parameter calculation model based on the structure of a curve𝑑𝑑 is consistent with FigureFigure 4.4. The schematic map of the width (w) and depth (d) of a bend. the incremental LoD detection approach. 2.1.3. Graphical Unit of Humanmade Features 2.1.3. Graphical Unit of Humanmade Features Humanmade features tend to have regularly1 geometricd shapes. Each side of a building polygon = svo is approximatelyHumanmade parallel features or tend orthogonal to have toregularly another geometric side of the shapes. building. Each The side LoD of a of building such a building polygon(2) MlR ( *1000) polygonis approximately can be measured parallel or by orthogonal regular graphical to another units side (such of the as building. convexity, The concavity, LoD of such stair, a and building gap). Thesepolygon units can can be easily measured be identified by regular using graphical computational units (such geometry as convexity, techniques. concavity, stair, and gap). where is the actual scale denominator (as opposed to the nominal scale denominator); is the TheseThe units approaches can easily to identifyingbe identified building using computational polygon basic graphicalgeometry units techniques. exist mostly in the references LoD parameter (measured in meters), and is the minimum distinguishable distance (in on geographicalThe𝑀𝑀 𝑅𝑅approaches generalization. to identifying For example, building Xu polygon and Long basic identified graphical the localunits model exist constructedmostly𝑙𝑙 in the by the references continuous on geographical four points of generalization. a building polygon For𝑑𝑑 𝑠𝑠𝑠𝑠𝑠𝑠example, [22]; Samsonov Xu and and Long Yakimova identified [23 ]the recognized local model the partialconstructed characteristic by the continuous model by using four points the orthogonality of a building of polygon continuous [22]; sides Samsonov and length and diYakimovafferentiation [23] forrecognized line features, the partial and then characteristic proposed respective model by simplification using the orthogonality strategies for of di ffcontinuouserent models sides in map and generalization.length differentiation Wang and for Leeline [ 24features,] and Wang and etthen al. [25proposed] summarized respective building simplification polygon partial strategies pattern for recognitiondifferent models methods in thatmap are generalization. based on the relationshipsWang and Lee between [24] sidesand Wang and angles. et al. [25] summarized building polygon partial pattern recognition methods that are based on the relationships between Herein, we suggest some partial models for building polygon identification based on polygon sides and angles. characteristics such as angles between continuous sides and side length variations. In the calculation Herein, we suggest some partial models for building polygon identification based on polygon of the included angles between two continuous sides, we consider clockwise to be negative and characteristics such as angles between continuous sides and side length variations. In the calculation counterclockwise to be positive (angles range from 90to 90 ). In Figure5, we calculate the included of the included angles between two continuous sides,− we◦ consider clockwise to be negative and → → → → → anglescounterclockwise of vector quantities to be positiveAB, BC (anglesand CD range. The from included -90°to angle 90°). formed In Figure anticlockwise 5, we calculate from theAB includedto BCis positive,angles of and vector the included quantities angleAB formed, BC and clockwiseCD . The from includedBC→ to CD→ angleis negative. formed Usinganticlockwise the included from angles AB constructedto BC is positive by continuous, and the sides included and the angle length formed differentiation clockwise of thefrom continuous BC to CD sides, is the negative. identification Using ofthe the included partialgraphical angles constructed unit of a polygon by continuous can be obtained. sides and Table the length1 lists thedifferentiation conditional descriptionsof the continuous and methodssides, the for identification various graphical of the units; partial the graphical fifth column unit provides of a polygon detailed can parameter be obtained. values Table for 1 di listsfferent the graphicalconditional units descriptions of building and polygons. methods for various graphical units; the fifth column provides detailed parameter values for different graphical units of building polygons.

Figure 5. Schematic map of the size and direction of the intersection angle. Figure 5. Schematic map of the size and direction of the intersection angle.

Table 1. Graphical units, identification methods and LoD parameters of artificial building polygons.

No. Pattern Example Regularity Minimal detail distance (d)

The signs of the middle four The minimal detail distance consecutive turning angles is the shorter length of edge 1 convex are positive, negative, positive 23 and edge 34, i.e., d = and negative; their angle MIN(d23, d34), where d23 value is approximately 90 and d34 are the lengths of

ISPRS Int. J. Geo-Inf. 2020, 9, x FOR PEER REVIEW 6 of 16

2.1.3. Graphical Unit of Humanmade Features Humanmade features tend to have regularly geometric shapes. Each side of a building polygon is approximately parallel or orthogonal to another side of the building. The LoD of such a building polygon can be measured by regular graphical units (such as convexity, concavity, stair, and gap). These units can easily be identified using computational geometry techniques. The approaches to identifying building polygon basic graphical units exist mostly in the references on geographical generalization. For example, Xu and Long identified the local model constructed by the continuous four points of a building polygon [22]; Samsonov and Yakimova [23] recognized the partial characteristic model by using the orthogonality of continuous sides and length differentiation for line features, and then proposed respective simplification strategies for different models in map generalization. Wang and Lee [24] and Wang et al. [25] summarized building polygon partial pattern recognition methods that are based on the relationships between sides and angles. Herein, we suggest some partial models for building polygon identification based on polygon characteristics such as angles between continuous sides and side length variations. In the calculation ISPRSof the Int. included J. Geo-Inf. 20angles20, 9, x FORbetween PEER REVIEWtwo continuous sides, we consider clockwise to be negative7 ofand 16 counterclockwise to be positive (angles range from -90°to 90°). In Figure 5, we calculate the included angles of vector quantities AB , BC and CD . The included angle formed anticlockwise from AB ISPRS Int. J. Geo-Inf. 2020, 9, x FOR PEER REVIEW degrees, and the angles of  edge 23 and edge 34,7 of 16 to BC is positive, and the included angle formed clockwise from BC to CD is negative. Using vector 12 and vector 56 are respectively. the included angles constructed by continuous sides and the length differentiation of the continuous degrees,close andto 0 thedegrees. angle s of edge 23 and edge 34, sides,ISPRS Int. the J. Geoidentification-Inf. 2020, 9, x FORof the PEER partial REVIEW graphical unit of a polygon can be obtained. Table 1 lists7 ofthe 16 conditional descriptions and methods forvector various 12 graphical and vector units; 56 are the fifth columnrespec providestively. detailed The signs of the middle four parameter values for different graphical unitsclose of building to 0 degrees. polygons. consecutivedegrees, and turn theing angle angless of edge 23 and edge 34, The minimal detail distance arevector negative, 12 and positive vector, positive 56 are respectively. ISPRS Int. J. Geo-Inf. 2020, 9, x FOR PEER REVIEWThe signs of the middle four is the shorter length of edge7 of 16 concave andclose negative to 0; degrees.their angle 2 consecutive turning angles 23 and edge 34, i.e., d = value is approximately 90 The minimal detail distance are negative, positive, positive degrees, and the angles of is theedge shorterMIN(d23, 23 and length d34).edge of 34, edge concave Thedegrees,and signs negative andof the the; theirmiddle angles angle four of 2 vector 12 and vector 56 are 23 andrespec edge tively.34, i.e. , d = vectorsconsecutivevalue 12 is andapproximately turn56 areing close angles 90 to 0 close to 0 degrees. The minimal detail distance ISPRS Int. J. Geo-Inf. 2020, 9, x FOR PEER REVIEWare negative,degrees. positive , positive MIN(d23, d34). 7 of 16 degrees, and the angles of is the shorter length of edge concave and negative; their angle 2 vectorsThe signs 12 and of the56 aremiddle close four to 0 23 and edge 34, i.e., d = valueThe issigns approximately of the four 90 consecutivedegrees, degrees.and turn the ing angle angless of edgeMIN(d23, 23 and edged34). 34, consecutivedegrees, and turn theing angles angles of The minimal detail distance arevector negative, 12 and positive vector, 56positive are respectively. vectorsareThe negative,12 signs and 56of positive,theare fourclose to 0 is the shorter length of edge concave Figure 5. Schematic map of theand sizeclose negative and to direction 0 degrees.; their ofangle the intersection The minimal angle. detail distance ISPRS Int. J. Geo-Inf. 2020, 9, 4102 negative, negativedegrees. or positive, 23 and edge 34, i.e., d = 7 of 16 consecutivevalue is approximately turning angles 90 is the shorter edge length of ISPRS3 Int.notch J. Geo- Inf. 2020, 9, x FOR PEER REVIEW negative, positive, positive; MIN(d23, d34). 7 of 16 Table 1. Graphical units, identification methodsThedegrees,are signs negative, an ofandd LoDthe the positive, middleparameters angles four of of artificialthe notch building, i.e., polygons.d = MIN(d12, Thetheir signs angle of value the four is The minimal detail distance negative,vectorsconsecutive 12 negative and turn 56 areingor positive closeangles to ,0 d23). Table 1. Graphical units, identificationapproximatelyconsecutive turn methods 90ing degrees, angles and LoDisThe parameters the minimal shorter ofedgedetail artificial length distancebuilding of polygons. No.3 Pnotchattern Example arenegative,degrees, negative,R andpositiveegularity positive the ,angle positive , positives of ; Minimaledge 23 detail and distanceedge 34, (d) are negative,degrees. positive, theis the notch shorter, i.e., lengthd = MIN(d12, of edge concave andvectorand edgestheir negative 12 12 angleand and vector; valuetheir 23 are angle56is short are respectively. No. Pattern2 Example RegularityThe23 minimal and edged23). detail 34, i.e. distance, d = Minimal Detail Distance (d) negative,approximatelyvalue is negative approximatelyedges. 90 or degrees, positive 90 , The signsTheclose signs ofto the0 ofdegrees. middle the four four isThe the minimal shorter detailedge length distance of 3 notch negative,The signspositive of the, positive middle;four consecutiveMIN(d23, turning d34). angles are The minimal detail distance is the shorter length of andconsecutiveconsecutivedegrees, edges 12and andturn turn the 23ing ingangles are angles angles short of isthe the notch shorter, i.e. ,length d = MIN(d12, of edge The fourpositive, consecutive negative, turning positive and negative; their angle value is edge 23 and edge 34, i.e., d = MIN(d23, d34), where 1 convex vectorsThe signstheir 12 andangleofedge the 56 s.valuemiddle are close is four to 0 1 convex are positive,areapproximately negative, negative, positive, 90 positive degrees, and the23 and angles edged23). of vector34, i.e. 12, d and = d23 and d34 are the lengths of edge 23 and edge 34, consecutiveapproximatelyangles’ valuesturn 90ing degrees,are angles The minimal detail distance negative,and negative negativedegrees.; their vector or anglepositive 56 are, closeTheMIN(d23, to minimal 0 degrees. d34), detail where distance d23 respectively. approximatelyand edges 12 and 90 degrees,23 are short the Theis the minimal shorter detail edge distancelength of 3 multiplenotch - areThenegative,value negative, fourThe is consecutive approximately signs positive positive of the, positive, middle turningpositive 90 four; is consecutiveand the shortestd34 are turning theedge’s lengths angles length, areof 4 symbols areedge alternating,s. and isthe the notch shorter, i.e. length, d = MIN(d12, of edge stepconcave stair andTheangles’t negativeheirnegative, signs angle values of; positive, their valuethe are fourangle positiveis andi.e. negative;, d = MIN their (d12, angle d23, value d34, is The minimal detail distance is the shorter length of 2 concave2 edges 23 and 45 are short The23 minimal and edged23). detail 34, i.e. distance, d = approximatelyconsecutivevalueapproximately is approximately turn 90 degrees,ing 90 angles degrees, 90 the and the anglesd45). of vectors 12 and edge 23 and edge 34, i.e., d = MIN(d23, d34). multiple- Theapproximately four consecutive 90 degrees, turning is the shortestMIN(d23, edge’s d34). length, 4 edgesymbolsdegrees,ares (i.e. negative, are ,and the alternating, themiddle positive, angles56 edges areand of close to 0 degrees. step stair and edges 12 and 23 are short i.e.The, dminimal = MIN (d12,detail d23, distance d34, allangles’ areThe short values signs edges). of are the four consecutive turning angles are vectorsnegative,edges 12 23 negativeand andedge 56 45 s.are areor close positive short to 0, The minimal detail distance approximatelynegative, 90 positive, degrees, negative, the negativeis the shorter or positive,d45). edge negative, length of The minimal detail distance is the shorter edge 3 notch3 multiplenotch - edgenegative,s (i.e. ,degrees.positive the middle , positive edges; is the shortest edge’s length, 4 Thesymbols anglepositive, are between alternating, positive; vector theirand 12 anglethe value notch is approximately, i.e., d = MIN(d12, 90 length of the notch, i.e., d = MIN(d12, d23). step stair The allfourtheir are consecutive angle short valueedges). turningis i.e., d = MIN (d12, d23, d34, edgesand 23 vectoranddegrees, 45 34 are is and short edges 12 and 23 are shortd23). edges. approximatelyTheangles’ signs valuesof the90 degrees, fourare The minimald45). detail is the edgeapproximatelys (i.e., the middle 0 degrees, edges The minimal detail distance andapproximatelyTheconsecutive edgesangle between12The turnand 90 four ing 23degrees, vectorconsecutive are angles short 12 the turningshorter angles’ edge values length are of the 5 multiplespike - edgesallapproximatelyand are23 and vectorshort 34 edges). 34are 90is short degrees, theis symbols the shortest are alternating, edge’s length, and The minimal detail distance is the shortest edge’s 4 multiple-step4 stair symbolsare negative, areedge alternating, s.positive, and acute angle, i.e., d=min(d23, step stair edges, andedges the 23 angle and 45 between are short edgesThei.e.The (i.e.,,minimal d minimal = the MIN middle detail (d12, detail edges d23,distance is the alld34, length, i.e., d = MIN (d12, d23, d34, d45). negative,approximately negative 0 ordegrees, positive , d34) Theedges angle 23 between and 45 are vector shortare 12 shortisshorter edges). the shorter edge edge length length of the of 35 notchspike vectorThenegative,edges four 12 23 consecutiveand positive and vector 34 are, positive 34 turningshort is an ; d45). edges (i.e., the middle edges acutethe notch angle, i.e., i.e., d, =d =MIN(d12,min(d23, edges,theirangles’ andacute angle thevector valuesangle. angle value 34 areisbetween is all are short edges). The minimal detail is the approximatelyThe angle between 0 degrees, vector 12 andThe vector minimal 34d23).d34) is detail approximately distance approximatelyvectorapproximately 12 and vector 90 90 degrees, degrees, 34 is an the shorter edge length of the The minimal detail is the shorter edge length of the 5 spike5 multiplespike - Theedges angle0 23 degrees, between and 34 edges are vector short 23 and 12 34 areis shortthe shortest edges, and edge’s the anglelength, 4 andsymbolsThe edges angleacute are 12 between alternating,and angle. 23 vector are short and 12 acute angle, i.e., d=min(d23, acute angle, i.e., d=min(d23, d34) step stair edgeands, vectorand betweenthe 34 angle is an vector betweenacute 12 and vectori.e., d 34 = is MIN an acute (d12, angle. d23, d34, Dull edgesand 23 edgeandvector 45s. 34are is short d34) 6 vectorangle, 12 edge and 23 vector is the 34 short is an The minimald=d45). d23 detail is the corner Theedgeapproximately angles (i.e. between, the middle 0 degrees,vector edges 12 edgeTheand ,four and vector acute consecutiveedges 34 angle. 12is anand turningacute 34 are shorter edge length of the 5 Dullspike edgesallThe are 23 angle shortand between34 edges). are short vector 12 and vector 34 is an acute angle, 6 Dull corner6 angle,angles’ edgelongedge values23edge is 23 s.theis are theshort short edge,acute and angle edgesd=,12 i.e.d23 and, d = 34min(d23, d= d23 corner edges, and the angle between The minimal detail distance approximatelyedgeTheThe , angle angleand edges betweenbetween 90 12 degrees, and vectorvector 34 arethe1212 long edges. d34) multiple- Thevectorand overall vector 12 and shape 34 vector is isan a acute34ladder is an is theThe shortest minimum edge’s detail length, 4 Dull symbolsand longare vector alternating, edge 34s. is and 6 step stair aboveangle, theedgeacute sector 23 angle. is; thethe shortfour distancei.e.The, d =minimal MIN is dthe= (d12,d23 lengthdetail d23, isof d34, thethe ISPRS Int.Curvecorner J. Geo-Inf. 2020, 9, x FOR PEER REVIEW edgesapproximately 23The and overall 45 0 are degrees, shape short isa ladder above the sector; the four8 of 16 The minimum detail distance is the length of the 7 Curve steps7 edgeconsecutive, and edges turning consecutive12 and angles’ 34 are turning shortestshorter angles’ valuesedgeedged45). including length are of the shortest edge including the sector arc segment, i.e., d 5 stepsspike TheedgeTheedges overall sangle (i.e. 23, between andtheshape middle 34 isare vectora laddershort edges 12 The minimum detail values arelong approximately edges. approximately 90 sectoracute 90 degrees. anglearc segment,, i.e., d=min(d23, i.e., d = = MIN(d12, d23, d34, d45, d67) edgeaboveandalls, and vectorare the theshort sector 34 angle isedges).; tanhe between acute four distance is the length of the Curve degrees. MIN(d12,The minimum d23,d34) d34, detail d45, d67 ) 7 Dull vectorconsecutive 12 and turning vector angles’34 is an shortest edge including the 6 steps Theangle, overall edge shape 23 is is the a ladder short distanceThe minimum isd the= d23 minimum detail corner Thevalues angle areacute betweenapproximately angle. vector 9012 sector arc segment, i.e., d = The minimum detail distance is the minimum value Perforate Aedge roundabove, and holethe edges sector in the 12; t middleandhe four 34 are of valuedistance between is the the length d value of the of 8 Perforated8 Curve anddegrees vectorA round 34. is hole in the middleMIN(d12, of the d23, building. d34, d45, d67) between the d value of the building profile and the 7 d consecutivethelong building turning edges.. angles’ theshortestThe building minimal edge profile includingdetail and is the the steps Theapproximately angle between 0 degrees, vector 12 diameter of the hole, i.e., d = MIN (d23, 2r) values are approximately 90 diametersectorshorter arc edgeof segment, the length hole, i.e. i.e.of ,,the dd == 5 spike Theedgesand overall vector 23 and shape 34 34 is are anis a acuteshort ladder The minimum detail Dull degrees. MIN(d12,acute angleMIN d23,, (d23i.e. d34,, d, 2r= min(d23,d45) , d67) 6 edgeangle,aboves, and edge the the sector 23 angle is ;the t hebetween short four distance isd the= d23 length of the This tablecornerCurve identifies the basic graphical units of building polygons, which are used to calculated34) the LoD parameters for further analysis. 1This7 table identifies the basic graphical unitsedgevectorconsecutive of, and building12 andedges turningvector polygons, 12 and 34 angles’ 34is which anare areshortest used to edge calculate including the LoD the steps parameters for further analysis. values areacutelong approximately edgeangle.s. 90 sector arc segment, i.e., d = degrees. MIN(d12, d23, d34, d45, d67) The angleoverall between shape is vector a ladder 12 The minimum detail 2.2. Minimum Representative Scale andabove vector the sector 34 is ;an the acute four distance is the length of the CurveDull 67 To calculate the minimum representativeconsecutiveangle, edgesize ofturning23 ais feature,the angles’ short we mustshortest construct edged= d23 aincluding dataset forthe cornersteps geographical features (including natural edgevalues and, and are artificialedges approximately 12 andfeatures) 34 are90 on somesector scales,arc segment, identify i.e. , thed = mi nimum graphical units by using the approachlongdegrees detailed edge.s. in Section MIN(d12,2.1, and calculate d23, d34, the d45 LoD, d67 ) parameters of each graphical unit. For natural curve features, which include abundant basic graphical units (leaf bends), the smallestThe leaf overall bend shapetheoretically is a ladder repres ents Thethe mostminimum detailed detail level. Due to the need for smooth and aestheticabove representation, the sector; theredundant four pointsdistance are isoften the length included of the in Curve natural7 curve features. In a map, very fewconsecutive parts of turningthe curve angles’ converge shortest together edge and including are visually the steps acceptable. In this section, we calculate values the minimum are approximately representative 90 scalesector for arc a setsegment, of leaf i.e. bends, d = using mathematical statistics. degrees. MIN(d12, d23, d34, d45, d67) The process we use to calculate the minimum representative scale consists of the following steps. First, we divide the LoD parameters for all the leaf bends of each curve into quartiles. Identifying the three levels of parameters as Q1, Q2, and Q3, we calculate the average value of all parameters smaller than the lower quartile point (Q1) and use this value as the LoD parameter of the curve feature. Next, we eliminate outliers (normally, greater than Q3+1.5*(Q3-Q1) or less than Q1-1.5*(Q3-Q1)). Following this, we can calculate the LoD parameter and estimate the actual scale of a curve using Equation (2). If the nominal scale of the map is known, we can compare it with the actual scale. For each curve, the average value of the LoD parameter is used to evaluate the overall detail level of all features. The standard deviation of the detail parameter can be used to differentiate the detail levels among different curves in the dataset. When the number of leaf bends of a curve is not large enough, we can combine the leaf bends of all the curves and calculate the average value of all parameters below the lower quartile point (Q1). This average value can be used as an overall LoD parameter for this kind of curve. Because basic graphical units are not abundant in polygons depicting artificial features such as buildings, it is inappropriate to analyze the LoD parameter for each feature. Instead, we analyze the basic graphical unit parameters of all building polygons, eliminate outliers, and calculate the average value (Ave) and mean square error (δ) of all parameters for the lower quartile point (Q1). The average value can be a LoD parameter for this batch of building polygons. After we determine the actual scale of the building polygons (1: ), we can compare it to the nominal scale (1: ). We must set an allowed scale variation threshold value t (0 < t < 1; t = 0.1 in this paper). When the 𝑀𝑀𝑅𝑅 𝑀𝑀𝑁𝑁 actual scale meets the (1 ) < < (1 + ) criterion, it is considered consistent with the

nominal scale. 𝑁𝑁 𝑅𝑅 𝑁𝑁 The mean square error− 𝑡𝑡, δ,∗ can𝑀𝑀 be 𝑀𝑀used to describe𝑡𝑡 ∗ 𝑀𝑀 LoD consistency. The greater the mean square error is, the greater the difference in the LoDs among the features will be. When δ is less than or equal to 20% of the average value ( 0≤≤δ 0.2* Ave ), the LoD can be considered consistent. If δ is greater than 20% of the average value and less than or equal to 30% of the average value (i.e., 0.2*Ave≤≤δ 0.3* Ave ), consistency is considered moderate. If δ is greater than 30% of the average value (i.e.,δ ≥ 0.3* Ave ), consistency is poor.

3. Experiment and Analysis

3.1. Contour Lines For our analysis, we selected 50 contour lines from a 1:5K topographical map. Table 2 provides

ISPRS Int. J. Geo-Inf. 2020, 9, 410 8 of 16

2.2. Minimum Representative Scale To calculate the minimum representative size of a feature, we must construct a dataset for geographical features (including natural and artificial features) on some scales, identify the minimum graphical units by using the approach detailed in Section 2.1, and calculate the LoD parameters of each graphical unit. For natural curve features, which include abundant basic graphical units (leaf bends), the smallest leaf bend theoretically represents the most detailed level. Due to the need for smooth and aesthetic representation, redundant points are often included in natural curve features. In a map, very few parts of the curve converge together and are visually acceptable. In this section, we calculate the minimum representative scale for a set of leaf bends using mathematical statistics. The process we use to calculate the minimum representative scale consists of the following steps. First, we divide the LoD parameters for all the leaf bends of each curve into quartiles. Identifying the three levels of parameters as Q1, Q2, and Q3, we calculate the average value of all parameters smaller than the lower quartile point (Q1) and use this value as the LoD parameter of the curve feature. Next, we eliminate outliers (normally, greater than Q3 + 1.5 (Q3 Q1) or less than Q1 1.5 (Q3 ∗ − − ∗ Q1)). Following this, we can calculate the LoD parameter and estimate the actual scale of a curve − using Equation (2). If the nominal scale of the map is known, we can compare it with the actual scale. For each curve, the average value of the LoD parameter is used to evaluate the overall detail level of all features. The standard deviation of the detail parameter can be used to differentiate the detail levels among different curves in the dataset. When the number of leaf bends of a curve is not large enough, we can combine the leaf bends of all the curves and calculate the average value of all parameters below the lower quartile point (Q1). This average value can be used as an overall LoD parameter for this kind of curve. Because basic graphical units are not abundant in polygons depicting artificial features such as buildings, it is inappropriate to analyze the LoD parameter for each feature. Instead, we analyze the basic graphical unit parameters of all building polygons, eliminate outliers, and calculate the average value (Ave) and mean square error (δ) of all parameters for the lower quartile point (Q1). The average value can be a LoD parameter for this batch of building polygons. After we determine the actual scale of the building polygons (1:MR), we can compare it to the nominal scale (1:MN). We must set an allowed scale variation threshold value t (0 < t < 1; t = 0.1 in this paper). When the actual scale meets the (1 t) MN < MR < (1 + t) MN criterion, it is considered consistent with the nominal scale. − ∗ ∗ The mean square error, δ, can be used to describe LoD consistency. The greater the mean square error is, the greater the difference in the LoDs among the features will be. When δis less than or equal to 20% of the average value (0 δ 0.2 Ave), the LoD can be considered consistent. If δis greater than 20% of the ≤ ≤ ∗ average value and less than or equal to 30% of the average value (i.e., 0.2 Ave δ 0.3 Ave), consistency ∗ ≤ ≤ ∗ is considered moderate. If δ is greater than 30% of the average value (i.e., δ 0.3 Ave), consistency is poor. ≥ ∗ 3. Experiment and Analysis

3.1. Contour Lines For our analysis, we selected 50 contour lines from a 1:5K topographical map. Table2 provides the basic descriptions for six of them. We constructed bend-level trees, obtained all the leaf bends, and calculated the LoD parameter for each curve. In Figure6, the Figure6A,B show the left and right leaf bends of the No.6 contour line with some close-ups in the rectangles. Table3 lists the corresponding parameters for the left and right leaf bends of the curve. In the table, ‘l’ is the LoD parameter. ISPRS Int. J. Geo-Inf. 2020, 9, x FOR PEER REVIEW 9 of 16 the basic descriptions for six of them. We constructed bend-level trees, obtained all the leaf bends, and calculated the LoD parameter for each curve. In Figure 6, the subfigures A and B show the left and right leaf bends of the No.6 contour line with some close-ups in the rectangles. Table 3 lists the corresponding parameters for the left and right leaf bends of the curve. In the table, ‘l’ is the LoD parameter.ISPRSISPRS Int.Int. J.J. GeoGeo--Inf.Inf. 2020,, 9,, xISPRSx FORFORISPRS Int.PEERPEER Int. J. Geo J.REVIEWREVIEW Geo-Inf.-Inf. 20 2020, 9,, 9x,, FORxx FORFOR PEER PEERPEER REVIEW REVIEWREVIEW 9 of 16 9 of9 of16 16

thethe basic descriptions forthethe sixbasic basic of them.descriptions descriptions We construct for for six six edof of them.bend them.-- levellevelWe We construct trees, trees,construct obtainobtaineded bended bend all-level -thelevel leaftrees, trees, bends, obtain obtain eded all all the the leaf leaf bends, bends, and calculated thethe LoDand parameterand calculate calculate forTabled d theeach the LoD LoDcurve.2. parameterParameters parameter In Figure for for 6 each,, the theeachfor curve.subfiguressubfigures curve.some In In Figureof FigureA theand 6, B 6experimentalthe, showthe subfigures subfigures the left A A and curves. and B Bshow show the the left left and right leaf bends ofand thetheand No.right right6 contourleaf leaf bends bends line of ofwith the the No. some No.6 6contour closecontour--ups line line in with the with rectangles.some some close close Table-ups-ups in 3 in thelistslists the rectangles. the therectangles. Table Table 3 3lists lists the the corresponding parameterscorrespondingcorresponding for the left parametersand parameters right leaf for for bendsthe the left leftof and theand# ofrightcurve. right leaf leafIn bendsthe bends table, of of the ‘ ‘lthel’’ isiscurve. curve.thethe LoD In In the the table, table, ‘l ’‘ lis’ isthe the LoD LoD # of parameter. parameter.parameter. # of left ISPRSparameter. Int. J. Geo-Inf. 2020parameter., 9#, 410of right # of # of left right9 of 16 No. thumbnail leaf No. thumbnail Tablepoints 2. Parameters for someTableTable of 2.the 2.Parameters Parametersexperimentalleaf for for somecurves. some of of the the experimental experimental curves. curves. points bends leaf # ofbends # of# of # of # of# of # of left # of# of left left bends bends # of Tableright 2.# of# Parameters of forrightright some # of of the# of experimentalleft right# of# of curves.# of# of left left rightright No. thumbnailthumbnail No.No. thumbnailthumbnailleafleaf No.leaf leafthumbnailthumbnail No.No. thumbnailthumbnail points pointsleafleafpoints leafleaf points bends leafleafpointspoints bendsbends leafleaf 1 2bends8399 2552 bendsbends 3491 4 24255 2085 2734 No. Thumbnail # of Points # of Leftbends Leaf Bends # of Rightbends Leafbends Bends No. Thumbnailbends # of Points # of Leftbendsbends Bends # of Right Leaf Bends 1 1 28399283991 1 2552 255234912839928399 4 2552 2552 349134913491 242554 4 4 2085 2734242552425524255 20852085 208527342734 2734 2 26392 2380 329 6 5 4190 381 489 2 2 26392263922 2380 2380329263926 5 2380 32963296 41905 5 381 4894190 4190 381 381489 489 2 263922 2380 329263926 5 2380 3296 4190 5 381 4894190 381 489

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Figureigure 6. The ((A)) and ((B)) FleafleafigureFigure bendsbends 6. 6.The andandThe (A fourfour()A and) and closeclose (B ()B --leafups) leaf bendsalong bends theand and No. four four 6 close contour close-ups-ups line along along in Tablethe the No. 1No... 6 contour6 contour line line in in Table Table 1. 1.

Table 3. The LoD parametersTableTable for 3. the 3.The The left LoD LoDand parameters rightparameters leaf bendsfor for the the of left theleft and No.and right 6 right contour. leaf leaf bends bends of of the the No. No. 6 contour.6 contour.

Left leaf bend LeftLeft leafRight leaf bend bendleaf bend RightRight leaf leaf bend bend NO. w(m) d(m) l(m)l(m)NO.NO. NO. w(m)w(m) w(m) d(m)d(m) d(m) l(m)l(m) l(m) l(m)NO.NO. w(m)w(m) d(m)d(m) l(m)l(m) 1 16.6 2.3 2.31 1 116.6 16.6 3.32.3 2.3 0.32.32.3 0.31 1 3.33.3 0.30.3 0.30.3 2 15.9 3.0 3.02 2 215.9 15.9 3.03.0 3.0 0.33.03.0 0.32 2 3.03.0 0.30.3 0.30.3 3 12.3 2.4 2.43 3 312.3 12.3 22.92.4 2.4 3.42.42.4 3.43 3 22.922.9 3.43.4 3.43.4 4 16.9 2.8 2.84 4 416.9 16.9 14.82.8 2.8 4.62.82.8 4.64 4 14.814.8 4.64.6 4.64.6 Figure 6. The5 (A21.7) and 4.9 (B ) leaf4.95 5 bends521.7 21.7 16.7 and4.9 4.9 four6.54.94.9 close-ups 6.55 5 16.716.7 along 6.56.5 the 6.56.5 No. 6 contour line in Table1. Figure 6. The6 (A30.5) and 14.9 (B ) 14.9leaf6 6 bends630.5 30.5 11.1 and14.914.9 four 2.714.914.9 close 2.76 6 -ups11.111.1 along 2.72.7 the 2.72.7 No. 6 contour line in Table 1. 7 28.9 13.6 13.67 728.9 3.613.6 0.313.6 0.37 3.6 0.3 0.3 Table7 3. The28.9 LoD13.6 parameters13.6 7 7 28.9 for3.6 13.6 the 0.3 left13.6 0.3 and 7 right3.6 leaf 0.3 bends 0.3 of the No. 6 contour. 8 9.4 4.2 4.28 8 89.4 9.4 4.64.2 4.2 0.34.24.2 0.38 8 4.64.6 0.30.3 0.30.3 Table9 3. The7.5 LoD1.7 parameters1.79 9 97.5 7.5 for2.31.7 1.7 the 0.21.7 left1.7 0.2and9 9 right2.32.3 leaf0.20.2 bends 0.20.2 of the No. 6 contour. 10 37.2 8.8Left 8.810 Leaf 10 1037.2 Bend 37.2 2.18.8 8.8 0.28.88.8 0.210 10 2.12.1 Right 0.20.2 Leaf 0.20.2 Bend 11 9.5 0.4 0.4 11 12.5 2.2 2.2 11 NO.9.5 0.4 w(m)Left0.4 11 leaf 11 d(m) 9.5bend 12.5 0.4 l(m) 2.20.4 2.2 11 NO. 12.5Right w(m) 2.2 leaf 2.2 d(m)bend l(m) … … … …… … …… … …… … …… … …… … …… …… …… 1 16.6 2.3 2.3 1 3.3 0.3 0.3 NO. w(m) d(m) l(m) NO. w(m) d(m) l(m) 21 15.916.6 3.02.3 3.02.3 1 2 3.3 3.0 0.3 0.3 0.3 0.3 3 12.3 2.4 2.4 3 22.9 3.4 3.4 2 15.9 3.0 3.0 2 3.0 0.3 0.3 4 16.9 2.8 2.8 4 14.8 4.6 4.6 53 21.712.3 4.92.4 4.92.4 3 5 22.9 16.7 3.4 6.5 3.4 6.5 64 30.516.9 14.92.8 14.92.8 4 6 14.8 11.1 4.6 2.7 4.6 2.7 75 28.921.7 13.64.9 13.64.9 5 7 16.7 3.6 6.5 0.3 6.5 0.3 86 9.430.5 4.214.9 14.9 4.2 6 8 11.1 4.6 2.7 0.3 2.7 0.3 9 7.5 1.7 1.7 9 2.3 0.2 0.2 7 28.9 13.6 13.6 7 3.6 0.3 0.3 10 37.2 8.8 8.8 10 2.1 0.2 0.2 118 9.59.4 0.44.2 0.44.2 118 4.6 12.5 0.3 2.2 0.3 2.2 ...... 9 7.5 1.7 1.7 9 2.3 0.2 0.2 10 37.2 8.8 8.8 10 2.1 0.2 0.2 11 9.5 0.4 0.4 11 12.5 2.2 2.2 In Figure7a–f, L, R, and LR correspond to the box plots of left bends, right bends, and all bends, … … … … … … … … respectively. The “+” signs in the ﬁgure represent outliers. Figure7 shows that there is a large range in the median of the detail level parameters among the six contour lines. It also illustrates the necessity of considering both left and right bends to evaluate the detail level of a curve. ISPRS Int. J. Geo-Inf. 2020, 9, x FOR PEER REVIEW 10 of 16

In Figure 7a-f, L, R, and LR correspond to the box plots of left bends, right bends, and all bends,

ISPRSrespectively. Int. J. Geo-Inf. The2020 “+”, 9 signs, 410 in the figure represent outliers. Figure 7 shows that there is a large10 range of 16 in the median of the detail level parameters among the six contour lines. It also illustrates the necessity of considering both left and right bends to evaluate the detail level of a curve.

Figure 7. Boxplots of the LoD parameters based on the left bends, right bends, and all bends for six contourFigure 7 lines. Boxplots (the “ +of” the symbol LoD representsparameters abnormal based on values). the left (bends,a) Boxplot right of bends, the LoD and parameters all bends forfor thesix No.1contour contour lines line.(the “+” (b) Boxplotsymbol represents of the LoD abnormal parameters values). for the ( No.1a) Boxplot contour of line.the LoD (c) Boxplotparameters of the for LoD the parametersNo.1 contour for line the. ( No.3b) Boxplot contour of line.the LoD (d) Boxplot parameters of the for LoD the parametersNo.1 contour for line the. No.4(c) Boxplot contour of line. the (LoDe) Boxplot parameters of the LoDfor the parameters No.3 contour for the line No.5. (d) contour Boxplot line. of the (f) LoD Boxplot parameters of the LoD for parameters the No.4 contour for the No.6line. ( contoure) Boxplot line. of the LoD parameters for the No.5 contour line. (f) Boxplot of the LoD parameters for the No.6 contour line. Figure8 is the boxplot of all of the bends (left and right) of six curves after removing outliers. This ﬁgureFigure shows 8 is the the boxplot lower quartile of all of of the the bends leaf bend (leftparameter and right) of of each six curves curve is after around removing 2.0 m. Weoutliers. used theThis average figure shows value the of all lower parameters quartile underof the theleaf lowerbend parameter quartile of of each each curve curve as is itsaround LoD parameter.2.0 m. We Theused third the rowaverage in Table value4 shows of all the parameters calculated under scale for the each lower curve. quartile Note thatof each the nominalcurve as scale its andLoD thoseparameter. calculated The third for the row 5th, in 6th,Table 8th, 4 shows 22nd, 26ththe calculated and 30th curvesscale for are each inconsistent. curve. Note The that average the nominal detail parameterscale and ofthose the 50calculated curves is for 1.554 the m, 5th, with 6th, a mean 8th, square22nd, 26th error andδ of 0.10530th curves m. Because are inconsistent. this mean square The erroraverage value detail is less parameter than 20% of ofthe the 50 averagecurves is (i.e., 1.5540 mδ, with0.2 a meanAve ), square we consider error δ the of LoD0.105 of m. the Because curve ≤ ≤ ∗ highlythis mean consistent. square error value is less than 20% of the average (i.e., 0≤≤δ 0.2* Ave ), we consider the LoD of the curve highly consistent.

Table 4. LoDs for the 50 contour lines and the consistency of the nominal scale and actual scale.

No. 1 2 3 4 5 6 7 8 9 10 LoD (m) 1.529 1.5 1.566 1.651 1.872 1.378 1.543 1.653 1.453 1.621

ISPRS Int. J. Geo-Inf. 2020, 9, x FOR PEER REVIEW 11 of 16

Scale denominator 5097 5000 5220 5503 6240 4593 5143 5510 4843 5403 Scale consistency YES YES YES YES NO NO YES NO YES YES No. 11 12 13 14 15 16 17 18 19 20 LoD (m) 1.394 1.603 1.743 1.464 1.583 1.632 1.39 1.567 1.583 1.445 Scale denominator 4647 5343 5810 4880 5277 5440 4633 5223 5277 4817 Scale consistency YES YES YES YES YES YES YES YES YES YES No. 21 22 23 24 25 26 27 28 29 30 LoD (m) 1.567 1.704 1.6 1.456 1.532 1.8 1.542 1.64 1.543 1.673 Scale denominator 5223 5680 5333 4853 5107 6000 5140 5467 5143 5577 Scale consistency YES NO YES YES YES NO YES YES YES NO No. 31 32 33 34 35 36 37 38 39 40 LoD (m) 1.542 1.621 1.543 1.456 1.432 1.532 1.502 1.542 1.632 1.432 Scale denominator 5140 5403 5143 4853 4773 5107 5007 5140 5440 4773 Scale consistency YES YES YES YES YES YES YES YES YES YES No. 41 42 43 44 45 46 47 48 49 50 LoD (m) 1.554 1.376 1.534 1.543 1.64 1.502 1.435 1.583 1.623 1.46 Scale denominator 5180 4587 5113 5143 5467 5007 4783 5277 5410 4867 Scale consistency YES YES YES YES YES YES YES YES YES YES The interval allowed by the actual scale is 1/4500–1/5500; the actual scale of 6 out of 50 curves is inconsistent with the nominal scale. The average value of the detail level parameters of the 50 curves is 1.554. The mean square deviation (δ ) is 0.105, which meets 0≤≤δ 0.2* Ave . In general, the consistency of LoD is high. The actual average scale of the 50 curves is 1/5180. This is consistent with ISPRS Int. J. Geo-Inf. 2020, 9, 410 11 of 16 the nominal scale.

Figure 8. Boxplots of the LoD parameters based on the leaf bends of the six curves (after removing Figure 8. Boxplots of the LoD parameters based on the leaf bends of the six curves (after removing the outliers). the outliers). Table 4. LoDs for the 50 contour lines and the consistency of the nominal scale and actual scale. 3.2. Buildings No. 1 2 3 4 5 6 7 8 9 10 Given two LoDmaps (m) of the 1.529same region 1.5 1.566 of a city, 1.651 one 1.872 having 1.378 a 1.543scale 1.653of 1:2K 1.453 and 1.621the other of 1:5K, we selected Scalean area denominator of 17,500 5097 m2 for 5000 our 5220 experiment. 5503 6240 Figure 4593 9a 5143shows 5510 the 1:2K 4843 experimental 5403 area, Scale consistency YES YES YES YES NO NO YES NO YES YES where we detected the basic graphical unit of the building polygon utilizing the approach detailed in No. 11 12 13 14 15 16 17 18 19 20 Section 2.1.3. Figure 9b shows the details of an area corresponding to the red box in Figure 9a. Each LoD (m) 1.394 1.603 1.743 1.464 1.583 1.632 1.39 1.567 1.583 1.445 red solid areaScale represents denominator the 4647 smallest 5343 graphical 5810 4880 unit 5277detected. 5440 Figur 4633e 9 5223d show 5277s the 4817 results detected for the sameScale area consistency from the YES 1:5K YESmap. YESUsing YES the YESquartile YES method YES YESdescribed YES in YES Section 2.2, we analyzed the dataNo. for these 21two 22different 23 scales 24 (Figure 2526 10). 27The mean 28 LoDs 29 30for the building polygons on theLoD 1:2K (m) map is 1.5670.55 m. 1.704 Given 1.6 a map 1.456 distance 1.532 of 1.8 0.3 1.542 mm, we 1.64 determine 1.543 1.673d that the actual Scale denominator 5223 5680 5333 4853 5107 6000 5140 5467 5143 5577 map scale is 1:1850Scale consistency for the 1:2K YES map. NO In contrast, YES YES the YESLoD for NO the YES 1:5K map YES is YES 1.350 NO m, and its actual scale 1:4500. The twoNo. actual scales 31 are 32 both 33 with 34in the 35 tolerance 36 range 37 of 38 the map 39 scales. 40 LoD (m) 1.542 1.621 1.543 1.456 1.432 1.532 1.502 1.542 1.632 1.432 Scale denominator 5140 5403 5143 4853 4773 5107 5007 5140 5440 4773 Scale consistency YES YES YES YES YES YES YES YES YES YES No. 41 42 43 44 45 46 47 48 49 50 LoD (m) 1.554 1.376 1.534 1.543 1.64 1.502 1.435 1.583 1.623 1.46 Scale denominator 5180 4587 5113 5143 5467 5007 4783 5277 5410 4867 Scale consistency YES YES YES YES YES YES YES YES YES YES The interval allowed by the actual scale is 1/4500–1/5500; the actual scale of 6 out of 50 curves is inconsistent with the nominal scale. The average value of the detail level parameters of the 50 curves is 1.554. The mean square deviation (δ) is 0.105, which meets 0 δ 0.2 Ave In general, the consistency of LoD is high. The actual average scale of the 50 curves is 1/5180. This is≤ consistent≤ ∗ with the nominal scale.

3.2. Buildings Given two maps of the same region of a city, one having a scale of 1:2K and the other of 1:5K, we selected an area of 17,500 m2 for our experiment. Figure9a shows the 1:2K experimental area, where we detected the basic graphical unit of the building polygon utilizing the approach detailed in Section 2.1.3. Figure9b shows the details of an area corresponding to the red box in Figure9a. Each red solid area represents the smallest graphical unit detected. Figure9d shows the results detected for the same area from the 1:5K map. Using the quartile method described in Section 2.2, we analyzed the data for these two diﬀerent scales (Figure 10). The mean LoDs for the building polygons on the 1:2K map is 0.55 m. Given a map distance of 0.3 mm, we determined that the actual map scale is 1:1850 for the 1:2K map. In contrast, the LoD for the 1:5K map is 1.350 m, and its actual scale 1:4500. The two actual scales are both within the tolerance range of the map scales. ISPRS Int. J. Geo-Inf. 2020, 9, x FOR PEER REVIEW 12 of 16 ISPRS Int. J. Geo-Inf. 2020, 9, 410 12 of 16 ISPRS Int. J. Geo-Inf. 2020, 9, x FOR PEER REVIEW 12 of 16

Figure 9. Building polygon experimental data and detected graphical units. (a) 1:2K building Figurepolygon 9. dataBuilding. (b) Graphical polygon units experimental within the data red andbox detectedin Figure graphical9a. (c) 1:5K units. scale (buildinga) 1:2K buildingpolygon Figure 9. Building polygon experimental data and detected graphical units. (a) 1:2K building polygondata. (d) data.Close (-bup) Graphical of graphical units units within within the redthe red box box in Figure in Figure9a. ( 9cc). 1:5K scale building polygon data. polygon(d) Close-up data of. ( graphicalb) Graphical units units within within the redthe boxred inbox Figure in Figure9c. 9a. (c) 1:5K scale building polygon data. (d) Close-up of graphical units within the red box in Figure 9c.

Figure 10. Boxplot of LoDs for the building polygons from 1:2K (a) and 1:5K (b) maps. Figure 10. Boxplot of LoDs for the building polygons from 1:2K (left) and 1:5K (right) maps. 3.3. Scale Inconsistency Detection Figure 10. Boxplot of LoDs for the building polygons from 1:2K (left) and 1:5K (right) maps. 3.3. ScaleAs for Inconsistency the whole Detection map in VGI data, some geographic features that are not consistent with the actual scale may be mixed into the map data uploaded by volunteers, resulting in the phenomenon As for the whole map in VGI data, some geographic features that are not consistent with the 3.3.that Scale the geospatialInconsistency information Detection does not match the real map scale. The method proposed in this actual scale may be mixed into the map data uploaded by volunteers, resulting in the phenomenon paperAs can for detect the whole the scale map inconsistency in VGI data, caused some bygeographic the large-scale feature datas that in theare small-scalenot consistent map. with For the that the geospatial information does not match the real map scale. The method proposed in this actualdetection scale of may natural be geographicalmixed into the features, map data in the uploaded ﬁrst step, by we volunteers, selected some resulting river in elements the phenomenon in Sichuan paper can detect the scale inconsistency caused by the large-scale data in the small-scale map. For that the geospatial information does not match the real map scale. The method proposed in this Provincethe detection of China of natural with ageographical nominal scale feature of 1:2,000,000s, in the asfirst the step, original we select data,ed and some obtained river elements the left leaf in paper can detect the scale inconsistency caused by the large-scale data in the small-scale map. For detailSichuan level Province parameters of China and rightwith leafa nominal detail levelscale parameters of 1:2,000,000 contained as the original in each river,data, respectively,and obtained then the the detection of natural geographical features, in the first step, we selected some river elements in calculatedleft leaf detail the map level scale parameters corresponding and right to the leaf actual detail situation level afterparameters integration; contained the second in each step river was, Sichuan Province of China with a nominal scale of 1:2,000,000 as the original data, and obtained the torespectively, select the riverthen datacalculate of thed samethe map area scale with corresponding a nominal scale to ofthe 1:750,000 actual situation as the “wrong after integration; data” and left leaf detail level parameters and right leaf detail level parameters contained in each river, selectthe second some step of them was to to replace select the originalriver data data of withthe same a scale area of 1:2,000,000,with a nominal so that scale the of mixed 1:750,000 data as could the respectively, then calculated the map scale corresponding to the actual situation after integration; be“wrong considered data” asand the select VGI some data uploadedof them to by replace volunteers; the original in the lastdata step, with we a scale calculated of 1:2,000,000, the actual so mapthat the second step was to select the river data of the same area with a nominal scale of 1:750,000 as the scalethe mixed of the data mixed could data, be and conside comparedred as it withthe VGI the resultsdata uploaded in the ﬁrst by step. volunteers; Figure 11in demonstratesthe last step, thewe “wrong data” and select some of them to replace the original data with a scale of 1:2,000,000, so that experimentalcalculated the data. actual map scale of the mixed data, and compared it with the results in the first step. the mixed data could be considered as the VGI data uploaded by volunteers; in the last step, we Figure 11 demonstrates the experimental data. calculated the actual map scale of the mixed data, and compared it with the results in the first step. Figure 11 demonstrates the experimental data.

ISPRS Int. J. Geo-Inf. 2020, 9, 410 13 of 16 ISPRS Int. J. Geo-Inf. 2020, 9, x FOR PEER REVIEW 13 of 16

Figure 11. Replacing some large-scale geographic features into a small-scale map. (a) Original small-scale map. Figure 11. Replacing some large-scale geographic features into a small-scale map. (a) Original (b) Mixed map. (The red polylines are the original small-scale geographic features with a scale of 1:2,000,000; small-scale map. (b) Mixed map. (The red polylines are the original small-scale geographic features the blue polylines represent “Wrong Data” on a large scale, with a scale of 1:750,000). with a scale of 1: 2,000,000; the blue polylines represent “Wrong Data” on a large scale, with a scale of 1:Through 750,000.) the analysis of the results in Table5, it can be found that the nominal scale of the original map is 1:2,000,000, which is close to the actual calculated scale of 1:1,997,362. When some large-scale Through the analysis of the results in Table 5, it can be found that the nominal scale of the geographic features are mixed into the original map, the level of detail in the map will change to a original map is 1:2,000,000, which is close to the actual calculated scale of 1:1,997,362. When some large extent, resulting in the reduction of the level of detail parameter l. Therefore, the actual scale of large-scale geographic features are mixed into the original map, the level of detail in the map will changethe mixed to a map large calculated extent, resulting by the method in the reduction in this paper of the will level also of change detail parameter greatly, from l. Therefore, 1:1,997,362 the to actual1:819,442, scale so of this the method mixed canmap be calculated used to detect by the the method scale inconsistency in this paper in will natural also geographicalchange greatly, features. from 1:1,997,362 to 1:819,442, so this method can be used to detect the scale inconsistency in natural Table 5. Parameters for the original map and the mixed map. geographical features. Original Map Mixed Map ParametersTable 5. Parameters for the original map and the mixed map. L R LR(LoD) L R LR(LoD) Original map Mixed map ParametersQ1 (m) 2951.554 2986.015 2972.630 1274.871 1243.072 1255.090 Q2 (m) 4344.659L 4411.786R 4392.251LR(LoD) 3234.362L 3149.018R 3207.992LR(LoD) Q1 (m)Q3 (m) 2951.5546201.038 2986.0156386.454 6313.4612972.630 5316.6511274.871 5441.5231243.072 5385.743 1255.090 l (m) 1976.980 2018.262 1997.362 813.389 825.634 819.442 Q2 (m) 4344.659 4411.786 4392.251 3234.362 3149.018 3207.992 Nominal scale 1:2,000,000 1:2,000,000 & 1:750,000 Q3Calculated (m) scale6201.038 6386.454 1:1,997,362 6313.461 5316.651 1:819,4425441.523 5385.743 l (m) 1976.980 2018.262 1997.362 813.389 825.634 819.442 Nominal scale 1:2,000,000 1:2,000,000 & 1:750,000 For the detection of building scale inconsistency, we chose a building group in a certain area Calculated scale 1:1,997,362 1:819,442 of China with a scale of 1:30,000 as the original data. The procedure was the same as that of the natural geographic element detection. Firstly, the actual map scale was calculated by the detail level For the detection of building scale inconsistency, we chose a building group in a certain area of parameters of the left and right leaves of each building; secondly, large-scale map data with a nominal China with a scale of 1:30,000 as the original data. The procedure was the same as that of the natural scale of 1:10,000 was selected to replace the buildings in the scale of 1:30,000; ﬁnally, the actual scale of geographic element detection. Firstly, the actual map scale was calculated by the detail level parametersthe “VGI like of data” the left was and calculated. right leaves of each building; secondly, large-scale map data with a nominalThe scale experimental of 1:10,000 results was selected are the sameto replace as those the ofbuildings the natural in the geographical scale of 1:30,000; features, finally, and the actualdata obtained scale of the in Table “VGI6 likeprove data that” was the methodcalculated has. a good discrimination e ﬀect on the inconsistency of building scale in the map. For the original building map, the nominal scale is 1:10,000, and the calculatedThe experimental scale is 1:9339; results these are are the close same to each as those other. of When the natural some buildings geographical from thefeatures large-scale, and map theare mixeddata obtained into the in original Table map,6 prove the actualthat the scale method of the has map a isgood increased discrimination from 1:9339 effect to about on the 1:2600, inconsistencyresulting in an of inconsistent building scale map in scale. the map. For the original building map, the nominal scale is 1:10,000, and the calculated scale is 1:9339; these are close to each other. When some buildings from the large-scale map are mixed into the original map, the actual scale of the map is increased from 1:9339 to about 1:2600, resulting in an inconsistent map scale.

Table 6. Parameters for maps in Figure 12.

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Parameters OriginalTable map 6. ParametersMap for for replacement maps in Figure Mixed 12. map 1 Mixed map 2 Q1 (m) 17.157 3.969 4.259 4.212 ParametersQ2 (m) Original24.495 Map Map for Replacement7.128 Mixed8.043 Map 1 Mixed7.752 Map 2 Q1Q3 (m)(m) 17.15742.157 3.96911.467 4.25912.893 14.925 4.212 LoDQ2 (m) (m) 24.4959.339 7.1282.473 8.0432.662 7.7522.626 NominalQ3 (m) scale 42.1571:30,000 11.4671:10,000 12.8931:30,000 & 1:10,000 14.925 LoD (m) 9.339 2.473 2.662 2.626 CalculatedNominal scale scale 1:30,0001:31,130 1:10,0001:8243 1:8 1:30,000873 & 1:10,0001:8753 Calculated scale 1:31,130 1:8243 1:8873 1:8753

Figure 12.12. ReplacReplacinging some large-scalelarge-scale buildingsbuildings into aa small-scalesmall-scale map.map. (a) Original small-scalesmall-scale buildings.buildings. ( (bb)) Large scale buildings for replacement.replacement. ( c)) Mixed map 1.1. ( d) Mixed map 2.2. (The red polygons represent the original small-scalesmall-scale geographic building buildingss with with a a scale scale of of 1: 1:30,000, 30,000, the blue polygons represent “Wrong“Wrong Data”Data” onon aa largelarge scale,scale, withwith aa scalescale ofof 1:10,000).1: 10,000.)

4. Conclusions Previous research on spatial data quality has focused primarily on integrity [26,,27],], positionposition accuracy [28,29], attribute accuracy, shapeshape similaritysimilarity [[3030],], andand somesome otherother aspectsaspects [[31]31].. These studies have rarely focused on LoD consistency. Given the ubiquity of data collection and the diversification diversiﬁcation of data processing, maps are often produced using various data and a wide range of data processing methods for a wide range of purposes.purposes. In In order to ensure that a map is an objectiveobjective and scientiﬁcscientific representation of the observable world, consistency analysis on data LoD is critical. To perform this analysis and resolve LoD inconsistencies, users must calculate the actual representative scale of map features and then classify or processprocess map features based on the LoD.LoD. This will ensure that the scale deviation of map features is within tolerance limits.limits. Consistency evaluation of the LoD for digital map products is also a vital part of the quality control process. This paper presents a method for detecting the basic graphical units of different

ISPRS Int. J. Geo-Inf. 2020, 9, 410 15 of 16

Consistency evaluation of the LoD for digital map products is also a vital part of the quality control process. This paper presents a method for detecting the basic graphical units of different geographical features using both computational geometry and statistics. From these graphical units, we can obtain feature LoDs and assess the deviation of the LoD from a nominal scale to measure its consistency. The method proposed in this paper is based on the map relationship between distinguishable distance and its minimum graphical unit, with consideration of human visual limitations. Furthermore, this method works well for both natural linear features and building polygons. However, it is worth noting that we cannot use the method in this paper to evaluate the inconsistency of geographical features in all cases. This is because, for some physical geographical features, such as smooth slopes in high mountains and transition zones between rock terrain, there may be a situation where smooth and rugged contours exist at the same time. In this case, the inconsistency of LoD can actually reflect the characteristics of the objective world very well. Therefore, in the next research process, we need to make some improvements to the methods in this paper, so that physical geographical features that can reflect the actual geographical features well will not in fact be judged as an LoD inconsistency. Besides, we need to further study the definition of the basic graphic units of other types of features and the detection of their shapes and try to select different statistical models to achieve the calculation of the comprehensive LoD parameters of massive data sets, so as to achieve the detection of map inconsistency on the big data scale.

Author Contributions: Formal analysis, Jia Xiao; resources, Pengcheng Liu; writing—original draft, Pengcheng Liu; writing—review and editing, Jia Xiao. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Fundamental Research Funds for the Central Universities, grant number CCNU30106190454 and Open Research Fund Program of Key Laboratory of Digital Mapping and Land Information Application Engineering, grant number ZRZYBWD201909. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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