Revista Cartográfica 103 | julio-diciembre 2021 | Artículos

ISSN (impresa) 0080-2085 | ISSN (en línea) 2663-3981


Este es un artículo de acceso abierto bajo la licencia CC BY-NC-SA 4.0

Positional quality assessment based
on linear elements

Evaluación de calidad posicional basada en elementos lineales

Antonio Tomás Mozas-Calvache1

Recibido 14 de diciembre de 2020; aceptado 21 de febrero de 2021


This study describes the current state of the art related to the assessment of the positional accuracy of spatial databases based on lines. The use of this type of element of spatial databases has increased in recent years because of the current possibilities of acquisition and sharing data of routes, roads, etc. Nowadays, users are also contributors and this supposes that the spatial quality of data acquired and shared by non-experts must be assessed as is the case with data produced by institutions and enterprises. In this context, several methods based on lines have been developed up to this time for several purposes. This study reviews these methods, analyzing their characteristics, measures, etc. In addition, more than 30 applications of these methods are also summarized. These applications include control of generalized and digitized lines, control of spatial databases, analysis of the displacement of lines between dates, etc.

Key words: lines, positional accuracy, quality, spatial databases, assessment.


Este estudio describe el estado actual del arte relacionado con la evaluación de la exactitud posicional basada en líneas. El uso de este tipo de elementos de las bases de datos espaciales se ha incrementado en los últimos años debido a las posibilidades actuales de adquisición e intercambio de datos espaciales de rutas, carreteras, etc. Hoy en día, los usuarios también son contribuyentes y esto supone que la calidad espacial de los datos, adquiridos y compartidos por personas no expertas, deben evaluarse como se hace con los datos espaciales producidos por instituciones y empresas. En este contexto, se han desarrollado varios métodos basados en líneas hasta este momento para varios propósitos. Este estudio repasa estos métodos, analizando sus características, métricas, etc. Además, también se resumen más de 30 aplicaciones de estos métodos desarrollados hasta el momento. Las aplicaciones incluyen el control de líneas generalizadas y digitalizadas, el control de bases de datos espaciales, el análisis del desplazamiento de líneas entre fechas, etc.

Palabras clave: líneas, exactitud posicional, calidad, bases de datos espaciales, evaluación.

1. Introduction

This article summarizes the main methods and applications developed, until this moment, for assessing and/or controlling the positional accuracy of spatial databases based on lines. Positional accuracy is one of the main components of quality related to spatial databases (Mozas-Calvache & Ariza-López, 2011). Other components include attribute accuracy, temporal accuracy, logical consistency and completeness (ISO, 2013). This study is focused on positional accuracy and more specifically on its assessment based on linear elements. A line can be defined by a set of ordered vertexes that are connected by segments. The use of lines supposes an important alternative to the traditional controls based on points both independently or in a complementary way. There are several methods and standards published by several authors and institutions during the last decades which use a sample of well-distributed points (USBB, 1947; ASCE, 1983; ASPRS, 1990; FGDC, 1998) to determine the positional accuracy of cartographic products. In general, the coordinates of these points are compared to those obtained from more accurate sources using several metrics (e.g. Root Mean Squared Error, RMSE). However, there are several aspects to be considered related to the positional accuracy of spatial databases. Firstly, there are more types of elements in a spatial database such as lines or polygons. In fact, lines suppose the large group in a spatial database (Cuenin, 1972) and they usually have a good spatial distribution over any area (Mozas-Calvache & Ariza-López, 2014). Secondly, the analysis of these elements can improve the results of the assessment because of their own spatial characteristics. Lines contain a great deal of geometrical information defined by a large quantity of vertexes (Mozas & Ariza, 2011). Despite the fact that lines are commonly well-distributed and well-defined on a map, producers and institutions have traditionally used points to check the positional accuracy. Maybe this selection was caused by the best definition of these elements, both in the database and in reality, and the ease of determination of the coordinates using classical surveying, including static observations performed using Global Navigation Satellite System (GNSS). Nowadays the acquisition of lines from a more accurate source has achieved a great improvement thanks to the evolution of the acquisition devices and to the development of kinematic measuring techniques. Lines are acquired much more comfortably and quickly Ruiz-Lendínez et al. (2009). Definitely, the use of linear elements can be considered for assessing the positional accuracy of spatial databases. This premise has been analysed during recent years by several authors.

The quality requirement of spatial databases has undergone a great development during the last few decades. This tendency has been based mainly on the growing concern of producers and users for the quality of products and services and the increase of the availability of the geographic information (GI) on the Internet, which implies a greater demand of data quality on the part of producers and users. An example of this increase of availability on the Internet is the development and establishment of Spatial Data Infrastructures, which allow users to access GI easily. As a consequence, the demand of GI has expanded to those user segments who were not used to these type of products in the past. In addition the global distribution of mobile devices, which has greatly eased positioning measurement, and consequently the development of location-based services, has obviously contributed to this expansion. Nowadays, this evolution supposes a change of paradigm because GI is no longer solely generated by traditional producers. Thus, users are producing, sharing and consuming geospatial data continuously. Goodchild (2007) suggested the term Volunteered Geographical Information (VGI) to include those data produced by citizens in this context. As an example, VGI includes routes and tracks surveyed and shared by users on applications such as OpenStreetMap, Wikiloc, etc. Consequently, we must consider that a large amount of VGI data are composed of lines. Obviously, the positional quality of this data must also be controlled because users require a level of quality similar to that demanded for data produced by institutions. This large amount of GI continually added and updated requires new mechanisms to control their quality feasibly and rapidly. In this context, the methods based on lines have a wide field of application. Antoniou & Skopeliti (2015) gave a more detailed description of measures and indicators of VGI quality.

During recent years the necessity of using lines to assess spatial databases has definitely intensified, both generally (a complete database) and particularly (several sets of elements represented by lines). Some examples of these particular assessments are the control of digitized lines, simplified lines, matching between vector datasets, the determination of displacements between dates, etc. Essentially, any application where two sets of lines must be compared spatially. This necessity has allowed the development of several methods for controlling the positional accuracy of a sample of these elements. In this study most of these methods are described in the methodology section and their main applications are described in the results section. Previously to their description it is necessary to understand how they have been developed based on the concept of uncertainty of lines based on their spatial characteristics. In this context several uncertainty models have been described until now for segments, and consequently for lines (Figure 1). The first model used to describe the uncertainty of digitized lines was the concept of the epsilon band. This was initially proposed by Perkal (1956) and subsequently adapted by several authors (Chrisman, 1982; Blakemore, 1984; Goodchild, 1987; Hunter & Goodchild, 1995; Leung & Yan, 1998; Kronenefeld, 2011). The concept of the epsilon band supposes the true position of a line included inside a certain displacement (epsilon) from the position of the digitized line. Geometrically, it is defined by two lines parallel to the most probable position and tangential to the circular error at the extreme vertexes of each segment (Figure 1a). This supposes a normal-circular distribution of error at the extremes of the segment. From the first definition the epsilon band model has been improved to include variable width in the intermediate points of the segment (Caspary & Scheuring, 1993) (Figure 1b), nearer to a theoretical model of probability distribution (Winter, 2000) (Figure 1c). Caspary & Scheuring (1993) proposed a simplified error band with the minimum positional error shown in the midpoint of the segment (Gil de la Vega et al., 2016) (Figure 1b). In addition to these models, Shi & Liu (2000) proposed the G-band considering that infinite points compose the line. They assumed that the errors at the extreme points follow normal distributions. In addition, the errors in these points can be correlated to each other and the intermediate points of the segment are stochastically continuous to each other (Figure 1d). Other models represent uncertainty with probability distribution functions (Heuvelink et al., 2007; Wu & Liu, 2008). Gil de la Vega et al. (2016) carried out a more detailed description of these models and their adaptation to the 3D case.

Figure 1. Uncertainty models of segements and lines: a) epsilon band; b) and c) error band; d) G-band; e) error band of a line.

In addition to these uncertainty models, several studies were undertaken to analyse positional effects caused by the simplification process based on the geometry of lines. These studies described interesting metrics for performing a positional control. McMaster (1986) described several metrics such as the percentage change of length, the ratio of change in the number of coordinates, the difference in average number of coordinates per inch and the ratio of change in the standard deviation of the number of coordinates per inch, the percentage change of angularity and the ratio of change in the number of curvilinear segments. McMaster (1986) included two comparative metrics in order to analyse the displacement between both lines (original and simplified). These metrics can be considered basic to the development of positional control methods. The first is the vector displacement, calculated by the sum of the length of all vectors divided by the length of the original line. The second is the areal displacement, which is determined by the sum of the areas of the polygons formed between the original and the simplified lines divided by the length of the original line. Jasinski (1990) extended these metrics to include the error variance, the average segment length and the average angularity. Veregin (2000) described the uniform distance distortion based on the distortion polygons (areas) enclosed by both lines. Ramirez & Ali (2003) described other interesting metrics such as the bias factor, the distortion factor, the fuzziness factor and the generalization factor.

Another aspect to consider when assessing the positional accuracy of any spatial database based on lines is related to the sample of the elements to be used. The selection of a sample of lines must consider the characteristic of the population in order to obtain similar results using the selected sample. Ruiz-Lendínez et al. (2009) described three phases: establish a population of interest, estimate the size of the population and determine the size of the sample. In this sense, Ariza-López et al. (2011) studied the size of a sample of lines based on a simulation process using roads. In this process they used several methods to assess positional accuracy based on lines.

Traditionally, the assessment of the positional accuracy of the spatial databases has been carried out using points both in 2D and in 3D. The application of the different metrics to heights supposed in the majority of cases a simple addition of the Z coordinate to the metrics applied to the horizontal component. However, the study of the vertical accuracy (heights) was usually carried out independently with respect to the horizontal accuracy (USBB, 1947; ASCE, 1983; ASPRS, 1990; FGDC, 1998). Among others, this was caused by the different spatial behaviour and representation of heights (e.g. contour lines, Digital Elevation Models, etc.). In the case of linear elements, the first metrics were developed to assess the positional accuracy of lines obtained from digitizing or generalization. So initially, the control of the vertical accuracy was not implemented. However, with the development of 3D spatial databases and the availability of control lines obtained directly from GNSS kinematic surveys (defined by 3D vertexes), Mozas-Calvache & Ariza-López (2015) included the vertical component in this type of assessment by adapting some metrics based on lines. In most cases the inclusion of the vertical component supposed the development of a new procedure for adapting previous metrics developed for 2D.

This study reviews the methods and applications based on lines developed during recent decades to assess the positional accuracy of a set of lines or a complete spatial database. The use of these types of elements is widely justified considering the current development of technology and the large amount of spatial data available on the Internet.

2. Methods

The firsts studies published on the assessment of the positional accuracy of lines were developed to analyse digitized and generalized lines. After that, several methods were developed to assess the positional quality of spatial databases. Previous studies have included some classifications of these methods, such as Atkinson-Gordo & Ariza-López (2002), Santos et al. (2015) and Gil de la Vega et al. (2016). In this study, we are going to update and review them following the classification shown by Mozas-Calvache & Ariza-López (2015) and Gil de la Vega et al. (2016), which differentiated between methods based on line-to-line distance measures (EBM, ADM, HDM, VIM, VIM-V) (Figure 2) and methods based on buffered lines (SBOM, DBOM) (Figure 3).

All methods are based on a set of lines to be assessed (XL) and the homologous set of lines obtained from a more accurate source used as reference (QL).

Figure 2. Methods based on line-to-line distance measures: a) EBM; b) ADM and HDM; c) VIM; d) VIM-V.

Figure 3. Methods based on buffered lines: a) SBOM; b) DBOM.

2.1. Methods based on line-to-line distance measures

2.1.1. Epsilon Band Method (EBM)

The Epsilon Band Method was based directly on the metrics developed by McMaster (1986). It consists of the determination of the enclosed area between both lines XL and QL (S=S1+S2+S3 in Figure 2a) and division by the length of the line to be controlled (XL). The calculation of the enclosed area is based on the sum of all sub-areas determined by two consecutive crosses between both lines. This metric initially was applied to the study of generalized lines but Skidmore & Turner (1992) adapted it to control the displacement between two lines. Several authors have also used this method to obtain a mean displacement between both lines, such as Veregin (2000) and Mozas & Ariza (2010). In addition, Mozas-Calvache & Ariza-López (2015) expanded the use of this method from 2D to 3D. In this case, the calculation of the area was based on a 3D triangulation between both lines (XL and QL).

2.1.2. Average Distance Method (ADM)

This method was described by McMaster (1986) to study the positional quality of the generalization process and was applied by Mozas-Calvache & Ariza-López (2010) to control the positional accuracy of a set of roads. It is based on the calculation of the Euclidean distance between the vertexes of the XL line to the nearest point of the QL line and vice versa (see red and grey arrows in Figure 2b). The mean displacement between both lines is obtained considering the average distance determined. So the Euclidean distance is calculated from each vertex of a line to the nearest point of the other line. This nearest point on the other line can be defined by a vertex or by an intermediate point of a segment. In this case, the distance is perpendicular to the segment in that point (Figure 2b). Mozas-Calvache & Ariza-López (2015) expanded this method to 3D, including height values in all calculations.

Lawford (2010) developed an adaptation of this method. In this case, the line is divided by a defined interval of length. The points determined by these intervals are used to obtain the values of the distances (offsets) with respect to the other line. Therefore they used a set of points distributed along the XL line, with a certain separation between them to obtain the distances to the other line (QL line). This method allows us to obtain a probability distribution of displacements. The result included the value of the 90th absolute percentile of offset distances.

2.1.3. Hausdorff Distance Method (HDM)

The Hausdorff distance gives a measure of proximity between two lines (XL and QL). Hangouët (1995) described some properties of this metric such as asymmetry, orthogonality, sensitivity, tricks and tangency. Abbas et al. (1995) used this metric to assess linear elements. The Hausdorff distance indicates the maximal distance between any vertex of one line to the other and vice versa. The value of the Hausdorff distance is the greatest of both maximum values of distances. As in the case of previous methods, Mozas-Calvache & Ariza-López (2015) expanded the use of this metric to 3D lines. This method has been applied in matching procedures of linear elements Xavier et al. (2016).

2.1.4. Vertex Influence Method (VIM)

Mozas & Ariza (2011) developed the vertex influence method. It consist of the determination of a mean value of displacement between both lines. First, the Euclidean distances between the vertexes of the QL line with respect to the XL line are calculated. Each distance is weighted by the length of the two segments adjacent to the implicated vertex (Figure 2c). So the final value of displacement is obtained from a weighted mean: summing up all values obtained in each vertex and dividing this sum by the double of the length of the QL line. Using this method the displacements (distances) from vertexes with longer adjacent segments are emphasized with respect to those that have less length.

An interesting metric added by Mozas & Ariza (2011) was the study of the components of the displacement vectors that define the distances. So they analysed each component (X and Y) of the displacement vectors independently. The goal was to detect positive and negative increments in these components and consequently to detect the presence of biases in the displacement between both lines. Similar to the case of the distances, the components values were also weighted by the length of the segments adjacent to the implicated vertex.

Mozas-Calvache & Ariza-López (2015) expanded both metrics to the 3D case. The result of this adaptation allows the determination of a mean 3D displacement between both lines.

2.1.5. Vertex Influence Method by Vertexes (VIM-V)

This method was derived from the previous one. In this case, Mozas-Calvache & Ariza-López (2018) suggested the application of the VIM method to calculating the distances and components between the vertexes of the QL line and their homologous vertexes of the XL line. The main goal was the determination of biases because lines can hide these type of displacements when the direction is coincident to the direction of the line. The method includes a previous procedure for determining homologous points in the XL line to those vertexes of the QL line. It consists of the adaptation of the turning function (Arkin et al., 1991) to identifying homologous vertexes using three parameters based on distance and angles. After that, an interpolation in the XL is carried out to obtain points in this line that match those vertexes of QL which are not homologous. Finally, the VIM method is adapted using the calculations between homologous vertexes (from QL to XL) instead of between the vertexes of QL to the XL line (Figure 2d).

2.2. Methods based on buffered lines

2.2.1. Single Buffer Overlay Method (SBOM)

Goodchild & Hunter (1997) described the Single Buffer Overlay Method. The idea is to apply the concept of the uncertainty band directly to the assessment of positional accuracy. It consists of the generation of a buffer with a defined width around the QL line and the calculation of the percentage of the length of the XL displayed inside this buffer. A probability distribution is obtained with the percentage of inclusion obtained after increasing the buffer’s width. Considering this method, Mozas-Calvache & Ariza-López (2010) added a secondary measure based on the percentage of vertexes included inside the buffers after the increase of the width. Analogically to the case of the length, they obtained a probability distribution of inclusion of vertexes depending on the width. This method has been widely applied to controlling spatial databases, to assessing VGI data, in matching procedures, etc.

2.2.2. Double Buffer Overlay Method (DBOM)

Tveite & Langaas (1999) described the Double Buffer Overlay Method. It consists of the generation of a buffer (with a defined width) around each line (XL and QL) and the analysis of the possible situations derived from their spatial behaviours and intersections. After that, the buffers’ width is increased in a similar way to the previous method in order to obtain a distribution of results. So they examined 4 types of areas generated by the buffers: outside both buffers, inside the QL buffer and outside the XL buffer, inside the XL buffer and outside the QL buffer and inside both buffers. Using these measures, they proposed several metrics for each buffer width (e. g. average displacement, oscillation, etc.). Finally, a distribution function is obtained for these measures based on the buffers’ width.

3. Applications and results

During the last decades some studies have used positional control methods based on lines to assess sets of linear elements or spatial databases. In this study we summarize the main applications carried out and their results. To be coherent with the evolution of the methods, in this summary the applications are described ordered by their date of appearance.

4. Discussion

This study has summarized the main methods and applications carried out to assess the positional accuracy of spatial databases based on lines. As has been previously described, there is a large variety of methods, and as consequence, a great number of applications of them. This variety is mainly based on the large amount of features of reality represented with this type of element, their spatial behaviour, the evolution of the uncertainty models and the demand of certain results depending on the application. Therefore, there is a large set of methods for providing a mean value of displacement between both linear datasets based on the calculation of distances (EBM, ADM and VIM) and buffers (DBOM). The selection of the method to be used will depend on the data and the application itself. The use of VIM will provide more extended results because of the possibility of analysing displacement vectors. In fact, it can be used to detect some types of systematic displacements based on the study of the components of these vectors. If a maximum value of displacement is demanded, the use of HDM is the most recommendable option. An alternative is the use of SBOM considering a high percentage of inclusion. However, the HDM is easily applicable when the calculation of distances is carried out using other methods, such as the ADM or the VIM. The use of methods based on buffered lines is recommended when a percentage of inclusion of a certain uncertainty is demanded. This is one of the reasons why these methods are widely used to study the quality of OSM lines. In addition, other adaptations of these methods can be implemented in some particular cases, such as the VIM-V to determine systematic displacements or to obtain the displacement of well-defined vertexes of lines considering the spatial behaviour of these elements (e.g. the study of a landslide between two dates). This supposes an important improvement with respect to methods based on points, because the behaviour of the line is considered.

5. Conclusions

The summary of the methods and applications described in this article has shown the current state of the art related to the assessment of positional accuracy based on lines. About seven methods, some of them with specific adaptations, and about 35 applications have been summarized to show readers this alternative type of assessment of positional accuracy, which is unknown by general users of GI. From the assessment of a complete spatial database to the simple detection of displacements and the control of the matching of elements these applications have contributed to the evolution and development of the methods described previously. The selection of the method to be used is in some cases complicated because each one has a diverse grade of difficulty in being implemented and shows different results and interpretations. Therefore, the selection of the method or methods to be applied will depend on several factors, but we can highlight the type of result to be obtained (mean displacement, maximum displacement, vector displacement, percentage of inclusion, uncertainty at a certain probability, displacement of homologous vertexes, etc.) to establish the most convenient method.

The future of this type of assessment is interesting, considering the current volume of acquisition, sharing and use of GI based on lines. Therefore, the development of new methods or adaptations of those previously described is expected and the extension of the applications to different purposes and fields.


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