Indoors Positioning Based on Spatial Relationships in Locality Description

Spatial relationships exist in our daily communication and provide many locality clues, but minimal attention has been devoted to positioning localities indoors with spatial relationships extracted from locality descriptions. Locality description generally contains spatial relationships (i.e., topological, distance, and direction relations) and reference objects (ROs). For locality description positioning, distance and direction relation convey more clue than topology and combined together can provide much clue to positioning. Explicit and implicit features are inherent characteristics of locality descriptions. Based on the analysis of locality descriptions, a classification scheme of indoor locality descriptions is provided. To address the positioning locality with spatial relationships, we propose a two-state method. First, the located region is obtained according to the fuzzy region of spatial relations in locality description. Second, the probability distribution is determined, which is a joint probability function that consists of relative direction and distance (quantitative or qualitative distance) membership functions. The relative direction membership function is conducted based on human’s direction cognition is related to angular information. The trapezoid quantitative distance membership function is based on indoor cognitive experiment. The qualitative distance membership function for indoor ROs is based on minimum Euclidean distance and stolen area. From the cognition and computation perspectives, the visible boundary met some restrictions is provided and a sampling method is defined. To obtain the unique positioning locality, a principle is proposed. We evaluate our method by conducting an indoor cognitive experiment and demonstrate that a positioning accuracy of 3.50 m can be achieved with semantically derived spatial relationships.

communication [8]. The development of cognitive technology, natural language understanding, and voice recognition technology has made communication between people and machines more mature [9], [10]. The interactions between people and machines will become more intelligent and easier with time. Natural language interactions via voice or text will be the future direction of LBS [10].
As one form of natural language, locality descriptions exist in our daily communication. In general, locality descriptions include reference objects (RO), spatial relationships (SR), and target objects (TO). Any named feature can be called a reference object. Spatial relationships in Geographic Information System (GIS) commonly involve three components, namely, distances, directions and topological relations [11]. Compared to GIS, locality descriptions focus more on direction and distance relations than topological relations [12]. Describing locality, distance and direction together provides more locality cues, which avoids ambiguities or uncertainties [13].
In recent years, although positioning methods based on locality descriptions have made great progress [14]- [15], there are still deficiencies in the study of spatial relation uncertainty modelling and computational methods for complex locality description scenes. In this paper, to address the bottleneck problem of locality description positioning, an indoor shopping market is taken as an example to study uncertainty modelling and a positioning method based on the spatial relationships in indoor locality descriptions. This paper proposes a computational method for indoor positioning with spatial relations extracted from locality descriptions. Specifically, we provide a modelling method for indoor spatial relations, i.e., relative direction, quantitative distance and qualitative distance. Our main contributions are threefold: (1) From a modelling perspective, this paper provides modelling methods for relative direction, quantitative distance and qualitative distance relations, which are based on fuzzy set and can be used for indoor positioning.
(2) From a methodological perspective, the paper contributes a two-stage computational method that includes obtaining a located region and describing a probability distribution, which are based on an indoor locality description analysis and classification.
(3) From an application perspective, this paper proposes an innovative use of spatial relations in locality descriptions for indoor positioning, which provides a theoretical basis and technical support for indoor positioning based on voice location descriptions.
The remainder of this paper is organized as follows. Section 2 reviews related works on indoor positioning, locality descriptions and spatial relations (i.e., distance and direction relations). Section 3 presents indoor spatial relations modelling, including relative direction, quantitative distance and qualitative distance. Section 4 provides the architecture of indoor locality description and the resulting positioning method. Examples are provided to demonstrate our method in Section 5. Finally, Section 6 summarizes the study and discusses future research directions.

II. RELATED WORK A. INDOOR POSITIONING
Recent years have witnessed much work on the development of indoor positioning technologies. Earlier work on indoor positioning focused on single positioning technology, such as iBeacon, Wi-Fi, camera, radio frequency identification [16]- [19]. iBeacon-based and RFID-based positioning method can provide a high-precision positioning for pedestrians in an indoor environment. But the high-precision positioning depends on the layout density of equipment, and their costs are always very high for a wide coverage area [20]. Due to the worldwide availability of Wi-Fi access point, the Wi-Fi fingerprint localization technology has become the most popular indoor positioning (AP) method [21]. However, it is still expensive to maintain a Wi-Fi-based positioning system because of the working status of Wi-Fi APs and the unexpected changes in the position [22]. Camerabased (i.e., visual-based) positioning method can be divided into recognition-based image geo-localization method and geometric matching-based method [23], which can provide enough positioning accuracy in intricate indoor environment. But this method is complex in algorithm.
With the improvement in computing and storage performance on smart phones, the sensor-fusion-based indoor positioning method attracts lots of attention. Li et al. [24] provided a hybrid pedestrian navigation algorithm based on investigation of different combinations of pedestrian deadreckoning (PDR), Wi-Fi fingerprinting, and magnetic matching (MM), which overcame the single-source drawback. To deal with the complex spatial topology and RF transmission environment, an indoor scene constrained method for localization is proposed in [25], which integrates cameras, Wi-Fi and inertial sensors. The fusion results show that the accuracy and stability of the system are better than those of an independent positioning source. However, it not only increases positioning and technical costs, but also increases the power consumption of the positioning terminal and decreases the service capacity.
Based on the above statement, indoor positioning technology has disadvantages of complicated algorithms, unstable positioning, high cost. Therefore, exploring new positioning source accounts for positioning accuracy and cost is meaningful.

B. LOCALITY DESCRIPTION
Locality description answers a ''where'' question [26]. Humans describe localities in a hierarchical manner, whether the location is indoors or outdoors [27]. Locality description reflects a human's direct or indirect interaction with the environment and uncertain is inevitable. There are reference objects, spatial relations and target objects in locality descriptions [11], [15]. Generally, there is at least one VOLUME 8, 2020 reference object and its spatial relations in a locality description [28]. Spatial relations include topological relations, distance relations and direction relations, which convey locality clues [11]. Compared to distance and direction relations, topological relations convey rough locality information and can be reflected by the distance and direction relations [11]. To avoid ambiguities, distance and direction are usually combined in locality descriptions, such as ''object A is 100 m in front, and object B is 50 m to the right'' [13]. However, locality descriptions are complex, either explicitly or implicitly [27], such as ''object A is 100 m in front, and object B is to the right'' implies that ''object A is 100 m in front, object B is to the right and object B is near me''.
Another area of research on locality description is the positioning method. The most commonly used method is the ''Point'' method, which uses a single coordinate to locate a feature whether it is polyline or polygon [29]. Wieczorek [29] developed a Point-Radius method to combine all kinds of uncertainty into a ratio that ignores the shape of the RO. The shape method is conceptually simple, but the geometry that it represents is complex and excludes the original locality description. Guo et al. [11] proposed the concept of uncertainty of a field to represent the uncertainty of ROs and spatial relationships, and presented a probability method to georeference localities. Considering the shape of the RO, Liu et al. [15] proposed a probability method and refined operation that combine a set of uncertainties. Since then, many scholars have performed related research [14], [30], [31].

C. SPATIAL RELATIONS 1) DIRECTION RELATION
Prior studies have modelled direction relations from various perspectives. At least three methods can be identified from the related literatures. One method is attentive to the absolute direction, which is based on an absolute spatial reference system (ASRS) [32]. The absolute direction can be qualitative (e.g., north) and quantitative (e.g., 45 • ) [15], of which quantitative direction is mainly used in special applications (e.g., species data, earthquake data) [14], [30]. The cardinal direction relationship (CDR) is one of the most influential relationships, which divides the space into four or eight cones according to different contexts. Since then, many direction models have been developed for different applications and scales [33]. The second method focuses on relative direction. Krishnapuram [34] argued that the cones that one person searches from one direction to another are related to angular information, and the distance is unimportant during the process. Based on this, relative direction membership functions (that is, right, left, above, etc.) were defined. Bloch [35] extended the ''between'' relation to medical images and defined the concept of visibility. Accordingly, ''surround'' and ''along'' relations are modelled based on fuzzy sets in image understanding [36], [37]. The third method focuses on the computation of the direction relation, which has always been the focus of many scholars. A histogram of angles is proposed to calculate the relative directions of objects [38]. To overcome the drawbacks of a histogram of angles, which has computational cost and is only suited for raster data, Matsakis [39] utilized a histogram of force. Since then, this notion has been better developed [40].

2) DISTANCE RELATION
Distance relations in locality descriptions can be divided into qualitative distances and quantitative distances. Qualitative distance refers to terms such as ''near'' or ''far'', for which different people have different understandings according to different scales. Given the conceptual membership function of the qualitative distance ''near'', Liu argued that the greater the distance target object (TO) from the reference object (RO), the less likely ''near'' would be described. Additionally, without considering other factors, there is an intersection point between the conceptual membership function of ''near'' and ''far'' [15]. Krishnapuram proposed a ''near'' and ''far'' membership function and argued that ''near'' is a complement to ''far'' [34]. Based on a human-participants study, Worboys discussed the vague spatial relation ''near'' in environmental space and presented three approaches (i.e., fuzzy nearness, distance measures and four-valued logic) to analyse it [41]. After a discussion of context factors that affect the relationship between qualitative distance in linguistic and metric distance measures, Yao applied Ordered Logit Regression (OLG) to proximity modelling [42]. Schockaert collected unstructured and semi-structured data that were available on the web and designed a computational framework to represent the near relation based on a fuzzy set [43]. Brennan drew on a Generalized Voronoi Diagram to qualitatively represent the ''near'' relation [44]. Building on this work, Gong provided a mixed membership function that is based on Voronoi stolen area and Euclidean distance to model near relations, but did not apply it further [45]. Wang et al. [46] provided the near relations used in landmark reference system (LRS), but not discussed other complex description scenes and only discussed the LRS in one floor.
Compared to qualitative distance, quantitative distance is a numeric distance used in practice and uncertainty is an inherent characteristic. In the work proposed by Liu et al. [15], the uncertainty of quantitative distance due to measurement error or record imprecision is comprehensively discussed. The quantitative distance in locality descriptions for species data or Search and Rescue (SAR) confirms to record imprecision, so Barve [47] and Doherty et al. [14] used this model to locate places. Conducting a cognitive experiment, Wang et al. [13] determined that the uncertainty of numeric distance derived from measurement error is consistent with a normal distribution.
Based on the above analysis of prior research, the method of modelling distance relations can be summarized into three components: (1) Cognitive experiments, which are based on subject experiments but are labour-intensive and time-consuming; (2) geographic information retrieval (GIR), which focuses on unstructured data on the web but is limited to the density of the population; and (3) spatial geometry, which is simple but has no practical application.

III. MODELLING INDOOR SPATIAL RELATIONS
The modelling method for indoor spatial relations is based on fuzzy set and spatial cognition. A fuzzy set in a universe X is formally defined as a mapping U from X to the unit interval [0, 1]. For relative direction, it is based on human's direction cognition is related to angular information. For quantitative distance, it is based on cognition experiments and conforms to normally distribution. For near relation, it is based on stolenregion and Euclidean distance, which consistent with fuzzy set.

A. RELATIVE DIRECTION
Indoors, where there is a lack of absolute references, people frequently tend to use relative directions. Additionally, people's orientations are related to angles [34]. For example, when turning from the front to left-front, the angle is approximately 45 • . Based on such spatial cognition, we provide a relative distance membership function P RelDir ( ), for which the function should be different according to different contexts.
Equation (1) [34] is based on the assumption that the space is divided into eight cones, i.e., front, left, right, back, front-left, front-right, back-left, and back-right.
Equation (2) is based on the assumption that the space is divided into four cones, i.e., front, left, right and back.
In Figure 1, the space is divided into four and eight cones according different contexts. The dashed lines are the center lines of corresponding cones, and each center line in assigned a number in the clockwise direction. In Equations (1) and (2), the parameter path ( ) is the minimum path between two different center lines of corresponding cones. For the place description ''object A is 50 m in front, object B is in the leftfront'', the path ( ) = 2.

B. QUANTITATIVE DISTANCE
Quantitative distance in locality description is based on visual cognition and measurements, which confirms a normal distribution. According to [13], cognitive experiments are conducted to model quantitative distances (10 m, 30 m, 50 m). In the experiments, each group distance has two marked points (i.e., straight or inclined, which is almost 45 • ), and participants are asked to stand at the marked points to describe the distances between the marked points and reference objects. The participants have different backgrounds, and their ages range from 15 to 52. All related data are collected, and overestimated data are eliminated. After normal tested (Figure 2), each group's cognitive distance conforms to a normal distribution ( Figure 3).
To model the quantitative distance, we provide a trapezoid function (P Dis (d)) [46] based on fuzzy set, which is intuitive-  ness, robustness, and computation efficiency.
The paraments n and k are the deviations from the right distance, and m and l can be derived from the normal distribution of quantitative distance. and v 6 . The neighbouring area of RO A not only consists of circle arcs but also the segments of its neighbours (as shown in Figure 5). (Figure 6).

Definition 3. Stolen-region: The stolen-region refers to the region that is part of the Voronoi region of the original RO but now belongs to the Voronoi region of TO, after a new site is inserted into the existing Voronoi diagram
Based on stolen-region and Euclidean distance [45], the near relation membership function for polygon ROs, P Near (t, RO i ), is provided: In the equation defined above, t represents TO, t ∈ NeighbourArea(RO i ), and min d(t, RO) is the squared minimum distance between t and RO i . Reg k represents the region stolen from RO k by t. The equation holds the following four  constraints, which are based on fuzzy set mapping of the near relation to interval [0, 1]. In the four constraints, x and y denote the coordinate of TO t. Constraint 1 indicates that the position of the t can be described by the ROs neighbour to t. Constraint 2 implies that when t is at the position of ROi, its probability is 1 . Constraint 3 shows the boundary of the near relation of ROi. Constraint 4 means that the near relation membership function of ROi is continuous. Constraint 1: Constraint 2: Constraint 3: Constraint 4: The features of indoor and outdoor locality descriptions are similar. Combining distance and direction relations provides more locality clues to avoid ambiguous descriptions. From the cognitive linguistics perspective, there are at least one and at most three reference objects with related spatial relationships in locality descriptions (Figure 7). However, one reference in a locality description (i.e., scene 1) cannot be known with regard to its relative direction and thus does not provide many clues with regard to positioning. Locality descriptions are complex, either implicitly or explicitly [27]. VOLUME 8, 2020   From the cognitive psychology perspective, when the RO lacks a related quantitative distance, the RO tends to be near the human [28], [48]. For this situation, we consider the near relation as a potential location clue to assist in positioning. Based on the above analysis, we focused on scenes 2 and 3 and divided the locality description into explicit and implicit according to the complete of quantitative distance in the locality description, as shown in Tables 1 and 2.
Explicit Locality Description: The quantitative distance and direction relations of reference objects in the locality description all are equipped. Implicit Locality Description: At least one reference object related to the quantitative distance in the locality description is lost.
We developed a two-stage computational method that takes ROs and their related spatial relations as inputs and then outputs the positioning location. The first stage is to obtain the located region by the intersection of distances in locality description, and the second stage is to describe the probability in the located region. The method was designed based on the theory of probability and spatial cognition. The joint probability is a statistical measure that calculates the likelihood of two events occurring together. In our method, the joint probability is employed to integrate all position clues in position description. The position clues can be divided into distance and direction relations. The distance and direction probability in located region can be calculated according to the number of direction and distance relations in the classification scheme of indoor locality descriptions. In other words, the positioning method is a joint probability function consists of distances and directions membership functions. Spatial cognition is a branch of cognitive psychology that studies how people acquire and use knowledge about their spatial environment, including location, distance, direction, pattern, and movement. To calculate the direction and distance between ROs, visible boundary, i.e. the segment of RO and can be seen from the current location, is introduced considering spatial cognition to improve the computation efficiency [35]. Figure 8 shows the overall architecture of the method. Figure 8 shows the overall architecture of the method. We focus on the computational method and it is given as follow. Figure 9. Definition 6. Located Region: The region in which the locality description may be located. It is the intersection of the distance region in the explicit or implicit locality description (the quantitative distance and qualitative distance regions respectively correspond to the fuzzy band and neighbouring area), denoted as Located Region. This is shown in Figure 9.   The acquisition of the visible boundary of RO for quantitative and qualitative distances are different. The former can be obtained by the intersection of a circle with its upper distance as the radius and can describe the location as a center with a boundary of its reference object. Based on the segment, visible and cognition restrictions are conducted. The acquisition of the visible boundary of qualitative distance is directly conducted, with visible and cognition restrictions on the boundary of the reference object.

Definition 5. Fuzzy Band: A band with outer and inner rings around the RO corresponds to upper and lower of quantitative distances, denoted as a Fuzzy Band. This is shown in
Visible Restriction: The visible boundary should not be occluded, as shown in Figure 10.
Cognition Restriction: The visible boundary should meet the Pareto principle, for which approximately 80% of effects originate from 20% of the cause, whether from a cognition or algorithm perspective. As shown in Figure 11, the space is divided into 8 cones, and each cone occupies 45 • . The occupation angle of the portion of the visible boundary meets restriction 2 in that cone should be approximately 9 • .
When there are two reference objects in a locality description and the angle between the two direction descriptions of the two reference objects is not equal to 180 • , this is unacceptable from a positioning perspective because there are two locations. To solve this problem, a principle based on direction relations in the locality description is given as follows.
Principle: As illustrated in Figure 12, the space is divided into eight directions, and each direction is assigned numbers from 1 to 8 in a clockwise direction. The path (a) is the path between two direction lines. Take the locality description ''my front 50 m is RO A , and my right is RO B '' as an example. Lines 1 and 3 respectively connect RO A and RO B to the positioning locality (i.e., t 1 and t 2 ). The unique positioning locality should meet the requirement that the direction from front (1) to right (3) is clockwise, and path (a) = 2, that is, t 1 .
When calculating the quantitative distance and relative direction of the positioning locality (i.e., t) to the visible VOLUME 8, 2020   boundary, we should sample the visible boundary. However, some segments should be eliminated from visible restrictions to cognition restrictions. To be consistent with cognition, we provide a sampling method (algorithm 1) as follows:

Algorithm 1
Step 1: Equally divide the angle between the positioning locality and the end points of the visible boundary. Obtaining the sampling points of the visible boundary meets the visible restriction.
Step 2: Sort the sampling points' polar coordinates. Filter the sampling points with the recursive method. Two points (i.e., clockwise and anticlockwise) are eliminated until the cognition restriction is met.

End for
Here, we use the implicit locality description (i.e., ''object A is 50 m in front, object B is (near) to the left, object C (near) is to the front-left'') as an example to illustrate the probability calculation process in the located region.
Without loss of generality and explaining our method clearly, we regard a polyline and polygon as a set of points, namely, A = {a 1 , a 2 , . . ., a n }. A, B and C are ROs, and T = Located_Region(A,B,C). The Visible_Bdy(A), Visi-ble_Bdy(B) and Visible_Bdy(C) are the visible segments of A, B and C, respectively, that meet the restrictions. We let dir(A,t,B) denote the angle between t and two visible segments of A and B, and dir(t,A) denote the distance from t to Visible_Bdy(A). The other dir(A,t,C), dir(B,t,C), dis(t,B) and dis(t,C) are the same.
The process is as follows.
Step 1: Calculate the probability of relative direction in the located region; P rel dir (t) = P rel dir (A, t, B)P rel dir (B, t, C) where P rel dir (A, t, B) and P rel dir (B, t, C) are the membership degrees that map dir(A,t,B) and dir(B,t,C) via the relative direction membership functions in Equation (1) or (2).
Step 2: Calculate the probability of quantitative distance in the located region; A) is the membership degree that maps the dis(t,A) via the quantitative distance membership function Equation (3).
Step 3: Calculate the probability of qualitative distance (i.e., near) in the located region; where P qual dis (t, B) and P qual dis (t, C) are the membership degrees that map the dis(t,B) and dis(t,C) via the qualitative distance membership function in Equation (4).
Step 4: Positioning; The calculation process is based on a joint probability function according to the spatial relation clue in the locality description. In other words, the process should be adjusted according to different spatial relation clues in the locality description.

V. CASE STUDY A. PROCEDURE
The cognitive experiment is conducted in a shopping mall, which offers a sufficient number of participants. To better verify our experiment, two floors (F1 and F7) of the shopping mall within a 45 m visual space are selected as our experimental ground. All participants participate once. Their ages range from 15 to 54, and they have different backgrounds. Before the experiment, all participants are told to describe context.
Describe context: If you lost your family or friends, there is a phone that can translate your locality description into a locality, and your family or friends can find you easily. Please describe your locality with distance, direction or both.

B. PARAMETERS
We adopt a 98% confidence interval as the β and γ of the quantitative distances (i.e., 10 m, 30 m, 50 m). α and δ can be obtained by their normal distribution. The upper and lower bounds of quantitative distances are not particularly critical but must not be too small or large. Bounds that are too small will not reflect the influence area, and bounds that are too large will increase the calculation complexity. Here, a 70% confidence interval is regarded as the upper and lower bounds of quantitative distances. The related parameters of other quantitative distances can be obtained by interpolation.
For relative direction, the range of parameter a is [2], [5] multiplied by path ( ). The angles that meet the cognition restriction is 10 • and 20 • for the 8 and 4 relative direction models, respectively. Lacking additional contextual information, we cannot determine how many cones the participants potentially divide the space. However, the direction relation ''left-front'' inferred that the space was divided into 8 cones. Thus, without additional contextual information, we use a 4-cone relative direction membership function.

C. EXPERIMENTAL RESULTS
We collected over 521 data, of which 361 data met the positioning requirement. Then, the related data were divided into explicit and implicit locality descriptions, as provided in Section 4.1. For convenience, Explicitly-Scene2 and Explicitly-Scene3 respectively denote the two and three reference objects in the explicit locality description. Implicitly-Scene2-1 denotes that there are two reference objects in the implicit locality description, and there is one reference object that lacks quantitative distance. As shown in Figure 14, the related data are marked as points. The collected locality description is shown in Table 3.
From the marked point in Figure 14 the locality description including quantitative distance (i.e., Explicitly-Scene2, Explicitly-Scene3, Implicitly-Scene2-1, Implicitly-Scene3-1, and Implicitly-Scene3-2) appears to be in a relatively spacious place, where there is a lot of space for participants to make distance judgements. However, the distribution of locality descriptions without quantitative distances (i.e., Implicitly-Scene2-2 and Implicitly-Scene3-3) is uniform. In general, the locality description for the space where there are enough reference objects and the distribution of explicit and implicit locality descriptions confirm human spatial cognition. Figures 15 and 16 respectively show the positioning results for Implicitly-Scene3-1 in F1 and Implicitly-Scene3-3 in F7. The locality descriptions are ''front 15 m is TISSOT, left 15 m is ZuoKY, left-front is I DO'' and ''front is FuCQ, right is RongLiJ, left-front is KeDuoCS''. Darker colour correspond to greater probability. The red point represents the positioning locality, i.e., the maximum probability point or the center point of the maximum probability region. The blue point denotes the participants' standing locations. From the results, the red point relative to the reference objects in the locality description is consist with the relative direction description.
To verify the positioning accuracy, the positioning cure and empirical cumulative distribution functions (CDFs) of the 7 scenes are illustrated in Figure 17. The positioning error is the distance from the maximum probability point or the center point of the maximum probability region to the participant locality (i.e., marked point). Table 6 shows the error statistics of the explicit and implicit locality descriptions. As shown in Figure 18, considering qualitative distance (i.e., near) in an implicit locality description, the positioning accuracy is improved, which reflects that participants potential selected nearby reference objects. The positioning accuracies are respectively improved by 0.54 m, 1.18 m, 0.70 m, 0.84 m and 1.08 m for Implicitly-Scene2-1, Implicitly-Scene2-2, Implicitly-Scene3-1, Implicitly-Scene3-2, Implicitly-Scene3-3 ( Table 4). The positioning accuracies of Implicitly-Scene2-2 and Implicitly-Scene3-3 showed obviously better improvements than the other three scenes because the quantitative distances of reference objects are not all equipped. In general, considering qualitative distance (i.e., near), the positioning accuracy is improved by 0.85 m.
Uncertainty is the intrinsic characteristic of spatial cognition and is inevitable. The large positioning error comes VOLUME 8, 2020  from unclear cognition and might be summarized as follows: First, via unclear quantitative distance cognition. The quantitative distance description of the reference object (RO) is too small or large ( Figure 19). As shown in Figure 19, the quantitative distance description of the RO 2 participant standing at TO 1 is 15 m but is actually 8 m. Second, via unclear direction cognition. Participants standing in adjacent positions have the same reference objects but different direction relation descriptions. As shown in Figure 19, a participant standing at TO 1 describes his locality as ''front is RO 2 , left is RO 3 '', but another participant standing at TO 2 adjacent to TO 1 describes his locality as ''right-front is RO 2 ,    left-front is RO 3 ''. The angles of reference objects in the locality descriptions are different (the former is 90 • , the latter is 45 • ). This may result from the different orientations associated with coming or standing ( Figure 19). Third, via qualitative distance. The reference object that lacks a quantitative distance description is obviously farther away than VOLUME 8, 2020   the reference objects that surround the participants. This may result from personal preferences or the fact that it is blocked by obstacles (e.g., concrete columns). As shown in Figure 19, the locality description for a participant standing at TO 1 is ''front 15 m is RO 2 , left is RO 3 ''. However, RO 3 is further than RO 1 from TO 1 . Additionally, large positioning errors may result from the combined addition of unclear distance (quantitative or qualitative) and direction cognition.
The positioning errors are grouped by the number of reference objects in the locality descriptions, and positioning error statistics are shown in Table 5. The positioning errors for locality descriptions with two and three reference objects are almost the same, which reflects that more reference objects in a locality description do not contribute to improving positioning accuracy. Figure 20 shows the overall positioning accuracy and empirical cumulative distribution function (CDF). The maximum and minimum positioning errors are 7.06 m and 0.74 m, respectively, and the mean positioning error is 3.50 m when using spatial relations in locality descriptions. The 35.15% and 87.46% accuracies are respectively 3 m and 5 m. Compared to current costly and complex indoor positioning technologies whose positioning accuracies are 3-5 m based on common smartphones, our positioning method is economical and its accuracy is acceptable. Additionally, if other data are considered, such as indoor roads and escalators, such data can generate spatial constraints on the positions of participants and improve the positioning accuracy to some extent.

VI. CONCLUSIONS AND FUTURE WORK
Spatial relations in locality descriptions convey locality clues and can support positioning. After a great deal of analyses (cognition, expression and computation), locality descriptions are divided into two categories (i.e., explicit and implicit) and are then subdivided according to the number of ROs. Based on this assumption, this paper presents a twostage computational method for indoor positioning with spatial relations extracted from indoor locality descriptions. The first stage focuses on the located region, which is obtained via the intersection of quantitative or qualitative distances in explicit or implicit locality descriptions. The second stage aims to describe the probability distribution in the located region, which is a joint probability function consisting of direction and distance (quantitative and qualitative distances) membership functions. Additionally, from the computational and cognition perspectives, some restrictions are provided. The study achieves a positioning accuracy of 3.50 m.
The contributions of this paper are apparent from three perspectives. From the perspective of modelling, this paper proposes a modelling method for relative direction, quantitative distance and qualitative distance based on cognition and geometry computations. From the perspective of methodology, this work presents a two-stage computational method that includes obtaining a located region and describing a probability distribution in that region. As indicated by the experiment results, the method has an acceptable performance in indoor positioning compared to using traditional indoor positioning technology. From the perspective of application, this paper presents an innovative use of spatial relations extracted from locality descriptions for indoor positioning.
The proposed approach for indoor positioning with spatial relations has limitations and can be improved by future work. First, the classification scheme of indoor locality descriptions provided in the paper servers as an example of the complexity of locality descriptions and is not considered to be a final scheme. Second, the qualitative distance (i.e., near) is based on the assumption all weights of ROs are the same. If the weights of all ROs differ, the ability of multiplicatively weighted Voronoi diagrams to answer the multiplicatively weighted near relation is worth being discussed. Third, it is undeniable that there are other kinds of ROs in locality descriptions, such as escalators and pillars, which can narrow the located region to improve positioning accuracy. Future researches should model these ROs and their related spatial relations. HONG FAN was born in Hunan, China. She received the Ph.D. degree in GIS and remote sensing from China Wuhan University, in 2001. She is currently a Professor, as well as a Doctoral Supervisor with Wuhan University. Moreover, she is a Geographic Information System Expert. She has been involved in teaching and research in cartography and geographic information system for a long time. Her major works and achievements are made in smart geographic information service, qualitative geographic information retrieval, and spatial-temporal big data mining.
LUYAO WANG was born in Hubei, China. He is currently pursuing the Ph.D. degree with the State Key Lab for Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University. His research interests are the urban economy, GIS, sustainable development, and consumer activities.
RENZHONG GUO was born in Jiangsu, China. He received the Ph.D. degree in geography from France Franche-Comté University, in 1990. He has been involved in research and development of cartography, GIS, and construction strategy of the digital city for a long time. Great achievements are also being made in theories and methods of the geographical information system, as well as information engineering of land resource management. He is a member of the Chinese Academy of Engineering, Professor, Doctoral Supervisor at Shenzhen University, and Geographical Information Engineering Expert.
XIAOMING LI was born in Shandong, China, in 1984. He received the Ph.D. degree in photogrammetry and remote sensing from the Wuhan University of China, in 2011. He is currently an Associate Professor with Shenzhen University. He is mainly involved in research on 3D GIS and VGE, and the city big data management. YOU LI received the Ph.D. degree in geographic information science from the Wuhan University of China, in 2017. He is currently holding a postdoctoral position at the Research Institute for Smart Cities, School of Architecture and Urban Planning, Shenzhen University. His main research interests include LiDAR processing and object recognition from LiDAR data.
XING ZHANG received the B.E. and Ph.D. degrees in geographic information science from Wuhan University, Wuhan, China. He is currently with the Shenzhen Key Laboratory of Spatial Information Smart Sensing and Services, Shenzhen University. His research interests include mobile navigation, visual cognition, ubiquitous computing, and intelligent transportation.
WENQUN XIU was born in Shandong, China. He is currently a Doctor, Senior Engineer, Lecturer, and is holding a postdoctoral position with Tsinghua University. He is also an Associate Researcher with the Chinese Academy of Sciences, where he is currently the Manager of the Information Research and Development, Shenzhen Urban Public Safety and Technology Institute. He is mainly involved in the smart city, big data, and cloud computing. VOLUME 8, 2020