A fuzzy semantic spatial partitioning model of regions and applications in understanding remote sensing data

Abstract

The division of the internal structure and external space of geographical entities is the premise of the analysis of topological relations and directional relations. Currently, most division methods are crisp, which does not conform to human cognitive habits. Occasionally, users want to integrate their meaning by vague natural language when they understand spatial scenarios; however, natural language suffers from uncertainty due to the effects of individual characteristics and the context of environmental factors. To handle uncertainties in spatial conceptions of regional features, the semantic spatial partitioning model for regions based on computing with words is proposed. The structure of a region is divided into several parts using fuzzy logic according to people’s cognitive habits, and then, a detailed direction model of regions is proposed. The proposed method is applied to understand a remote sensing dataset, and the results show that the proposed method can enrich the understanding of images while conforming to human cognitive habits.

Keywords

IT2FS semantic spatial partitioning CWW spatial relation remote sensing

1 Introduction

Geographical information systems (GISs), which are computer systems for capturing, storing, querying, analysing, and displaying geospatial data, are widely used to solve problems related to society or the natural environment in agriculture, municipal administration, transport management and other fields [24]. One of the predicted abilities of intelligent geographical information systems (IGISs) is that they will be able to manage perception information, allowing for the realization of natural language interaction between users and the IGIS [46]. This capability will hopefully provide objective and correct support for a variety of decision-making processes through intelligent spatial analyses and reasoning mechanisms that conform to human cognitive and thinking habits [11]. Natural language text data, such as network texts and historical geography documents, contain abundant geospatial information and have become important data sources for IGISs [37, 43]. However, traditional spatial data represented by coordinates and the methods for describing spatial relations based on geometry are essentially different from those described using natural language; in other words, there is a large gap between spatial data and natural language, and current GISs have limited ability to deal with natural language. This problem poses a great challenge to integrating natural language into IGISs.

1.1 Crisp spatial partitioning in GISs

To establish the relationship between traditional spatial relations and natural language vocabulary, a great deal of research has been performed. Egenhofer and Sharif [31] established three measurement indices of topological relations within natural language: splitting, alongness, and closeness; the relations among 64 English topological terms and nine intersection models [32] were established using these indices. Sharif et al. [2] improved the measurement indices of topological relations between lines and regions, and they calibrated 59 English topological predicates. Nedas et al. [22] used a similar method to establish measurement indices for topological relations between lines. Xu [16] studied the English natural language comprehension of topological relations between two linear objects through cognition experiments, and they found that the geometry of geographical objects and topological relations between geographical features are the main effects of natural language descriptions of spatial relations. Deng et al. [30] studied the relationships between spatial relation descriptions and natural language descriptions and established relationships between them. Zhang et al. [49] established the corresponding relations between Chinese language topological relational predicates and RCC-8 topological relations. Du et al. [43] classified 69 topological relation words between line objects and regions into seven categories using a random forest algorithm based on four splitting metrics and six alongness metrics.

These methods are aimed at crisp objects, which are divided into three partitions – interior, boundary and exterior – and this partition method has been adopted by a number of software packages, including ArcGIS and MapInfo. Because a crisp entity model is an abstraction of the real world, such expressions are often inconsistent with human cognition and the nature of the real world [44]. For example, both point and line features in the real world have some extension, but in geometry, all point objects are expressed by their coordinates, and line objects have no width. This type of hard division is inconsistent with human cognition as well; for example, the boundary of a linear object is represented by two end points in nine intersection models. Although this approach is convenient in topological analysis, its defects are clear [23]. Direction analysis requires spatial partitions as well, but all directional concepts in natural language are vague; for example, “the south of China” is a fuzzy extension. Vagueness is one of the inherent attributes of direction concepts [9, 19]. Guo and Cui [9] indicated that there are some faults in the existing direction relation models. The most common fault is that most models are crisp in their spatial partitioning, including the popular cone-based model [45] and the MBR model [40]. Second, the membership grade for each cardinal direction cannot be accurately expressed in the current fuzzy direction models.

1.2 Fuzzy spatial partitioning in GISs

One of the common characteristics of the aforementioned methods is that they use natural language to explain crisp spatial relations or to query results, which are processes of decoding spatial relations or querying results in natural language. However, the encoding process from natural language to spatial relations or queries is defective, and this process has been ignored by most researchers. Consequently, subjective consciousness cannot be effectively integrated into the spatial analysis process. The largest and most fundamental reason for this defective coding process is that the aforementioned spatial topological and direction models are crisp partitions of space or spatial objects and consequently cannot naturally express human meaning or semantics. Some words or language variables indicating spatial locations and relations cannot be adequately expressed by the existing models and methods. To solve these problems, Guo and Shao [11] adopted the concepts of spatial partitioning [33], fuzzy relations and computing with words (CWW) [12, 21] to model fuzzy spatial partitions of line objects based on fuzzy spatial modelling [1, 35] while integrating subjective human consciousness into the spatial analysis process in the initial stages. In addition, the semantics of typical spatial relations are expressed as language variables, which are more suitable for human cognition than crisp spatial partitions. However, these types of models describe fuzziness using continuous membership values and cannot address the errors from the membership degree.

Two features can induce higher order vagueness in GISs. First, many fuzzy geographical objects, such as forests and wetlands, exist. Such fuzzy geographical features or phenomena have higher order fuzziness attributes, usually due to the inherent uncertainty of geographical phenomena, the presence of observation errors, the complexity of their distributions, and the polymorphisms of geographical phenomena at different spatial scales [7 , 38]. The second feature that can induce higher order vagueness is that the semantics of the predicates and quantifiers used to represent spatial locations and the relationships of crisp or fuzzy objects usually have some uncertainty due to the effects of individual characteristics and the context of environmental factors (for example, scale, behaviour type, topological proximity, and accessibility) [48]. Semantic uncertainty includes individual and interpersonal uncertainty [12]. For example, in special tasks, the spatial range of the distance word “Near” for different people can be different; therefore, different people can use different classical fuzzy sets to express the meaning of “Near”. Even for the same person, the spatial ranges of the two expressions “near the bus station” and “near China” are different.

To overcome defaults of crisp spatial partitions and to integrate natural language into current GISs, the goal of this article is to propose a fuzzy semantic partitioning model for regional features based on CWW, where the interpersonal uncertainty of words can be considered. A geographical perceptual computing model is updated using interval type-2 fuzzy sets (IT2 FS) to address interpersonal uncertainties in section 2, Then, the structure of a region is divided by adopting the closeness definition; thus, it is a fuzzy partition. In section 3, a detailed direction model based on IT2 FS is proposed to manage detailed directional concepts. In section 4, we discuss the directional relations between IT2 FRs using the detailed direction model. In section 5, we provide a case study in which the proposed semantic spatial partitioning model is used to understand remote sensing data, and section 6 concludes the study.

2 The semantic division of regions

The structural division of crisp or fuzzy spatial objects is a fundamental component of spatial relations. Different partitions of a geographical feature have their own semantics, for example, the core and boundary of a fuzzy region. However, people usually focus on some special part of a region, for example, the water area beyond the boundary zone in a reservoir, the water area beyond the centre zone in a reservoir, or the water area between the centre and the boundary zone in a reservoir. In other words, people can integrate semantics with geographical features using natural language, and IGISs should have the ability to address such problems in different applications. To achieve this goal, we review some basic concepts in subsection 2.1, and the geographical perceptual computing model is modified based on interval type-2 fuzzy sets. In subsection 2.2, the semantic partitioning of regions is proposed.

2.1 Background

The interval type-2 fuzzy set (IT2 FS), which is a special case of the type-2 fuzzy set (T2 FS) initially introduced by Zadeh [29], is adopted to manage the uncertainty of semantics in this study. The T2 FS requires an undesirably large number of computations, and the secondary membership value or function is also difficult to handle; thus, it is comparatively difficult to apply to real problems. Therefore, the IT2 FS is a more reasonable choice [12, 21].

2.1.1 Interval type-2 fuzzy sets

Definition 1 (IT2 FS) [13]. an IT2 FS $\tilde{A}$ in the universe X ≠ φ is given by the following: $\tilde{A} = {((x, u), μ_{\tilde{A}} (x, u) = 1) | x \in X, u \in U \equiv [0, 1]}$ (1)

where U is the universe of discourse for the secondary variable u. Note that since U is a subset of [0,1], it could be a singleton {u_x}, singleton set ${u_{x}^{1}, u_{x}^{2}, \dots, u_{x}^{n}}$ , interval [A^- (x) , A⁺ (x)], pairwise disjoint interval set and a combination of numbers and pairwise disjoint intervals [6]. In this paper, we focus on the case of an interval, and then, for the sake of convenience, the IT2 FS is represented as $\tilde{A} (x) = [A^{-} (x), A^{+} (x)]$ (0 ≤ A^- (x) ≤ A⁺ (x) ≤ 1), and A^- (x) and A⁺ (x) are the lower membership function (LMF) and upper membership function (UMF), respectively. The footprint of uncertainty (FOU) of an IT2 FS $\tilde{A}$ is the uncertainty in the primary memberships $\tilde{A}$ , and it has always had the connotation of a geometric object bounded by a UMF and LMF.

Direct and superdirect images are adopted to model some relations of interval type-2 fuzzy regions, and in classical fuzzy set theory, the direct image (R ↑ _TA) and the superdirect image (R ↓ _IA) of a type-1 fuzzy set A in X under a fuzzy relation R in X are the fuzzy sets in X, which are defined as follows: $(R ↑_{T} A) (y) = sup_{x \in X} T (R (x, y), A (x))$ (2) $(R ↓_{I} A) (y) = inf_{x \in X} I (R (x, y), A (x))$ (3)

where T is the t-norm, and I is the implicator. I_W (x, y) = min(1, 1 - x + y) is selected along with the Lukasiewicz t-norm in this paper. Sup and inf denote the supremum and infimum of a set, respectively.

2.1.2 Interval type-2 fuzzy region

A number of fuzzy spatial object models have been widely discussed by researchers [10 , 47]. Models based on type-1 fuzzy sets have no ability to address membership value errors; thus, we have proposed a higher order fuzzy spatial object model based on an IT2 FS to overcome this fault [10]. As the most significant fuzzy spatial object, the interval type-2 fuzzy region (IT2 FR) is defined by LMF $\underline{S} (x, y)$ and UMF $\bar{S} (x, y)$ .

Definition 2. (Simple IT2 FR). If an IT2 FR satisfies the following properties –

(1)it is convex; (2) the LMF and UMF are both upper semicontinuous; (3) it is closed; and (4) its closure is compact-then the set is termed a simple interval type-2 fuzzy region (Simple IT2 FR). An IT2 FR is closed if the LMF and UMF are upper semicontinuous, and property; (4) indicates that no hole exists in it.

The region can be simplified to a simple type-1 fuzzy region if the LMF is identical to the UMF and can be simplified into a simple region in crisp space when the LMF and UMF are constants equal to 1.

Definition 3. (Normal IT2 FR). A normal IT2 FR has two properties in addition to those of a simple interval type-2 fuzzy region: (1) it is an interval type-2 fuzzy number; and (2) the interior, core and exterior are disconnected regular closed sets.

2.1.3 Geographical perceptual computing

Computing with words, which was introduced by Zadeh [27 –29], has been widely used in market investment, decision-making analyses, automatic control designs, fuzzy queries, data mining, and other fields. According to the existing literature, CWW techniques can be classified into three categories: membership-function-based models, symbolic linguistic computing models and 2-tuple linguistic models. Each category of model has unique advantages and limitations [5 , 41]. Li et al. proposed a new CW framework based on the 2-tuple linguistic model. Such a CW framework allows people to address personalized individual semantics (PIS) to allow CWW to maintain the idea that words mean different things to different people [4 , 51], and the fuzzy envelopes of hesitant fuzzy linguistic terms sets are adopted to personalize individual semantics [3]. The geographic perceptual computing (GPC) model proposed by Guo and Shao [11] was inspired by the perceptual computer (Per-C) model based on CWW [12]. The core modules of the GPC model include the input, the geo-encoder, a geographical CWW Engine and the geo-decoder.

The input data of GPC model include raster data, vector data and words in natural language, while the input data of classical GISs are raster and vector data. The greatest difference between the GPC model and typical GISs is that the GPC model contains a Geo-Encoder module, the major task of which is to convert fuzzy spatial concepts and qualitative spatial relations words, such as “Core area”, “North”, and “Near to”, as fuzzy sets. In this manner, humans’ subjective consciousness could be integrated effectively into the GPC model, and then, spatial query, spatial analysis and spatial decision making in GISs would be more humane and intelligent than classical GISs. The core of the GPC module consists of the fuzzy spatial data model and the fuzzy spatial partition model. The geographical CWW engine module uses fuzzy logic to solve fuzzy spatial analysis problems. The geo-decoder module is used to convert fuzzy sets into natural language words. This paper modifies the GPC model with IT2 FSs (Fig. 1). In the Geo-Encoder module, spatial concepts, the fuzzy spatial data model and the fuzzy spatial partition model are operated with an IT2 FS to manage individual and interpersonal uncertainties in semantics. The geographical CWW engine module is built using interval type-2 fuzzy logic. The KM algorithm, which is used to calculate the centroid of an IT2 FS [14, 36], is adopted for defuzzification in the geo-decoder module.

Fig.1

Geographical perceptual computing model based on CWW.

2.2 Semantic structures of geographical regions

An IT2 FR is divided into a core, a boundary, and an exterior [8], and it is obvious that the core, boundary, and exterior are the semantics of the IT2 FR’s own partitions. However, users sometimes integrate some special semantics into these partitions. Figure 2 shows Qiushui Lake at Tianjin Normal University. The yellow circle indicates the location of two dragon boats. We can say that “there are two dragon boats on the boundary of the lake”. The red rectangle indicates the location of the library at Tianjin Normal University. People usually describe the location of the library as follows: “the library is located on the boundary of the lake”. The word “boundary” in these two sentences has different semantic meanings.

Fig.2

Qiushui Lake at Tianjin Normal University.

Figure 3 shows a case with some villages in Xiayin Town and Luozhuangzi Town. The yellow lines are administrative boundaries, but many people, especially tourists, do not know the locations of these lines and say, “villages like Xiyuzhuang and Kuliyu are on the boundary between these towns” and “the village Ershilipu is located at the centre of Luozhuangzi Town”. In these sentences, it is estimated that both “boundary” and “centre” can have different spatial ranges for different people, indicating that the meaning of one spatial structure of a region might be different for different people.

Fig.3

Villages in Luozhuangzi Town and Xiayin Town.

Different people might focus on different parts of a regional feature; sometimes, people might focus on the boundary, treating other parts inside the region as interior. For example, the boundary of a forest might be safer than the interior region; as a result, visitors might be warned to move only within the boundaries. Sometimes, people focus on the centre part of a region, for example, a city centre. At other times, people might focus on the core and boundary regions at the same time, and other parts might not be important and be ignored. Figure 4 shows the difference between the crisp, type-1 and type-2 structures of a polygon. Figure 4(a) is the crisp spatial partitioning method for a region, and the boundary of a crisp polygon is a polyline without width. However, there is no geographical spatial object with a boundary without width in nature; thus, this model coarsely describes geographical features. Figure 3(b) is the type-1 semantic structure, which uses core, inner, inside boundary, outside boundary and exterior to describe geographical features. Because “words can mean different things to different people”, interpersonal uncertainty cannot be managed using this method. It should be noted that the core and inner areas of a polygon are different in real tasks; for example, the core area of a city is the centre of a city, and the inner area of a city is the zone surrounded by its boundary. To manage the interpersonal uncertainties of words, the CWW is adopted here to address vague words, as shown in Fig. 4.

Fig.4

Crisp, type-1 and type-2 semantic structures of a crisp region.

To model the semantic structure of a polygon, the notion of “closeness” is adopted in this study, and the semantic structures of a polygon could be modelled as IT2 FSs. Schockaert [44] conducted some research on this topic by defining closeness for crisp and type-1 fuzzy regions. To manage the interpersonal uncertainty of semantics (for example, different people have different meanings of “closeness” in different situations,) closeness is extended by an IT2 FS. For two points, p and q, in Euclidean space that are close to each other, closeness is defined as ${\tilde{C}}_{(\tilde{a}, \tilde{b})} (p, q) = [C_{(\underline{a}, \underline{b})} (p, q), C_{(\bar{a}, \bar{b})} (p, q)]$ , where $\tilde{a} = [\underline{a}, \bar{a}]$ , $\tilde{b} = [\underline{b}, \bar{b}]$ , $C_{(\underline{a}, \underline{b})} (p, q) = {\begin{matrix} 1 & d (p, q) \leq \underline{a} \\ \frac{\underline{b} - d (p, q)}{\underline{b}} & \underline{a} < d (p, q) \leq \underline{b} \\ 0 & d (p, q) > \underline{b} \end{matrix}$ , and ${\bar{C}}_{(\bar{a}, \bar{b})} (p, q)$ is calculated by $\bar{a}, \bar{b}$ and has a similar expression to $C_{(\underline{a}, \underline{b})} (p, q)$ . ${\tilde{C}}_{(\tilde{a}, \tilde{b})} (p, q)$ is depicted in Fig. 5.

Fig.5

Relationship between the degree at which points p and q are close to each other and their distance d (p, q).

Definition 4. (Semantic Structure of Regions). the semantic structure of a crisp region, type-1 or IT2 FR could be expressed as a geographical linguistic variable, and it is defined as

T₁ (X, GT) = T₁ (Structure, Region) = Core + Inner + Inside boundary + Outside boundary + Exterior (2) where X is the name of the geographical language variable “Structure”, and GT is a type of geometrical reference object, and it is “Region” here. The core area is the centre area of a region, and the inside boundary is a narrow ring on the inside of the border, while the inverse is true for the outside boundary. The inner area of a region is any part of the polygon surrounded by the inside boundary. There is a band between the core area and inside boundary that is part of the inner area called the Inner-Core area. The semantics for the Core, Inner, Inside boundary (IB), Outside boundary (OB), Inner-Core area (IC) and Exterior of a region can be expressed as $Core (A) = (C_{1} (p, q) ↓ A) (q) = inf_{p \in A, q \in X} I (C_{1} (p, q), A)$ (4) $Inner (A) = (C_{2} (p, q) ↓ A) (q) = inf_{p \in A, q \in X} I (\tilde{C_{2}} (p, q), A)$ (5) $IB (A) = A - Inner (A)$ (6) $OB (A) = (C (p, q) ↑ A) (q) - A = sup_{p \in A, q \in X} T (R (p, q), A) - A$ (7) $IC (A) = Inner (A) - Core (A)$ (8)

$SExterior (A) = 1 - (C (p, q) ↑ A) (q) = 1 - sup_{p \in A, q \in X} T (R (p, q), A)$ (9) where A can be a crisp, type-1 or type-2 fuzzy region. It is clear that the inner area and inside boundary are subsets of A, the core area and IC area are subsets of the inner area, and the outside boundary is a subset of the exterior of A; therefore, we call this area SExterior. In some situations, the IC area can contain the inside boundary, and thus, this area can also be expressed as IC₁ (A) = A - Core (A).

We can see that the maximum value of the intersection of IB⁺ (A) and Inner^- (A) is 0.5 and that the maximum value of the intersection of IB^- (A) and Inner⁺ (A) is 0.5 as well. This relationship indicates that the fuzzy area of IB⁺ (A) plus that of Inner^- (A) is equal to that of A, and the fuzzy area of IB^- (A) plus that of Inner⁺ (A) equals that of A, but the fuzzy area of IB (A) plus that of Inner (A) is not equal to the area of A. Different partitions have the following coverage properties.

The coverage of A is equal to the union of the coverages of Inner (A) and IB (A); that is, Cov (A) = Cov (Inner (A)) ⨿ Cov (IB (A)).

The coverage of A is equal to the union of the coverages of Core (A), IC (A) and IB (A); that is, Cov (A) = Cov (Core (A)) ⨿ Cov (IC (A)) ⨿ Cov (IB (A)).

The coverage of A is equal to the union of the coverages of Core (A) and IC1 (A); that is, Cov (A) = Cov (Core (A)) ⨿ Cov (IC1 (A)).

These three properties represent three different ways of understanding the substructures of regional features. Figure 6 shows the semantic structure of Luozhuangzi Town. We define the Inner, Inside boundary, Outside boundary, and Semantic exterior by closeness C_(200,1200), and the core area is defined by closeness C_(2000,3000), both in metres.

Fig.6

Semantic structure of a crisp boundary, including the (a) core; (b) inner; (c) inside boundary; (d) outside boundary; (e) semantic exterior; (f) area between the core and inside boundary; and (g) area between the crisp boundary and core.

3 Detailed direction model of regions

Cardinal direction models must also divide the space into several tiles [33 , 45], and several cardinal direction models under different reference frames [17] have been used for different applications. Four-direction and eight-direction relation models under direct and geomorphic reference frames have been widely used. Schneider et al. discussed the available qualitative cardinal direction models and their drawbacks based on four general design criteria – completeness, soundness, uniqueness, and generality – and three further object design requirements – shape consideration, converseness, and complex object support [34]. These models typically focus on external directional relationships. This section proposes an internal direction model based on the 8-direction fuzzy asymmetric model [9].

3.1 The 8-direction fuzzy asymmetric model

Definition 5. (4-Direction Model). The 4-direction model is constructed with four direction tiles: north, east, south and west (abbreviated N, E, S and W); it could be defined as a language variable: $T_{2} (X, GT {) = T}_{2} (4 - tiles, Region) = N + E + S + W$ (10)

where U = [0, 360) degrees, and the basic direction angles ϑ of the N, E, S and W tiles are 0, 90, 180, and 270, respectively. The azimuth angle from any point to the reference point is θ. The angle between the basic angle ϑ and the azimuth angle is α = |θ - ϑ|. The membership functions for each direction can be defined as follows: $μ_{Dir} (α) = - \frac{2}{π} α + 1, 0 \leq α \leq \frac{π}{2}, Dir \in {N, E, S, W}$ (11)

Definition 6. (8-Direction Fuzzy Asymmetric Model, EDFAM). The 8-direction fuzzy asymmetric model is constructed with eight direction tiles: north, east, south, west, northeast, southeast, northeast and northwest (abbreviated as N, E, S, W, NE, SE, SW and NW, respectively); and it could be abbreviated as EDFAM, as shown in Fig. 7. It could be defined as a language variable:

$\begin{matrix} T_{3} (X, GT {) = T}_{3} & (8 - tiles, Region) = N + E + S + W \\ + NE + NW + SE + SW \end{matrix}$ (12)

Fig.7

Eight-direction fuzzy asymmetric model. (a) Fuzzy 8-direction partition of a point; and (b) membership functions of each direction tile.

The membership functions for NE, NW, SE and SW can be obtained using the following method: μ_NE (α) = 2 × ∧ (μ_N, μ_E), μ_SE (α) = 2 × ∧ (μ_S, μ_E), μ_SW (α) = 2 × ∧ (μ_S, μ_W), μ_NW (α) = 2 × ∧ (μ_N, μ_W), where ∧ is the t-norm operator. Here, the minimum operator is adopted. The membership functions for each direction in the 8-direction fuzzy asymmetric model are shown in Equation (13): $μ_{θ} (α) = {\begin{matrix} - \frac{2}{π} α + 1, 0 \leq α \leq \frac{π}{2}, θ \in {N, E, S, W} \\ - \frac{4}{π} α + 1, 0 \leq α \leq \frac{π}{4}, θ \in {NE, SE, SW, NW} \end{matrix}$ (13)

The division angle between two direction tiles can be calculated using their membership functions. Other division angles can also be calculated with this method, as shown in Fig. 7. The membership value of each division angle is 0.67.

After the membership values of an angle belonging to each direction are calculated, Equation (13) is used to determine the direction of this angle. $μ_{θ} = max (μ_{A} (θ), μ_{B} (θ), μ_{C} (θ))$ (14)

where $\begin{matrix} A, B, C \in {N, NE, E, SE, S, SW, W, NW}, \\ A \neq B \neq C, μ_{θ} \in [0.67, 1] \\ . \end{matrix}$

3.2 Interval type-2 direction tiles

The direction tiles of IT2 FRs must be interval type-2 fuzzy, and if the semantic structure of a spatial object is modelled using an IT2 FS, the direction tiles are interval type-2 fuzzy as well, regardless of whether the reference object is crisp, type-1 or interval type-2 fuzzy. Exactly like IT2 FRs, interval type-2 direction tiles can be expressed by the LMF and UMF. If the reference object is an IT2 FR, and the membership value of any external point located in an interval type-2 fuzzy direction tile can be calculated with: $\begin{matrix} μ_{θ} (\tilde{R}, p) = max_{q \in sup p (\tilde{R})} T (μ_{\tilde{R}} (q), μ_{θ} (α_{qp})) \\ θ \in {N, NE, E, SE, S, SW, W, NW} \end{matrix}$ (15)

where $\tilde{R}$ is an interval type-2 reference region, p is any external point, q is any point located in the IT2 FR, and a_qp is the azimuth of the vector $\vec{qp}$ . Clearly, μ_θ () could have a single value if $\tilde{R}$ is a type-1 fuzzy object or a crisp object. The UMFs and LMFs of tile N and SE for an IT2 FR are shown in Fig. 8.

Fig.8

The upper and lower membership functions of tiles N and SE for an IT2 FR.

3.3 The fuzzy detailed direction model

The fuzzy detailed direction model is a combination of a semantic structure and the EDFAM, as shown in Fig. 9. The space can be divided into five parts – Core, IC area, Inside boundary, Outside boundary and SExterior – from the core of the polygon to the exterior of the polygon, as discussed in section 2. Combining these partitions and the EDFAM, we obtained 33 tiles, as shown in Fig. 10.

Fig.9

Explanation of the fuzzy detailed direction model.

Fig.10

Thirty-three parts of the fuzzy detailed direction model.

Definition 7. (Inner EDFAM). The EDFAM divides the inner area of a region into a core area and eight tiles, including an eastern part (EP), southeastern part (SEP), southern part (SP), southwestern part (SWP), western part (WP), northwestern part (NWP), northern part (NP) and northeastern part (NEP). These parts construct the following set: CS ={ C, EP, SEP, SP, SWP, WP, NWP, NP, NEP }. Then, the inner EDFAM could be defined as a language variable:

$\begin{matrix} T_{4} (X, GT {) = T}_{4} (CS 8 tiles, Polygon) = CP + NP + \\ EP + SP + WP + NEP + NWP + SEP + SWP \end{matrix}$ (16)

Definition 8. (Inside boundary EDFAM). The EDFAM divides the inside boundary into eight tiles: eastern inside boundary part (EIBP), southeastern inside boundary part (SEIBP), southern inside boundary part (SIBP), southwestern inside boundary part (SWIBP), western inside boundary part (WIBP), northwestern inside boundary part (NWIBP), northern inside boundary part (NIBP) and northeastern inside boundary part (NEIBP). These parts construct a set $IBS = {\begin{matrix} EIBP, SEIBP, SIBP, SWIBP, \\ WIBP, NWIBP, NIBP, NEIBP \end{matrix}}$ . Then, the inside boundary EDFAM could be defined as a language variable:

$\begin{matrix} T_{5} (X, GT {) = T}_{5} (IBS 8 tiles, Polygon) = NIBP \\ + EIBP + SIBP + WIBP + NEIBP + NWIBP + \\ SEIBP + SWIBP \end{matrix}$ (17)

Definition 9. (Outside boundary EDFAM). The EDFAM divides the outside boundary into eight tiles: eastern outside boundary part (EOBP), southeastern outside boundary part (SEOBP), southern outside boundary part (SOBP), southwestern outside boundary part (SWOBP), western outside boundary part (WOBP), northwestern outside boundary part (NWOBP), northern outside boundary part (NOBP) and northeastern outside boundary part (NEOBP). These parts construct a set $OBS = {\begin{matrix} EOBP, SEOBP, SOBP, SWOBP, \\ WOBP, NWOBP, NOBP, NEOBP \end{matrix}}$ . Then, the outside boundary EDFAM could be defined as a language variable:

$\begin{matrix} T_{6} (X, {GT) = T}_{6} (OBS 8 tiles, Polygon) = NOBP \\ + EOBP + SOBP + WOBP + NEOBP + NWOBP \\ + SEOBP + SWOBP \end{matrix}$ (18)

Definition 10. (Exterior EDFAM). The EDFAM divides the exterior space into eight tiles: eastern exterior part (ESEP), southeastern exterior part (SESEP), southern exterior part (SSEP), southwestern exterior part (SWSEP), western exterior part (WSEP), northwestern exterior part (NWSEP), northern exterior part (NSEP) and northeastern exterior part (NESEP). These parts construct the set $SES = {\begin{matrix} ESEP, SESEP, SSEP, SWSEP, \\ WSEP, NWSEP, NOBP, NESEP \end{matrix}}$ . Then, the exterior boundary EDFAM could be defined as a language variable:

$\begin{matrix} T_{7} (X, {GT) = T}_{7} (SES 8 tiles, Polygon) = NSEP \\ + ESEP + SSEP + WSEP + NESEP + NWSEP \\ + SESEP + SWSEP \end{matrix}$ (19)

The membership grade of the core area can be calculated using Equation (4), and the membership grades of other the 32 words in Equations (16)– (19) can be calculated from the intersection of the eight direction tiles of the core area and the IC area, IB area, OB area and SE area. This membership can be expressed as $\begin{matrix} mg (w) = p \cap t, w \in (CS \cup IBS \cup OBS \cup SES) and \\ w \neq C, p \in {IC, IB, OB, SE}, \\ t \in {E_{C}, S_{C}, W_{C}, N_{C}, {SE}_{C}, {SW}_{C}, {NE}_{C}, {NW}_{C}} \end{matrix}$ (20)

For example, the outside boundary EDFAM of Luozhuangzi Town contains eight tiles, including the outside boundary part (EOBP), SEOBP, SOBP, SWOBP, WOBP, NWOBP, NOBP and NEOBP, as shown in Fig. 11.

Fig.11

Eight direction tiles for the outside boundary of Luozhuangzi Town.

4 Direction relations between IT2 FR and interval type-2 fuzzy spatial objects under the detailed direction model

Spatial objects in geographical space can be classified into three types: crisp objects, type-1 fuzzy objects and type-2 fuzzy objects. Direction tiles can be classified into two types: type-1 fuzzy direction tiles and type-2 fuzzy direction tiles. Therefore, the direction relations can be classified into two types according to the type of direction tile.

4.1 Type-1 fuzzy direction relations

The type-1 fuzzy direction relation is the relation between crisp objects and type-1 fuzzy objects. All of the direction tiles are type-1 direction tiles. When the 33 tiles of a reference object are CS ∪ IBS ∪ OBS ∪ SES, the weight of the target object located in one tile of the reference object and can be calculated as follows: $\begin{matrix} p (θ_{R}, B) = \frac{\sum μ (θ_{R}) \cap μ (B)}{\sum μ (B)}, \\ θ_{R} \in CS \cup IBS \cup OBS \cup SES \end{matrix}$ (21)

where R is the reference object, and B is the target object. The relationship between R and B can then be determined from the maximum weight value.

4.2 Type-2 fuzzy direction relations

The type-2 fuzzy direction relationship indicates that one of the direction tiles and target objects must be type-2 fuzzy. Therefore, there are three scenarios:

(a) Direction tiles are type-2 fuzzy, and the target object is crisp or type-1 fuzzy. The direction tiles of crisp, type-1 and interval type-2 fuzzy objects can be modelled as an interval type-2 fuzzy set according to section 4; thus, the weight of the target object located in one direction tile can be expressed as $\tilde{p} = [\underline{p} (θ_{\tilde{R}}, B), \bar{p} (θ_{\tilde{R}}, B)]$ , and $\underline{p} (θ_{\tilde{R}}, B)$ and $\bar{p} (θ_{\tilde{R}}, B)$ can be calculated as follows: $\underline{p} (θ_{\tilde{R}}, B) = \frac{\sum \underline{μ} (θ_{\tilde{R}}) \cap μ (B)}{\sum μ (B)}, θ_{\tilde{R}} \in CS \cup IBS \cup OBS \cup SES$ (22) $\bar{p} (θ_{\tilde{R}}, B) = \frac{\sum \bar{μ} (θ_{\tilde{R}}) \cap μ (B)}{\sum μ (B)}, θ_{\tilde{R}} \in CS \cup IBS \cup OBS \cup SES$ (23)

where μ (B) is the membership degree of type-1 fuzzy or crisp object B, $\underline{μ} (θ_{R})$ and $\bar{μ} (θ_{\tilde{R}})$ are the lower and upper membership degrees, respectively, of an interval type-2 direction tile $θ_{\tilde{R}}$ of $\tilde{R}$ , and CS, IBS, OBS, SES are sets of interval type-2 direction tiles.

(b) The target object is interval type-2 fuzzy, and the direction tiles are type-1 fuzzy. In other words, the reference object is crisp or type-1 fuzzy, and the direction words are modelled by a type-1 fuzzy set. The weight of the target object located in one direction tile can then be expressed as $\tilde{p} = [\underline{p} (θ_{R}, \tilde{B}), \bar{p} (θ_{R}, \tilde{B})]$ , and $\underline{p} (θ_{R}, \tilde{B})$ and $\bar{p} (θ_{R}, \tilde{B})$ can be calculated as follows:

$\underline{p} (θ_{R}, \tilde{B}) = \frac{\sum μ (θ_{R}) \cap \underline{μ} (\tilde{B})}{\sum \bar{μ} (\tilde{B})}, θ_{R} \in CS \cup IBS \cup OBS \cup SES$ (24) $\bar{p} (θ_{R}, \tilde{B}) = \frac{\sum μ (θ_{R}) \cap \bar{μ} (\tilde{B})}{\sum \underline{μ} (\tilde{B})}, θ_{R} \in CS \cup IBS \cup OBS \cup SES$ (25)

where $\tilde{B}$ is an interval type-2 fuzzy object, and θ_R is a type-1 fuzzy direction tile.

(c) The reference and target objects are interval type-2 fuzzy, and thus, all the direction tiles are interval type-2 fuzzy as well. Similarly, the weight of the target object located in one direction tile can be expressed as $\tilde{p} = [\underline{p} (θ_{\tilde{R}}, \tilde{B}), \bar{p} (θ_{\tilde{R}}, \tilde{B})]$ , and $\underline{p} (θ_{\tilde{R}}, \tilde{B})$ and $\bar{p} (θ_{\tilde{R}}, \tilde{B})$ can be calculated as follows:

$\begin{matrix} \underline{p} (θ_{\tilde{R}}, \tilde{B}) & = & \frac{\sum μ (θ_{\tilde{R}}) \cap \underline{μ} (\tilde{B})}{\sum \bar{μ} (\tilde{B})}, \\ θ_{\tilde{R}} & \in & CS \cup IBS \cup OBS \cup SES \end{matrix}$ (26)

$\begin{matrix} \bar{p} (θ_{\tilde{R}}, \tilde{B}) & = & \frac{\sum μ (θ_{\tilde{R}}) \cap \bar{μ} (\tilde{B})}{\sum \underline{μ} (\tilde{B})}, \\ θ_{\tilde{R}} & \in & CS \cup IBS \cup OBS \cup SES \end{matrix}$ (27)

where $\tilde{R}$ and $\tilde{B}$ are the reference and target objects, respectively.

The calculation methods for the above three cases are clearly consistent in a broad sense, and the three scenarios and methods could be unified as scenario (c) because a crisp or type-1 fuzzy object can be treated as a special case of an interval type-2 fuzzy spatial object. Another problem should be noted: adjacent direction tiles can overlap with each other, as shown in section 6. Equations (20)– (26) cannot address this issue effectively; therefore, Equations (26) and (27) could be updated as

$\begin{matrix} \underline{p} (θ_{\tilde{R}}, \tilde{B}) & = & \frac{\sum μ (θ_{\tilde{R}}) \cap \underline{μ} (\tilde{B})}{\sum_{i = 1}^{N} \sum μ (θ_{\tilde{R}}) \cap \bar{μ} (\tilde{B})}, \\ θ_{\tilde{R}} & \in & CS \cup IBS \cup OBS \cup SES \end{matrix}$ (28)

$\begin{matrix} \bar{p} (θ_{\tilde{R}}, \tilde{B}) & = & \frac{\sum μ (θ_{\tilde{R}}) \cap \bar{μ} (\tilde{B})}{\sum_{i = 1}^{N} \sum μ (θ_{\tilde{R}}) \cap \underline{μ} (\tilde{B})}, \\ θ_{\tilde{R}} & \in & CS \cup IBS \cup OBS \cup SES \end{matrix}$ (29)

where N is the number of tiles in a detailed direction model.

The possibility method proposed by Xiao et al. [15] is adopted to calculate the maximal weights of all of the direction tiles.

5 Applications in understanding remote sensing data

In this section, we use the proposed method to analyse the spatial distributions and variations of wetlands in different seasons. The research area, the Baidagang wetland, is located in the Tianjin Binhai New Area with latitude of 38°41' to 38°50' and a longitude of 117°15' to 117°15', as shown in Fig. 12. The Beidagang Reservoir is in the south, and the north zone and contains mainly fish ponds, rivers, and marshlands. Figure 12(a) presents a Sentinel 2 remote sensing image from May 2016. May is at the end of the dry season; thus, there is very little water in the reservoir, and the main types of land cover shown are bare land, water, and vegetated land. Figure 12(b) presents a Sentinel 2 remote sensing image from August 2016, which is in the middle of the rainy season; therefore, the amount of water in the reservoir is more than that in the dry season, and the vegetation is growing very well. The main types of land cover are water and vegetated land, and vegetation is scarce near the dirt road and the transition zone between the water and the vegetated land. Overall, the water in the reservoir is lower because the North China Plain does not have much rainfall.

Fig.12

The Baidagang wetland in Tianjin.

Both images are classified into three classes by the IT2 FCM* algorithm [18]. The acquired datasets are converted from Level-1C to Level-2A by SNAP software and its plug-in component CEN2COR provided by the European Space Agency (ESA). Both datasets contain ten bands and have a resolution of 20 m. The two images are extracted using the crisp boundary of the reservoir. At the initialization phase, the classification number is set to 3, the termination criterion value ɛ is equal to 0.001, and the fuzzifiers m₁ and m₂ are set to 1.3 and 2.5, respectively, according to Fisher’s (2010) research on wetland classification. The results of the classification are shown in Fig. 13. Then, we can acquire the lower and upper membership grades of water for the two months, as shown in Fig. 14. Let A be an interval type-2 fuzzy set in a raster image having g rows and h columns of pixels. Each pixel has area Δ, and therefore, the area of A, FArea_A, can be calculated as

$\begin{matrix} {FArea}_{\tilde{A}} = [{FArea}_{{\tilde{A}}^{-}}, {FArea}_{{\tilde{A}}^{+}}] \\ = [\sum_{j = 1}^{g} \sum_{k = 1}^{h} ({\tilde{A}}^{-} (i, k)), \sum_{j = 1}^{g} \sum_{k = 1}^{h} ({\tilde{A}}^{+} (i, k))] \times Δ \end{matrix}$ (30)

Fig.13

Results of classification for two months.

The area of the water increased from May to August. Let $\tilde{A}$ be the water zone in May and $\tilde{A}$ be the water zone in August. The zone of increase can be calculated as

$\begin{matrix} ∥ & Incre (\tilde{A}, \tilde{B}) = [\max (0, B^{-} - A^{+}), \\ \max (0, B^{-} - A^{-}, B^{+} - A^{+})] \end{matrix}$ (31)

As seen in Figs. 13 and 14, the area lost is minimal; thus, we ignore the lost area in this paper.

Fig.14

The lower and upper membership grades of water at two months.

The increment zone from May to August can be obtained using Equation (31), and the fuzzy area of water in May and August and the increment zone can then be calculated and are shown in Table 2. Using the crisp method, the increment area is calculated by subtracting the area in August because it is minimal. Using the fuzzy method, the lower value in the interval number can be treated as the minimum area, and the upper value can be treated as the maximum area. This fuzzy statistics method is similar to Fisher’s method [39] and represents an advance for the scientific community concerned with the conceptualization of land cover classes. Then, we count the fuzzy area of α-cut sets of lower and upper membership grades using different α-values for May and August. When α>0.3, the change in area from May to August becomes stable and satisfies the criterion that the crisp area values be between the corresponding fuzzy area interval values and the area interval with the shortest length. In contrast, the original fuzzy area (α= 0) has greater uncertainty than the other results. The result is conservative when α= 0.5 because the membership degree of pixels belonging to water is equal to or greater than 0.5 only when the sum of the membership degrees of pixels belonging to other classes is less than 0.5; thus, it is clear that this pixel belongs to water. Therefore, α= 0.3 is meaningful, and the following discussion uses this value.

Table 1

Macro-areal statistics of the Beidagang Reservoir for May and August using the crisp and fuzzy methods

Method	Area in May (km²)	Area in August(km²)	Increment (km²)
Crisp	13.965	26.439	12.474
Fuzzy (α= 0)	[10.292, 27.114]	[21.528, 35.745]	[8.466, 15.760]
Fuzzy (α= 0.1)	[10.105, 20.984]	[21.357, 29.897]	[8.383, 14.696]
Fuzzy (α= 0.2)	[9.991, 15.593]	[21.184, 28.449]	[8.206, 13.558]
Fuzzy (α = 0.3)	[9.864, 14.241]	[20.979, 27.228]	[7.910, 12.306]
Fuzzy (α= 0.4)	[9.622, 13.501]	[20.757, 25.965]	[7.048, 11.51]
Fuzzy (α= 0.5)	[8.777, 13.333]	[19.603, 25.568]	[5.855, 10.723]

Table 2

Microscopic statistics of water area in different semantic partitions of the Beidagang Reservoir (km²) (α= 0.3)

Month	IB	Inner	Core	IC	IC1
May	[4.671, 6.949]	[5.504, 7.228]	[0.006, 0.073]	[4.952, 7.989]	[9.860, 14.188]
August	[6.962, 9.408]	[14.512, 17.645]	[1.348, 1.521]	[12.485,17.479]	[19.855, 25.792]

Now, we require the microscopic statistics; for example, we must calculate the area of the water located on the boundary or in the core zone, and then, the semantic structures are needed. We define the inner, inside boundary, and semantic exterior regions by closeness [C_(300,700), C_(350,900)], the core area is defined by closeness [C_(2400,3500), C_(2600,3800)] to manage interpersonal semantic uncertainty, and both are presented in metres. The semantic structures of the Beidagang Reservoir are built by Equations (4)– (9), and all of this partition is shown in Fig. 15.

Fig.15

The semantic structures of Beidagang Reservoir.

We can overlay the membership grades of water and these semantic structures using the intersection operation

$\begin{matrix} intersection (\tilde{A}, \tilde{B}) = [min (A^{-} (x), B^{-} (x)), \\ min (A^{+} (x), B^{+} (x))] \end{matrix}$ (32)

where $\tilde{A}, \tilde{B} \in IT 2 FS$ . Then, after calculating the areas of these intersection layers using Equation (32), we can calculate the microscopic statistics, which are shown in Table 2. The water area in the core zone is small both in the rainy season and in the dry season. Additionally, from the dry season to the rainy season, the water area in the IC (IC1) zone increased considerably. The change in water area in the inner boundary is not obvious.

Next, we can use the detailed direction model to determine the directional distribution of water. As discussed in subsection 2.2, there are three common ways to understand the substructure of a regional feature. To facilitate discussion, we choose the core zone as the reference object to build the detailed direction tiles for the IC1 zone of the Beidagang Reservoir. The lower and upper membership grades of each detailed direction tile in the IC1 zone are shown in Figs. 16 and 17, respectively.

Fig.16

Lower membership grades of each detailed direction tile in the IC1 zone.

Fig.17

Upper membership grades of each detailed direction tile in the IC1 zone.

The direction relation matrix is expressed as $Dir = | \begin{matrix} NWP & NP & NEP \\ WP & C & EP \\ SWP & SP & SEP \end{matrix} |$ (33)

Because tiles in the detailed direction model can overlap each other, each cell in the direction matrix is determined using Equations (28) and (29). The direction relation matrix for May is then ${Dir}_{May} = | \begin{matrix} [0 . 023, 0 . 044] & [0 . 035, 0 . 093] & [0 . 052, 0 . 121] \\ [0.002, 0 . 011] & [0, 0.003] & [0 . 233, 0 . 497] \\ [0 . 039, 0 . 127] & [0 . 135, 0 . 301] & [0 . 14 3, 0 . 315] \end{matrix} |$ .

We can see that the water zone is primarily distributed in the EP, SEP and SP, and the core zone has a small proportion, similar to WP. We use the same method to obtain the direction relation matrix for August, ${Dir}_{August} = | \begin{matrix} [0 . 121, 0 . 236] & [0 . 129, 0 . 24 6] & [0.089, 0 . 17 0] \\ [0 . 085, 0 . 164] & [0.009, 0.014] & [0 . 111, 0 . 214] \\ [0 . 066, 0 . 133] & [0 . 062, 0 . 12 3] & [0 . 046, 0 . 093] \end{matrix} |$ (34)

Comparing this matrix with that for May, the water body coverage of all of the partitions increases, and water is uniformly distributed on the eight surrounding direction tiles. The proportions of water in the NP, NWP, WP, SWP and core zone increase, while those in the other partitions decrease.

6 Conclusion

Artificial intelligence-related technologies have been widely used in geographic information sciences in recent years, including deep learning, fuzzy logic, natural language processing, and neural networks. Intelligent GISs should have the following abilities: (1) they should address high-order fuzzy geographical phenomena; (2) they should address fuzzy geographical concepts in natural language processing; and (3) they should integrate the user’s meaning. Intelligent GISs can express the user’s subjective ideas for different applications. As a new data source in intelligent GISs, natural language requires the expression and integration of the semantics of geospatial words. However, the fuzziness of spatial semantics poses a great challenge to intelligent GISs. In this paper, CWW is used to define geographical concepts, and then, the semantic partition model of region objects is studied. Three different ways of understanding the substructures of regional features are provided by this model. Then, a detailed direction model of region objects is defined, and this model is more appropriate for expressing human orientation cognition than the model proposed in the literature [42].

The research in this paper is a part of the study of geographical vocabulary expressions, and a great deal of geographical vocabulary is worth studying, including “along” and “between”. In different contexts, the meanings of these words can differ. Therefore, in subsequent studies, our research will focus on the expression of these words, as well as the semantic integration and application of different geospatial words. Hesitant linguistic uncertainty affects spatial decision-making processes, but it has not been discussed in the GIS domain. Therefore, we will perform hesitant linguistic spatial analysis based on hesitant fuzzy linguistic terms sets [3 , 50] in our future work.

Footnotes

Acknowledgments

The work in this paper was supported by grants from the Chinese National Nature Science Foundation (No. 41101352) and the Key Project of the Tianjin Natural Science Foundation of China (No. 17JCZDJC39700).

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