Indoors Locality Positioning Using Cognitive Distances and ... - MDPI

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Indoors Locality Positioning Using Cognitive Distances and Directions Yankun Wang 1,2 , Hong Fan 1,2, * and Ruizhi Chen 1,2 1 2

*

State Key Lab for Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, 129 Luoyu Road, Wuhan 430079, China; [email protected] (Y.W.); [email protected] (R.C.) Collaborative Innovation Center of Geospatial Technology, Wuhan University, Wuhan 430079, China Correspondence: [email protected]; Tel.: +86-186-2771-6767

Received: 17 October 2017; Accepted: 2 December 2017; Published: 7 December 2017

Abstract: Spatial relationships are crucial to spatial knowledge representation, such as positioning localities. However, minimal attention has been devoted to positioning localities indoors with locality description. Distance and direction relations are generally used when positioning localities, namely, translating descriptive localities into spatially explicit ones. We propose a joint probability function to model locality distribution to address the uncertainty of positioning localities. The joint probability function consists of distance and relative direction membership functions. We propose definitions and restrictions for the use of the joint probability function to make the locality distribution highly practical. We also evaluate the performance of our approach through indoor experiments. Test results demonstrate that a positioning accuracy of 3.5 m can be achieved with the semantically derived spatial relationships. Keywords: spatial relationships; membership function; positioning localities indoors; locality description

1. Introduction The importance of spatial relationships is well recognized in many domains, such as image interpretation, spatial reasoning, and spatial knowledge representation [1–4]. Spatial relationships in geographical information systems (GIS) commonly involve three components, namely, a distance relation (e.g., “near” and “100 m”), a direction relation that describes angle order, and a topological relation that describes intersections and neighborhoods [5]. Locality description contains one or more place names and spatial relationships. Any natural or artificial feature with a name can be used as a reference object (RO) [5–7]. Locality description can be derived from daily communication; for example, we could state that “object A is 100 m north, and object B is 50 m northwest” for outdoor cases or “object A is 50 m in front, and object B is 30 m to the right” for indoor cases. Humans tend to use relative direction to describe locality. Semantically derived direction and distance are frequently combined to perform a highly precise locality description [5]. The mapping mechanism between qualitative and quantitative data is a research issue not only in GIS but also in other domains [8]. The current work introduces a novel method of positioning localities indoors by locality description. Existing literature on this topic includes many meaningful references for positioning localities by locality description. To deal with uncertainties associated with point localities, Wieczorek [9] developed a method to combine different types of uncertainty into a point radius. Guo [10] considered the shape of ROs, refined Wieczorek’s method, and proposed a probabilistic approach for georeferenced localities for museum data collection. Liu [5] extended the work of Guo and developed a general method to position localities through spatial assertion. The author indicated that when distance is quantitative, which may be uncertain, the target object (TO) is distributed in the band surrounding the Sensors 2017, 17, 2828; doi:10.3390/s17122828

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2 of 17 is quantitative, which may be uncertain, the target object (TO) is distributed in the band surrounding the RO. A probability density function (PDF) was provided for absolute direction relations density to describe the probability of absolute a locality.direction Furthermore, refinement and RO. A probability function (PDF) wasdistribution provided for relations to describe integration were utilized for cases in which two or more ROs were involved in 2D Euclidean space. the probability distribution of a locality. Furthermore, refinement and integration were utilized for This method suitable for ROs various descriptions, but the indoor locality description cases in whichistwo or more werelocality involved in 2D Euclidean space. This method is suitable(i.e., for “object A is 50 m in front, and object B is 30 m to the right”) described in this work involves relative various locality descriptions, but the indoor locality description (i.e., “object A is 50 m in front, and direction In other direction relations, ROs, and descriptive target localityInshould object B isrelations. 30 m to the right”)words, described in this work involves relative direction relations. other be transformed from relative to absolute if the method proposed in [5] is used. Calculating words, direction relations, ROs, and descriptive target locality should be transformed from relativethe to relative angle of ROs (Figure 1) in and target locality can resolve this problem. absolute if the method proposed [5]descriptive is used. Calculating the relative angle of ROs (FigureSeveral 1) and studies have indicated histogram of anglesSeveral can represent fuzzyindicated relative that direction relations descriptive target localitythat canaresolve this problem. studies have a histogram of between [11,12]. However, ROs are crisp objects, and the region of target locality description angles canobjects represent fuzzy relative direction relations between objects [11,12]. However, ROs are crisp is a fuzzy object with an distribution, which inappropriate. venuncertain if such a distribution, relationship objects, and the region of uncertain target locality description is aisfuzzy object withEan is appropriate, the calculation cost awould be high. Accordingly,theother breakthroughs have been which is inappropriate. Even if such relationship is appropriate, calculation cost would be high. presented. The “between” relations provided in [1,13] indicate that human perception of the spatial Accordingly, other breakthroughs have been presented. The “between” relations provided in [1,13] relation that between two objects of is the closely related to between angular two information, which related conforms to the indicate human perception spatial relation objects is closely to angular trapezoid membership function. information, which conforms to the trapezoid membership function. N

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Figure 1. 1. Relative angle angle of of ROs. ROs. Figure

Our novel contribution is that we introduce a method to position localities indoors via Our novel contribution is that we introduce a method to position localities indoors via cognitive cognitive distances and directions. The method, which is based on a joint probability function, distances and directions. The method, which is based on a joint probability function, consists of consists of distance and relative direction membership functions. Definitions and restrictions are distance and relative direction membership functions. Definitions and restrictions are proposed to proposed to make the process of positioning localities conform to cognition. The test results make the process of positioning localities conform to cognition. The test results demonstrate that a demonstrate that a positioning accuracy of 3.5 m can be achieved with cognitive distances and positioning accuracy of 3.5 m can be achieved with cognitive distances and directions. directions. The paper is organized as follows: in Section 2, we review previous studies on spatial relationships. The paper is organized as follows: in Section 2, we review previous studies on spatial In Section 3, we conduct a recognition experiment on distance to construct a distance membership relationships. In Section 3, we conduct a recognition experiment on distance to construct a distance function and propose a membership function for relative direction. In Section 4, we develop a method membership function and propose a membership function for relative direction. In Section 4, we of positioning localities with a joint probability function on the basis of distances and directions. develop a method of positioning localities with a joint probability function on the basis of distances Examples are presented to demonstrate the method of positioning localities in Section 5, and the and directions. Examples are presented to demonstrate the method of positioning localities in conclusions and directions for future work are provided in Section 6. Section 5, and the conclusions and directions for future work are provided in Section 6. 2. Related Work 2. Related Work The definitions of spatial relations in existing literature are briefly presented in this section. The definitions of spatial relations in existing literature are briefly presented in this section. 2.1. Distance Relationship 2.1. Distance Relationship Distance relationship may be divided into qualitative and quantitative when describing Distance relationship may be statements divided into qualitative andqualitative quantitative when describing localities [5]. “Near”, “far”, and other are used to describe distance relationships, localities [5]. “Near,” anda rough other clue statements are used to describe qualitative distance and qualitative distance“far,” provides about locality. Worboys [14] conducted a cognitive relationships, qualitative provides a rough clue about Worboys [14] experiment on and the vague spatialdistance relation “near” in environment space andlocality. concluded that formal theories can be applied to reasoning with vague spatial notions. In [15], a statistical approach called

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conducted aa cognitive Sensors 2017, 17, 2828 of 17 conducted cognitive experiment experiment on on the the vague vague spatial spatial relation relation “near” “near” in in environment environment space space3 and and concluded that formal theories can be applied to reasoning with vague spatial notions. In [15], concluded that formal theories can be applied to reasoning with vague spatial notions. In [15], aa statistical statistical approach approach called called ordered ordered logit logit regression regression was was used used to to predict predict metric metric distance distance on on the the basis basis ordered logit regression was used to predict metric distance on the basis of corresponding context of of corresponding corresponding context context information information and and linguistic linguistic distance. distance. information and linguistic distance. Compared Compared with with qualitative qualitative distance, distance, quantitative quantitative distance, distance, also also called called semi-quantitative semi-quantitative Compared with qualitative distance, quantitative distance, also called semi-quantitative distance, distance, is utilized more frequently and thus conveys more accurate information on distance, is utilized more frequently and thus conveys more accurate information on locality locality [5]. [5]. Liu Liu is utilized more frequently and thus conveys more accurate information on locality [5]. Liu comprehensively comprehensively discussed quantitative distance and argued that distance may be uncertain comprehensively discussed quantitative distance and argued that distance may be uncertain discussed quantitative distance andimprecise argued that distance may be uncertain because of measurement because because of of measurement measurement errors errors or or imprecise records, records, and and different different uncertainties uncertainties possess possess different different errors or imprecise records, and different uncertainties possess different probability distributions; probability distributions; among these uncertainties, that caused by a measurement error (Figure probability distributions; among these uncertainties, that caused by a measurement error (Figure 2) 2) among these uncertainties,Considering that caused by a measurement error (Figure 2) hasisa consistent normal distribution. has has aa normal normal distribution. distribution. Considering that that our our cognition cognition of of spatial spatial distance distance is consistent with with that that Considering that our cognition we of spatial this distance is and consistent with that involving a measurement involving involving aa measurement measurement error, error, we adopt adopt this model model and discuss discuss it it in in Section Section 3.1. 3.1. error, we adopt this model and discuss it in Section 3.1.

Figure 2. Illustration Illustration of uncertain uncertain quantitative distance distance and error error following a normal distribution [5]. Figure Figure 2. 2. Illustration of of uncertain quantitative quantitative distance and and error following following a normal distribution [5].

2.2. Direction Relationship Relationship 2.2. Direction Direction Relationship Direction relationship can categorized relative front and (i.e., as as relative (i.e.,(i.e., front and and back)back) and absolute (i.e., north Direction relationship relationshipcan canbebe becategorized categorized as relative (i.e., front and back) and absolute absolute (i.e., north and south) when describing locality. Cardinal direction relationships (CDR) (Figure 3) and south) when describing locality. locality. CardinalCardinal directiondirection relationships (CDR) (Figure 3) (Figure are influential north and south) when describing relationships (CDR) 3) are are influential relationships abstracted from angle values that divide a space into cones. Describing aa relationships abstracted from angle values that divide a space into cones. Describing a locality influential relationships abstracted from angle values that divide a space into cones. Describing is locality is complicated because of the vagueness of the direction relationship. Many direction complicated because of the vagueness of the direction relationship. Many direction models have been locality is complicated because of the vagueness of the direction relationship. Many direction models been developed; these include cone based (Figure 3a) is developed; these models include conemodels based (Figure [16], which is applicable towhich point ROs; minimal models have have been developed; these models include3a) cone based (Figure 3a) [16], [16], which is applicable applicable to point ROs; minimal bounding rectangle (MBR) (Figure 3b) [17], which is suitable for linear or bounding rectangle (MBR) (Figure 3b) [17], which is suitable for linear or areal ROs; and internal to point ROs; minimal bounding rectangle (MBR) (Figure 3b) [17], which is suitable for linear or areal ROs; and internal cardinal direction (Figure 3c) [18], which refines the spatial relationship cardinal direction (Figurecardinal 3c) [18],direction which refines the3c) spatial when the TO relationship is inside an areal ROs; and internal (Figure [18], relationship which refines the spatial when the areal whenRO. the TO TO is is inside inside an an areal areal RO. RO.

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Figure (a) Cone-based CDR model; (b) MBR-based CDR model; (c) ICD model (the Figure 3. 3. (b)(b) MBR-based CDR model; (c) MBR-based MBR-based ICD ICD model (the Figure 3. (a) (a)Cone-based Cone-basedCDR CDRmodel; model; MBR-based CDR model; (c) MBR-based model dashed line is the reference object, and the solid lines are the boundaries of directions). dashed line is the reference object, and the solid lines are the boundaries of directions). (the dashed line is the reference object, and the solid lines are the boundaries of directions).

Fuzzy Fuzzy mathematical mathematical relations relations are are utilized utilized to to describe describe the the relative relative positions positions of of objects. objects. Deng Deng [11] [11] Fuzzyofmathematical relations are utilizedtotorepresent describe the the relative relative positions positions of of objects. objects. However, Deng [11] proposed the use of a histogram of angles proposed of the use of a histogram of angles to represent the relative positions of objects. However, proposed of the use of a histogram of angles to represent the relative positions of objects. Matsakis However, this this this approach approach has has aa high high computational computational cost cost and and considers considers only only the the raster raster date. date. Matsakis [12] [12] approach has a high computational cost and considers only the raster date. Matsakis [12] introduced introduced the histogram of force, which is superior to the histogram of angles and can process introduced the histogram of force, which is superior to the histogram of angles and can process the histogram of force, numerous which is superiorhave to the histogram of angleson and can process vector data. vector vector data. data. Since Since then, then, numerous studies studies have been been conducted conducted based based on this this notion notion [2,19,20]. [2,19,20]. Since then, numerous studies have been conducted based on this notion [2,19,20].

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Severalinteresting interestingrelations, relations, such such as as “left”, “left,” “between,” mentioned in [1]. Several “between”,and and“above,” “above”,were were mentioned in [1]. The authors argued that human perception of spatial positions between two objects is The authors argued that human perception of spatial positions between two objects is approximately approximately related to angle information and defined relative direction membership functions for related to angle information and defined relative direction membership functions for them. We use them. We use the “between” relationship (Figure 4) as an example. The degree to which object B is the “between” relationship (Figure 4) as an example. The degree to which object B is between two between two objects A and C is calculated based on point relations. For all points with a ∊ A, b ∊ B, objects A and C is calculated based on point relations. For all points with a ∈ A, b ∈ B, and c ∈ C, and c ∊ C, angle Θ at b with edges that connect a and c is calculated. A membership function is angle Θ at b with edges that connect a and c is calculated. A membership function is proposed to proposed to illustrate the degree to which b is between a and c. For extended objects, angle Θ is the illustrate degree to points which (a, b isb,between and c. For extended angle Θ is the average the angle averagethe angle over c). By aconsidering the shapeobjects, of an object, [15] developed over points (a, b, c). and By considering the shape ofofanvisibility. object, [15] developed “between” “between” relation proposed the definition Other concepts,the such as “along”relation and and proposed the definition of visibility. Other concepts, such as “along” and “surrounding” [21,22], “surrounding” [21,22], have been developed from these concepts. All concepts are based on an have been developed fromlines. these concepts. All concepts are based on an angle between two sight lines. angle between two sight

B A

C

Figure 4. Illustration of the definition “between” in [1]. Figure 4. Illustration of the definition “between” in [1].

3. Fuzzy Distance and Relative Direction Function 3. Fuzzy Distance and Relative Direction Function We conduct a cognitive experiment to construct a distance membership function and construct We conduct cognitive experiment to construct a distance membership function and construct a a fuzzy relativeadirection model. fuzzy relative direction model. 3.1. Cognitive Experiment and Fuzzy Distance Function 3.1. Cognitive Experiment and Fuzzy Distance Function A cognitive experiment on distance is conducted in a shopping market, which is an ideal A cognitive experiment on distance is conducted in aofshopping is an ideal indoor indoor environment that provides sufficient participants different market, ages andwhich backgrounds. During environment that provides sufficient participants of different ages and backgrounds. During the experiment, the participants are asked to stand at a marked point and describe the distancethe experiment, themarked participants are asked to stand at (shops a marked point and describe distance between between the point and cognitive objects in the market). Throughthe field work on the theshopping marked market, point and objects (shops of incognitive the market). Through field work shopping wecognitive established three groups distance experiments at 10,on 30,the and 50 m, market, we established three groups of cognitive experiments at 10, 30, andexperiment 50 m, with each with each group containing 40 participants. Wedistance established rules for the cognitive in group containing We established rules for the[15]. cognitive experiment in consideration consideration of40 theparticipants. factors that influence distance cognition Each group is expected to locate cognitive (one or two cognition other adjacent objects and isolated) to avoid thethree influence of of three the factors that objects influence distance [15]. Each group is expected to locate cognitive spatial distribution of objects. The orientation between cognitive objects of and participants is objects (one or two other adjacent objects and isolated) to the avoid the influence spatial distribution arbitrary (either positive or inclined) to simulate the real environment. Each participant is sampled of objects. The orientation between the cognitive objects and participants is arbitrary (either positive to prevent the cross of differentEach groups. or once inclined) to simulate theinfluence real environment. participant is sampled once to prevent the cross Data that deviate significantly from the correct distance are eliminated. For example, if the influence of different groups. correct is 10significantly m, but the cognitive distance is 30 m,are which deviates significantly the Data distance that deviate from the correct distance eliminated. For example, iffrom the correct correct distance and does not follow the normal distribution. As shown in Figure 5, cognitive distance is 10 m, but the cognitive distance is 30 m, which deviates significantly from the correct distance follows a normal distribution after normality examination. A large cognitive distance distance and does not follow the normal distribution. As shown in Figure 5, cognitive distance follows equates to a large deviation from the correct distance. a normal distribution after normality examination. A large cognitive distance equates to a large We will use fuzzy sets to represent the vague geographical knowledge. A fuzzy set in a deviation from the correct distance. universe X is formally defined as a mapping U from X to the unit interval [0, 1]. For x in X, U(x) is We will use fuzzy sets to represent the vague geographical knowledge. A fuzzy set in a universe called the membership degree of x in U, and reflects the extent to which x has the (fuzzy) property X is formally defined as a mapping U from X to the unit interval [0, 1]. For x in X, U(x) is called the that U is modeling. Fuzzy sets are particularly useful to represent fuzzy distance relations. membership degree of x in U, and reflects the extent to which x has the (fuzzy) property that U is modeling. Fuzzy sets are particularly useful to represent fuzzy distance relations.

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Figure 5. Normal distribution of fuzzy distance cognition. (a) 10 m; (b) 30 m; (c) 50 m. (a) (b) (c)

When Figure used 5.toNormal definedistribution fuzzy distance relations, the trapezoid function presents many of fuzzy distance cognition. (a) 10 m; (b) 30 m; (c) 50 m. Figure 5. Normal distribution of fuzzy distance cognition.and (a) 10intuitiveness m; (b) 30 m; (c) [23]. 50 m. Numerous advantages, such as computation efficiency, robustness, qualitative distance relationships defined with a trapezoid function When and usedsemi-qualitative to define fuzzy distance relations,can thebetrapezoid function presents many [5,11,23]. When used to define fuzzy distance relations, the trapezoid function presents many advantages, advantages, such as computation efficiency, robustness, and intuitiveness [23]. Numerous We α, β, semi-qualitative γ and efficiency, δ be non-negative numbers, and the is α ≤ Numerous β with ≤γ ≤ δ. the basis ofand the such as let computation robustness, and intuitiveness [23]. qualitative qualitative and distance relationships canorder be defined a On trapezoid function cognitive experiment, we model fuzzy distance as a non-isosceles trapezoid membership function semi-qualitative distance relationships can be defined with a trapezoid function [5,11,23]. [5,11,23]. μdis(d). We numbers, and and the the order order is is αα ≤≤ββ≤γ≤≤γ δ. ≤On δ. On basis of We let let α, α, β, β, γ γ and and δδ be be non-negative non-negative numbers, the the basis of the the cognitive experiment, model fuzzydistance distanceasasaanon-isosceles non-isosceles trapezoid trapezoid membership membership function cognitive experiment, wewe model fuzzy function 1 d  µμdis (d). dis(d).  (  )(d   )  d d≤γ   1 β≤  （d）   dis (1)   (β − 1α)(d − α)  d≤ d β α≤  d (1) µdis (d) =  (   )(d  )  γ −)(d δ)( d −)δ) γ≤ d ≤ δ    (( d     0 d  ,   d  （d）   dis (1) d ≤α,δd≤ d (   )(d0  )



An illustration the fuzzy distance function in  membership An illustration of theoffuzzy distance membership is shown Figure 6. 6. ,in  0 function disshown d Figure

An illustration of the fuzzy distance membership function is shown in Figure 6.

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αfuzzy β γ In Figure 6. 6. Illustration Illustration of of the the fuzzy distance distance membership function. function. (1), are the the δ Figure membership In Equation Equation (1), β β and and γ γ are deviation from from the the correct correct distance, distance, and and α α and and δδ may may be be derived derived from from the the fuzzy fuzzy distance distance distribution. distribution. deviation Figure 6. Illustration of the fuzzy distance membership function. In Equation (1), β and γ are the

3.2. Fuzzy Relative Direction Function 3.2. Fuzzy Relative Function deviation fromDirection the correct distance, and α and δ may be derived from the fuzzy distance distribution. Human perception perception of of the the spatial spatial relation relation between between two two objects objects is is closely closely related related to to angular angular Human 3.2. Fuzzy Relative Direction Function information [1]. For instance, a person could search a cone area by turning approximately 45° from information [1]. For instance, a person could search a cone area by turning approximately 45◦ from front Human to front–left; front–left; such aaof process does not not involve distance. the of relativetodirection direction perception the spatial relation between twoOn objects is closely angular front to such process does involve distance. On the basis basis of the therelated relative membership function for “right,” “left,” and “above” constructed in [1], we define the membership information [1]. For instance, a person search a constructed cone area byinturning 45° from membership function for “right”, “left”,could and “above” [1], we approximately define the membership function for fuzzy fuzzy relative relative direction as µnot ( ):) : distance. On the basis of the relative direction front to front–left; such a direction process does function for as Θ reldir(involve reldir

membership function for “right,” “left,” and “above” constructed in [1], we define the membership function for fuzzy relative direction as  reldir (  ) :

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1       1  path - () 4   8  reldir（）   path    8   4  1( a)  reldir（）     π π8 + 4 ×path(Θ) −Θ   8  a µreldir (Θ ) = 0   8 π8 −a    0 0 

  path    a () 4   path    a 4 a   ( ) path (  )     π 8 × path4(Θ) − Θ ≤ a     a 4   path  4  path (  )     π8 a ≤ π4 × ) Θ ≤ 4 path(Θ()−   88 π  path (  )    π 4 4× path(Θ) − Θ ≥ 88

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(2) (2) (2)

An relativedirection directionmembership membership function is provided in Figure Anillustration illustrationof of the the fuzzy fuzzy relative function is provided in Figure 7. 7. An illustration of the fuzzy relative direction membership function is provided in Figure 7.

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Figure7.7.Illustration Illustrationofofthe thefuzzy fuzzyrelative relativedirection directionmembership membershipfunction, function,Equation Equation(2). (2). Figure Figure 7. Illustration of the fuzzy relative direction membership function, Equation (2).

Asshown shownininFigure Figure8,8,path path(Θ)in inEquation Equation(2) (2)isisthe theminimum minimumpath pathbetween betweenthe thecenterlines centerlinesofof As (Θ) corresponding cones.For For example, fromfront front path (Θ) = 2. The visual fieldthe divided intoeight eight As shown incones. Figure 8, example, path (Θ) in from Equation (2) isleft, the minimum path between centerlines of corresponding totoleft, path visual field isisdivided into (Θ) = 2. The sectors, namely, “front, back, left, right, right front, right back, left front, and left back” or “north, corresponding cones. For back, example, from front left,right pathback, (Θ) = 2.left Thefront, visual field divided into eight sectors, namely, “front, left, right, right to front, and leftisback” or “north, south, south, west, east, northeast, northwest, southwest, and southeast”. sectors, namely, “front, back, left, right, right front, right back, left front, and left back” or “north, west, east, northeast, northwest, southwest, and southeast”. south, west, east, northeast, northwest, southwest,front(1) and southeast”. front(1) right-front(2) right-front(2) right(3) right(3)

back(5) back(5)

Figure 8. Illustration of path(Θ). The dashed lines are the centerlines of corresponding cones. Each centerline is assigned a number from 1 to 8 clockwise (e.g., front is assigned 1). Figure 8. Illustration of path (Θ). The dashed lines are the centerlines of corresponding cones. Each Figure 8. Illustration of path(Θ) . The dashed lines are the centerlines of corresponding cones. centerline is assigned a number from 1 to 8 clockwise (e.g., front is assigned 1). Each is assigned a number from into 1 to 8four clockwise (e.g., front is assigned 1). Thecenterline visual field can also be divided sectors, namely, “front, back, left, and right” or

“north, south, west, Then, the relative namely, direction“front, membership function changes The visual field canand alsoeast.” be divided intofuzzy four sectors, back, left, and right” or to The visual field alsoThen, be divided intorelative four sectors, namely, “front, back, left, and right” Equation (3): “north, south, west, andcan east.” the fuzzy direction membership function changes to or “north, south, west, and east”. Then, the fuzzy relative direction membership function changes to Equation (3): Equation (3):

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1       2  path (  ) -  reldir（）  4 1   π π4  a  + × path −Θ 2 ( Θ ) 4  µreldir (Θ) = 0π4 −a    

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  path    a () 2 π a    path (  )     − Θ 2 × path2 ≤a 4 (Θ) π  π a≤ 2 × path((Θ) )−Θ ≤ 4  path 2 4 π 2 × path(Θ) − Θ ≥ π4

(3) (3)

4. 4. Positioning Positioning Localities Localities Based Based on on Probability Probability Function Function The process of of obtaining obtaining the the location location region region is is introduced introduced in in this this section. section. A joint probability probability The process A joint function function is is proposed proposed to to describe describe the the probability probability distribution distribution of of locality locality description description in in the the region. region. Several definitions and a restriction are provided. Several definitions and a restriction are provided. 4.1. Location Region: Admissible Domain Definition 1. Fuzzy Fuzzy band: band: Uncertain Uncertain ring ring around around the the RO RO with with fuzzy fuzzy distance. distance. The fuzzy band usually Definition 1. comprises outer and inner rings with upper and lower distances, distances, as as shown shown in in Figure Figure 9. 9. Definition 2. Admissible domain: The fuzzy region in which the descriptive locality may be located; it is the Definition 2. intersection of the fuzzy bands of two or more ROs, ROs, as as shown shown in in Figure Figure 9. 9.

A1

A2

Figure 9. Definition Definition of of fuzzy fuzzy band band and and admissible admissible domain. domain. The fuzzy Figure 9. The blue blue bands bands correspond correspond to to the the fuzzy bands objects 1 and A2 Fuzzy_Band(A (e.g., Fuzzy_Band(A 1), Fuzzy_Band(A 2)); dashed the red dashed regions bands ofofobjects A1 A and A2 (e.g., ), Fuzzy_Band(A )); the red regions correspond 1 2 1,A2)). correspond to thedomain admissible (e.g., Admiss_Dom(A to the admissible (e.g.,domain Admiss_Dom(A 1 ,A2 )).

By convention, most locality descriptions with distances and directions contain at most three ROs By [9].convention, While describing localitydescriptions with one RO, relativeand direction relation is impossible to most locality withone distances directions contain at most three identify, so we don’t take this situation into consideration. Three fuzzy bands intersect in a unique ROs [9]. While describing locality with one RO, one relative direction relation is impossible to admissible the intersection of two fuzzy bands hasbands two admissible identify, so domain. we don’t However, take this situation into consideration. Three fuzzy intersect in domains, a unique which is unacceptable for positioning localities. A unique region is necessary to satisfy the admissible domain. However, the intersection of two fuzzy bands has two admissible domains, which requirement of positioning localities. is unacceptable for positioning localities. A unique region is necessary to satisfy the requirement of The process of obtaining a unique admissible domain is detailed hereafter. We assume that the positioning localities. sceneThe of locality description follows: “my front–right is A2, and my front–left 50 mthat is Athe 1.” process of obtainingisaasunique admissible domain50 is m detailed hereafter. We assume As shown in Figure 10, eight directions from front to left–front clockwise are assigned scene of locality description is as follows: “my front–right 50 m is A2 , and my front–left 50 m is A1 ”. corresponding numbers from 1 to 8. The the path betweenaretwo direction lines. The As shown in Figure 10, eight directions frompath(a) front to is left–front clockwise assigned corresponding admissible domains, intersected by the fuzzy bands of A 1 and A2, are Admiss_Dom(A1,A2)1 and numbers from 1 to 8. The path(a) is the path between two direction lines. The admissible domains, Admiss_Dom(A 1,A2)2. Line 8 and 2 connect A1 and A2 to the admissible domains respectively. The intersected by the fuzzy bands of A and A , are Admiss_Dom(A ,A )1 and Admiss_Dom(A ,A )2 . 1

2

1

2

1

2

Line 8 and 2 connect A1 and A2 to the admissible domains respectively. The unique admissible domain

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unique meet admissible domain should meet the requirement that the from to should the requirement that the direction from front–right (2)direction to front–left (8)front–right is clockwise(2)and 1 1 front–left (8) is clockwise and path(a) = 6, that is, Admiss_Dom(A1,A2) . path (a) = 6, that is, Admiss_Dom(A1 ,A2 ) . front(1) right-front(2)

Admiss_Dom(A1,A2)2 right(3)

left-front(8)

right-front(2)

A1

back(5)

A2

a left-front(8)

right-front(2)

Admiss_Dom(A1,A2)1

Figure 10. Illustration of the process of obtaining a unique admissible domain with two fuzzy bands: Admiss_Dom(A11,A22))11(the (thedirection directionof ofrotation rotationisismarked marked with with aa red red dashed dashed line). line).

4.2. Probability Distribution: Joint Function 4.2. Probability Distribution: Joint Probability Probability Function Definition 3. 3. Visible Visiblesegment: segment:The Thesegment segmentboundary boundary of of RO RO (Figure (Figure 11) 11) is is observed observed from from aa fuzzy fuzzy distance, Definition distance, which is consistent with spatial cognition. which is consistent with spatial cognition. When viewed viewedfrom froma afuzzy fuzzy distance, a segment anshould RO should be in the field visual field and When distance, a segment of anofRO be in the visual and possess possess the characteristic of visibility. From an algorithmic point of view, the visible segment the characteristic of visibility. From an algorithmic point of view, the visible segment reduces the reduces of thepoints number ofexplored points to[13]. be explored [13]. The to points belongsegment. to the visible segment. The number to be The points belong the visible The visible segment visible segment in Figure 11a contains theofentire objectdistance. A within a fuzzy given distance. in Figure 11a contains the entire boundary objectboundary A within aoffuzzy However, the However, given the restriction of visibility, the boundary within a fuzzy distance does not restriction of visibility, the boundary within a fuzzy distance does not completely belong to a visible completely belong11b,c. to a visible segment in Figure 11b,c. segment in Figure Certain restrictions (Figure 11) 11) that that are are consistent consistent with with cognition cognition should should be be proposed proposed when when Certain restrictions (Figure exploring points in the visible segment. exploring points in the visible segment. Restriction: The exceed aa concrete concrete angle angle based based on on different different cone-based cone-based Restriction: The angle angle of of sight sight should should not not exceed models, in which the the number number of cones could could be be 44 or or 88 [18]. [18]. Its value should based on on fuzzy fuzzy models, in which of cones Its value should be be set set based cognition and the Pareto principle that roughly 80% of effects originate from 20% of the cause. For cognition and the Pareto principle that roughly 80% of effects originate from 20% of the cause. example, in Figure 12, if the space is divided into eight cones, the angle of each cone is 45 degree. For example, in Figure 12, if the space is divided into eight cones, the angle of each cone is 45 degree. The occupation occupation angle angle of of the the red red line line in in its its cone cone should should be be about about 99 degree. degree. The

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A

bb

(a) (a)

B A

B

A

A A

Fuzzy distance Fuzzy distance

b b

Fuzzy distance Fuzzy distance

b

b

(b) (c) (b) (c) Figure 11. Definition of visible segment Visible_Seg(A) (red line). The red and blue lines form the Figure 11.11. Definition visible segment Visible_Seg(A) (red line). Thered red andblue blue lines form Figure Definition visible segment Visible_Seg(A) (red line). The and form boundary of A fromof aof fuzzy distance locality b. The blue line is the invisible segment, andlines the red linethethe boundary of A from a fuzzy distance locality b. The blue line is the invisible segment, and the red line boundary of A from a fuzzy distance locality b. The blue line is the invisible segment, and the red line is the visible segment. (a) The whole part; (b) due to invisibility itself; and (c) interrupted by RO B. is the visible segment. (a) The whole part; (b) due to invisibility itself; and (c) interrupted by RO B. is the visible segment. (a) The whole part; (b) due to invisibility itself; and (c) interrupted by RO B.

n n

A A

o Fuzzy distance o Fuzzy distance

m m k k

Figure 12. Restrictions of visible segment A; angle K should meet the restriction.

Figure Restrictionsofofvisible visible segment segment A; A; angle Figure 12.12.Restrictions angle K K should shouldmeet meetthe therestriction. restriction.

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A locality description generally contains two or three ROs with associated spatial relations. Refinement can be performed to handle cases in which more than one spatial predicate and RO are involved [5]. We regard an object as a set of points, namely, A = {a1 , a2 , . . . , an }. For a ∈ Visible_Seg(A), b ∈ Visible_Seg(B), and t ∈ T, we let dis(a,b) and dir(a,t,b) denote the distance and angle between two directions, respectively. A and B are ROs, and T represents the admissible domain. The Visible_Seg(A) and Visible_Seg(B) are the segment of A and B that meet the restriction. We use positioning localities with two ROs as an example. We assume that the unique admissible domain (Admiss_Dom(A,B) = T) has been obtained. The calculation of the locality probability distribution in the admissible domain is as follows: (1) (2)

We obtain Visible_Seg(A) and Visible_Seg(B) from locality t with upper fuzzy distance t ∈ Admiss_Dom(A,B). Refinement is performed to calculate distance probability Pdis (t) with two ROs. Pdis A (t) is the membership degree that maps the average dis(a,t) via the distance membership function Equation (1), that is, a ∈ Visible_Seg(A): Pdis(t) = PA dis(t) PB dis(t)

(3)

We calculate direction probability Pdir (t). PAB dir (t) is the membership degree that maps the average dir(a,t,b) via the relative direction membership function Equation (2), that is, a ∈ Visible_Seg(A) and b ∈ Visible_Seg(B): Pdir(t) = PAB dir(t)

(4)

(4)

(5)

Refinement is performed to calculate joint probability P(t) : P(t) = Pdir(t) Pdis(t)

(6)

Having two ROs is only slightly different from having three ROs. The process is as follows: (1) (2)

We obtain Visible_Seg(A), Visible_Seg(B), and Visible_Seg(C) from locality t with upper fuzzy distance t ∈ Admiss_Dom(A,B,C). We calculate distance probability Pdis (t) with Equation (1): Pdis(t) = PA dis(t) PB dis(t) PC dis(t)

(3)

We calculate direction probability Pdir (t) with Equation (2): Pdir(t) = PAB dir(t) PBC dir(t) PAC dir(t)

(4)

(7)

(8)

Joint probability P(t) is obtained with Equation (6):

This procedure introduces a calculation with more than one RO and spatial relation. For generality, we provide the PDF (Q(t) ) from Equation (6) on the basis of [5]: Q(t) =

Pdir(t) Pdis(t) ∑

i∈Admiss_Dom

Pdir(i) Pdis(i)

(9)

5. Case Study To illustrate the process of positioning localities with distance and direction, we conducted two groups of cognitive experiments on the basis of the scenes presented in Section 4.2. The cognitive experiments are conducted in the same shopping market mentioned in Section 3.1. The shopping

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market has about 45 m visual space. Before the cognitive experiments, we select two points arbitrarily and mark them as TO(A) and TO(B). During the cognitive experiments, the participants, standing at the marked points, are asked to look around and describe their positions with distances and directions (i.e., front, left–front, and back). To ensure reasonable spatial cognition, we select male and female Sensors 2017, 17, different 2828 11 of 17 participants with backgrounds, and their ages range from 20 to 60. The first step of positioning is to find the admissible domain. On the basis of the distance cognition and directions (i.e., front, left–front, and back). To ensure reasonable spatial cognition, we select experiment, we adopt a 98% confidence interval as the upper and lower bounds of fuzzy distance [24], male and female participants with different backgrounds, and their ages range from 20 to 60. and their 98% confidence intervals (i.e., 50 m) are (9.1, 12.2), and (49.5, 59.7), The first step of positioning is to 10, find30 theand admissible domain. On the(27.8, basis 38), of the distance respectively. the basis we of the cognition experiment, theasparameters (α,lower β, γ,bounds δ) of 15ofmfuzzy and 20 m cognitionOn experiment, adopt a 98% confidence interval the upper and distance(1) [24], their 98% intervals 10,are 30 and 50 m) are15 (9.1, andγ = 18, in Equation areand obtained byconfidence interpolation and(i.e., they as follows: m 12.2), (α = (27.8, 5.5, β38), = 13, (49.5, 59.7), the experiment, the parameters (α, β,of γ, parameter δ) of 15 m a in δ = 33.6) and 20 respectively. m (α = 7.4, On β =the 16,basis γ = of 25, δ cognition = 45.6). In these examples, the range and 20 m in Equation (1) are obtained by interpolation and they are as follows: 15 m (α = 5.5, β = 13, Equation (2) is [2, 5] multiplied by path(Θ) . γ = 18, δ = 33.6) and 20 m (α = 7.4, β = 16, γ = 25, δ = 45.6). In these examples, the range of parameter a An angle value should be determined to meet the restriction. Without additional contextual in Equation (2) is [2, 5] multiplied by path(Θ). information, we cannot determine which cone-based model the relationship “front” stands for [5]. An angle value should be determined to meet the restriction. Without additional contextual However, the relationship “left–front” represents themodel 8 cone-based model. Hence, forfor a direction information, we cannot determine which cone-based the relationship “front” stands [5]. relationship thatthelacks contextual information, we use 4 cone-based model. value that However, relationship “left–front” represents the the 8 cone-based model. Hence,The for angle a direction meets relationship the restriction be roughly 10◦ andwe 20◦use forthe the4 8cone-based and 4 cone-based respectively. that should lacks contextual information, model. Themodels, angle value that meets the restriction should be roughly 10° and 20° for the 8 and 4 cone-based models, respectively.

Example 1. Positioning with two ROs standing at point TO(A). As shown in Figure 13, the locality Example 1. Positioning with two ROs standing at point TO(A). As shown in Figure 13, the locality description is “front 20 m is TISSOT, and left 15 m is ZuoKY”. Figure 13b shows a local map that depicts description is “front 20 m is TISSOT, and left 15 m is ZuoKY.” Figure 13b shows a local map that depicts the the locality probability distribution in the admissible domain (dashed region). The TO(A) in Figure 13b is locality probability distribution in the admissible domain (dashed region). The TO(A) in Figure 13b is inside inside the the admissible admissibledomain, domain, because the deviation of distances and directions indescription locality description are small. because the deviation of distances and directions in locality are small. The The lower-middle theadmissible admissible domain dark colorthe reflects the most probable locality, lower-middle part part ofofthe domain with awith dark acolor reflects most probable locality, and its positionand its ◦ angle, which is consistent with spatial cognition. position relative to TISSOT and ZuoKY is nearly a 90which relative to TISSOT and ZuoKY is nearly at a 90°at angle, is consistent with spatial cognition.

(a)

(b) Figure 13. Positioning with two ROs at point TO(A) approximately 15.5 m away from TISSOT. The

Figurelocality 13. Positioning at point 15.5 m from TISSOT. description iswith “fronttwo 20 mROs is TISSOT, and TO(A) left 15 mapproximately is ZuoKY.” (a) Global; andaway (b) Local. The locality description is “front 20 m is TISSOT, and left 15 m is ZuoKY”. (a) Global; and (b) Local.

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Example with three ROsROs standing at point As shown Figure in 14,Figure the locality description Example2.2.Positioning Positioning with three standing at TO(B). point TO(B). Asinshown 14, the locality isdescription “front 20 is m“front is Watch, m is Playboy, left 30and m isleft ZuoKY”. Figure 14b indicates that the 20 mleft–front is Watch,30left–front 30 m isand Playboy, 30 m is ZuoKY.” Figure 14b indicates left admissible domain, domain, which has a dark hascolor, the most probable locality that meets spatial thatpart the of leftthe part of the admissible which hascolor, a dark has the most probable locality thatthe meets the relationship (i.e., distance and direction) from spatial cognition. The TO(B) in TO(B) Figure 14b is out of theisadmissible spatial relationship (i.e., distance and direction) from spatial cognition. The in Figure 14b out of the domain, because the because deviationthe of deviation distances of ordistances directionsorindirections locality description a bit large. admissible domain, in locality are description are a bit large.

(a)

(b) Figure 14. Positioning with three ROs at point TO(B) approximately 14.4 m away from Watch. The Figure 14. Positioning with three ROs at point TO(B) approximately 14.4 m away from Watch. locality description is “front 20 m is Watch, left–front 30 m is Playboy, and left 30 m is ZuoKY.” (a) The locality description is “front 20 m is Watch, left–front 30 m is Playboy, and left 30 m is ZuoKY”. Global; and (b) Local. (a) Global; and (b) Local.

To verify the positioning accuracy of the model, we conduct two groups of cognitive To verify(positioning the positioning accuracy ofthree the model, conduct two groups oflocality cognitive experiments experiments with two and ROs). we Tables 1 and 2 show the descriptions at (positioning with two and three ROs). Tables 1 and 2 show the locality descriptions at TO(A) and TO(B) TO(A) and TO(B) with two and three ROs, respectively. Positioning error is expressed as the with two and three ROs, respectively. Positioning expressed as maximum the distanceprobability of the maximum distance of the maximum probability point or error centeris point of the in the probability point of the maximum probability in the admissible region to the admissible point regionortocenter the known point TO(A) or TO(B). The positioning errors are shown in known Figures point TO(A) 15 and 16. or TO(B). The positioning errors are shown in Figures 15 and 16. Table 1. Locality description with two ROs. TO

Num

A

1 2 3 4

Name ZuoKY ZuoKY ZuoKY I DO

RO1 Distance 15 15 10 20

Direction front left front front

Name TISSOT TISSOT TISSOT ChaoHJ

RO2 Distance 15 15 15 25

Direction right front right left

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Table 1. Locality description with two ROs.

TO

Num

RO1

RO2

Name

Distance

Direction

Name

Distance

Direction

A

1 2 3 4 5 6 7 8 9 10

ZuoKY ZuoKY ZuoKY I DO ZuoKY Playboy I DO Playboy I DO ZuoKY

15 15 10 20 20 20 15 25 15 15

front left front front left right-front front front left-front front

TISSOT TISSOT TISSOT ChaoHJ TISSOT ZuoKY ChaoHJ ZuoKY TISSOT TISSOT

15 15 15 25 20 15 20 20 15 20

right front right left front front left left front right

B

11 12 13 14 15 16 17 18 19 20

Watch Watch Watch Watch Watch Watch Playboy Playboy ChaoHJ ChaoHJ

15 10 15 20 20 25 30 25 30 40

front front right-front front front front right-front right-front right right-front

Playboy Playboy Playboy Playboy ZuoKY ZuoKY ZuoKY ZuoKY I DO I DO

25 20 20 30 30 30 30 25 20 30

right-front right-front front right-front left left front front front front

Table 2. Locality description with three ROs.

TO

A

B

Num

RO1 Name

Distance

RO2 Direction

Name

Distance

RO3 Direction

Name

1

ZuoKY

15

left

TISSOT

15

front

2

ZuoKY

10

left

TISSOT

15

front

3

ZuoKY

15

front

TISSOT

15

left

4

ZuoKY

20

left

TISSOT

25

front

5 6 7 8 9 10

ZuoKY ZuoKY ZuoKY ZuoKY ZuoKY ZuoKY

15 15 20 15 20 15

left front left-front left-front left-front left-front

TISSOT TISSOT TISSOT TISSOT TISSOT TISSOT

20 15 20 15 20 15

front left right-front right-front right-front right-front

Hitomi Optician Hitomi Optician Hitomi Optician Hitomi Optician I DO I DO I DO Playboy Playboy Playboy

11 12 13 14 15 16 17 18 19 20

I DO I DO Watch Watch Watch Watch Watch Watch I DO Watch

30 25 20 15 20 20 10 15 30 15

front front front right right-front left front right-front right left

ChaoHJ ChaoHJ ZuoKY ZuoKY ZuoKY Playboy ZuoKY ZuoKY ChaoHJ ChaoHJ

30 30 30 25 30 30 25 30 35 25

left-front left-front left front left-front right-front left left-front left right-front

TISSOT TISSOT Playboy Playboy Playboy I DO Playboy Playboy TISSOT I DO

Distance

Direction

20

left-front

20

left-front

25

left-front

30

left-front

15 15 20 20 30 25

left-front left-front front front front front

20 15 30 25 30 30 25 30 20 25

right-front right-front left-front left-front front front left-front front front front

As shown in Figure 15, the maximum and minimum positioning errors without restriction are 8.2 and 0.76 m, respectively, and the mean positioning error is 4.43 m. The maximum and minimum positioning errors with restrictions are 7.1 and 0.73 m, respectively, and the mean positioning error is 3.48 m. As shown in Figure 16, the maximum and minimum positioning errors without restriction are 7.8 and 1.6 m, respectively, and the mean positioning error is 4.53 m. The maximum and minimum

12 Watch 10 front Playboy 20 right-front 17 Playboy 30 right-front ZuoKY 30 front 13 Watch 15 right-front Playboy 20 front 18 Playboy 25 right-front ZuoKY 25 front 14 Watch 20 front Playboy 30 right-front 19 ChaoHJ 30 right I DO 20 front 15 Watch 20 front ZuoKY 30 left 20 ChaoHJ 40 right-front I DO 30 front B 16 Watch 25 front ZuoKY 30 left Sensors 2017, 17, 2828 17 14 of 17 Playboy 30 right-front ZuoKY 30 front 18 Playboy 25 error right-front ZuoKY 25 front positioning 19 ChaoHJ 30 right I DO 20 front positioning10 errors with respectively, and positioning error is 20 restrictions ChaoHJare 6.1 and 40 1.3 m,right-front I DOthe mean30 front

3.26 m.

error(m)

error(m)

8 6

positioning error

10 4

No-Restriction

28

Restriction

06 4 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 num

No-Restriction Restriction

Figure 15. Positioning errors with two ROs: the dashed line indicates positioning errors with no 0 restriction, and positioning errors 1 the 2 3solid 4 5line6 indicates 7 8 9 10 11 12 13 14 15 16with 17 18restriction. 19 20

num As shown in Figure 15, the maximum and minimum positioning errors without restriction are 8.2 and 0.76 m, respectively, and the mean positioning error is 4.43 m. The maximum and minimum Figure errors withwith two two ROs: ROs: the dashed line indicates positioning errors with no restriction, Figure15. 15.Positioning Positioning errors the dashed line indicates positioning errors with no positioning errors with restrictions are 7.1 and 0.73 m, respectively, and the mean positioning error and the solidand linethe indicates positioning with restriction. restriction, solid line indicates errors positioning errors with restriction. is 3.48 m.

error（m） error（m）

As shown in Figure 15, the maximum and minimum positioning errors without restriction are positioning error 8.2 and 0.76 m, respectively, and the mean positioning error is 4.43 m. The maximum and minimum 9 positioning 8 errors with restrictions are 7.1 and 0.73 m, respectively, and the mean positioning error is 3.48 m. 7 6 positioning error 5 49 No-Restriction 38 Restriction 27 16 05 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 No-Restriction 3 num Restriction 2 116. Positioning errors with three ROs: the dashed line indicates positioning errors with no Figure Figure 16. Positioning errors with three ROs: the dashed line indicates positioning errors with no 0 restriction, solid line line indicates indicates positioning positioning errors errors with with restriction. restriction. restriction, and and the the solid 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 num Figures 15 and 16 indicate that the positioning errors are reduced by 0.95 and 1.27 m, respectively, 16. Positioning errors with three ROs: the is dashed line consistent indicates positioning errors but withcan no also whenFigure restrictions are considered. Angle restriction not only with cognition restriction, and the solid line indicates positioning errors with restriction. improve the positioning accuracy. In practice, the restriction can be adjusted or ignored according to ROs and space extent. Uncertainty is an inherent characteristic of spatial cognition [25]. Uncertainty in locality description originates from numerous sources, including external or internal factors, such as size, height, spatial distribution, task, and interest. As shown in Table 1 in numbers 6 and 8, different people possess different direction cognitions of the same scene. The distance descriptions in Table 1 (number 16) and Table 2 (number 9) are significantly large and cause considerable positioning errors. In a word, the more precise a locality description is with distances and directions, the more accurate the positioning is. Given the complex real environment and naïve cognition about distance and direction relations, the positioning accuracy exceeds 3.5 m, which is acceptable compared with that of complex and costly indoor positioning techniques [26] whose positioning accuracy is about 3–5 m when common smartphones are used.

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Context (e.g., spatial and semantic) is an important factor in locality description [27]. Positioning accuracy would improve if numerous contexts, such as locality description that is likely to occur on roads or the presence of infrastructures in the admissible domain, are available. 6. Conclusions and Future Work Positioning localities via locality description is a topic of next-generation GIS [8]. Resolution of positioning localities with place names and spatial relations facilitates the development of human-like geographic services that communicate with people intelligently about their everyday spatial needs [28]. To achieve positioning of localities indoors with cognitive distance and direction relationships semantically derived from locality description, we model relationships using the notions of admissible domain, visible segment, and restrictions. A joint probability function (i.e., distance and relative direction membership functions) is presented to describe the locality probability distribution in the admissible domain. The study demonstrates that the positioning accuracy exceeds 3.5 m within a 45 m visual space indoors. The contributions of this work are as follows: (1) (2) (3) (4)

The intersection of the rings around ROs is modeled to the region (i.e., admissible domain) for location description. Two regions are commonly found in two ROs; the unique region can be selected. A cognitive experiment based on distance is conducted to obtain the width of rings, and a distance membership function is constructed to describe how far a locality is from the RO. To access the degree-to-direction relationship of a locality relative to ROs, we develop a novel relative direction membership function that is consistent with human spatial intuition. A joint probability function based on distance and relative direction membership functions is provided to determine the position degree. For consistency with intuition, we provide the notion of visible segment and its restrictions.

The model proposed in this work is based on spatial cognition and visibility. The membership function for semi-quantitative distance is based on statistics and does not consider contextual data, such as task, personal reputation, background, interest, and hobbies, because these personal data are difficult to obtain. If these data are available, a membership function based on ordered logit regression can be conducted. Then, positioning accuracy would improve significantly. The positioning accuracy in this work is for 45 m of visual indoor space. The larger the visual space is, the lower the positioning accuracy is because a large cognitive distance corresponds to a large deviation from the correct distance. Similar to semi-quantitative distance, qualitative distance (i.e., “near”) is also frequently used in daily communication. Therefore, to make positioning localities with position description complete, a membership function based on qualitative distance will be used in our future work. Acknowledgments: This research is supported by the National Natural Science Foundation of China (Grant No. 41471323), the National Key Research and Development Program of China (Grant No. 2017YFB0503500), and the Specialized Research Fund of State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing of China. Authors would thank Huan Li for giving us profound guiding suggestions. Thanks also to Ting Zhang, Jia Zeng, Haixiang Wang, Yichao Guo and Ze Cheng for helping to collect the data for the cognition experiments. Author Contributions: Yankun Wang and Hong Fan conceived and designed the main idea and experiments; Yankun Wang performed the experiments; Yankun Wang wrote the paper. The author wishes to thank for Ruizhi Chen for supervision and mentoring support. All authors read and approved the final manuscript. Conflicts of Interest: The authors declare no conflict of interest.

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Krishnapuram, R.; Keller, J.M.; Ma, Y. Quantitative analysis of properties and spatial relations of fuzzy image regions. IEEE Trans. Fuzzy Syst. 1993, 1, 222–233. [CrossRef] Bloch, I.; Ralescu, A. Directional relative position between objects in image processing: A comparison between fuzzy approaches. Pattern Recognit. 2003, 36, 1563–1582. [CrossRef]

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