Fusing Noisy Fingerprints with Distance Bounds for Indoor Localization

Transcription

1 Fusing Noisy Fingerprints with Distance Bounds for Indoor Locaization Suining He 1 S.-H. Gary Chan 1 Lei Yu 2 Ning Liu 2 1 Department of CSE, The Hong Kong University of Science and Technoogy, Hong Kong, China 2 Schoo of Software, Sun Yat-sen University, Guangzhou, China Emai: 1 {sheaa, gchan}@cse.ust.hk, 2 {yuei@mai2, iuning2@mai}.sysu.edu.cn Abstract Fusing fingerprints with mutua distance information potentiay improves indoor ocaization accuracy. Such distance information may be spatia (e.g., via inter-node measurement or tempora (e.g., via dead reckoning. Previous approaches on distance fusion often require exact distance measurement, assume the knowedge of distance distribution, or appy narrowy to some specific sensing technoogy or scenario. Due to random signa fuctuation, wireess fingerprints are inherenty noisy and distance cannot be exacty measured. We hence propose Wi-Dist, a highy accurate indoor ocaization framework fusing noisy fingerprints with uncertain mutua distances (given by their bounds. Wi-Dist is a generic framework appicabe to a wide range of sensors (peer-assisted, INS, etc. and wireess fingerprints (Wi-Fi, RFID, CSI, etc.. It achieves ow errors by a convex-optimization formuation which jointy considers distance bounds and ony the first two moments of measured fingerprint signas. We impement Wi-Dist, and conduct extensive simuation and experimenta studies based on Wi-Fi in our internationa airport and university campus. Our resuts show that Wi-Dist achieves significanty better accuracy than other state-of-the-art schemes (often by more than 4%. Keywords Indoor ocaization, convex optimization, fusion, noisy fingerprint, distance bounds, measurement uncertainty. I. INTRODUCTION Indoor ocation-based service has attracted much attention in recent years due to its commercia potentia. The quaity of such service argey depends on the ocaization accuracy of users. Among a the current indoor ocaization techniques, fingerprint-based approach emerges as a promising one. Fingerprint-based indoor ocaization is usuay conducted in two phases. In the offine (survey phase, a site survey is conducted to coect the vectors of received signa strength indicator (RSSI at reference points (RPs of known ocations. The vectors of these RSSIs form the fingerprints of the site and are stored at a database. In the onine (query phase, a user (or a target sampes an RSSI vector at his position and reports it to the server. In traditiona fingerprinting, the server then compares the received vector with the stored fingerprints using some simiarity metric in the signa space (ike Eucidean distance in [1]. It then estimates the target position out of the RPs whose fingerprints cosey match the target s RSSI (termed the neighbors. This work was supported, in part, by Hong Kong Research Grant Counci (RGC Genera Research Fund (13, HKUST (FSGRF13EG1, and Nationa Natura Science Foundation of China (1424. Error in ocation estimation is inevitabe. This is due to random signa fuctuation in both offine and onine measurements. As targets are often considered independenty in the above traditiona approach, such measurement noise or uncertainty may ead to a disperse set of spatiay distant neighbors, which greaty degrades the ocaization accuracy. It has been observed that ocaization errors can be very high (more than m [2] under arge open indoor environment such as mas, train stations or airports. This is unsatisfactory for many appications. In order to reduce the estimation errors, one may incorporate, or fuse, wireess fingerprinting with mutua distance information. Embedding such information into the fingerprints can significanty reduce the dispersion, eading to substantia enhancement in ocaization accuracy. The distance information can be spatia, where the target estimates the distances to some of the nodes or beaconing devices in its neighborhood using, for exampes, Buetooth, Wi-Fi direct, utrasound, etc. Whie there have been impressive works on using spatia distance for ocaization, they often assume accurate distance measurement, resuting in rigid constraints over the fingerprints. The rigidity cannot be extended to the more reaistic scenarios when distance measurement is often uncertain with given upper and ower bounds. The distance information can aso be tempora, where the target estimates its dispacement over consecutive time instants (e.g., by step counter or inertia navigation system (INS provided in one s mobie phone. Previous fusion works in the area are often based on Bayesian approach, assuming some probabiity distribution in sensor measurement. In reaity, such probabiity distribution is often not known. Furthermore, these works cannot be easiy extended to the case of noisy sensor measurement over mutipe periods of time. Due to random signa fuctuations, wireess fingerprint is inherenty noisy and distance cannot be exacty measured. In this paper, we propose Wi-Dist, a nove indoor ocaization approach fusing noisy wireess fingerprints with uncertain mutua distances given by their bounds. Wi-Dist jointy considers distance bounds and noisy fingerprints to reduce indoor ocaization errors. It requires ony the first two moments (mean and variance of the fingerprint RSSI signas, and optimizes ocations based on Semi-Definite Programming (SDP. Using SDP reaxation, Wi-Dist soves the ocaization probem achieving exceent accuracy. Because Wi-Dist takes as input ony the upper and ower /1/$ IEEE 2

2 bounds of distances, it is not based on rigidity constraints or Bayesian approach, and hence does not require accurate distance measurement or knowedge of probabiity distribution. Using the bounds as constraints, Wi-Dist estimates the ocation by maximizing the overa simiarity with the fingerprint map consisting of random signas. It is appicabe to scenarios where distances may be asymmetric, in which case the upper (ower bounds can be obtained by using the arger (smaer vaue of the two directions. Wi-Dist is a generic framework appicabe to a wide range of sensing techniques, enabing indoor ocaization with adaptive spatia and tempora mobie sensing independent of how distance is measured. For exampe, it may empoy peer-to-peer spatia distances in the crowded region. For an unpopuated area, it may then switch to dead reckoning (INS for distance measurement. Though most of our discussion is in the context of Wi-Fi fingerprints (due to its ease of depoyment without extra infrastructure beyond the existing Wi-Fi one, Wi-Dist is genera enough to be extended to other wireess fingerprint signas such as RFID [3], [4] or channe state information (CSI []. We show in Figure 1 the overa architecture of Wi- Dist based on Wi-Fi fingerprints. The fingerprint database is initiaized by a site survey, storing pairs <ocation, RSSI vector> of RPs. In addition to the Wi-Fi RSSI vectors, a target measures the distance bounds and reports them to the ocaization server. Based on that and a difference metric for random Wi-Fi signas, the server constructs an SDP convexoptimization probem which eads to accurate ocaization of the target. We have impemented Wi-Dist based on Wi-Fi fingerprints on Android patforms and performed arge-scae simuation and testbed experiments in Hong Kong Internationa Airport (HKIA and our university campus. Using commony used indoor sensors of dead reckoning and peer-assisted measurement, we demonstrate that Wi-Dist achieves much higher accuracy than other approaches. The rest of this paper is organized as foows. After reviewing reated work in Section II, we present the ocaization probem of Wi-Dist in Section III, and SDP-based ocaization formuation in Section IV. Iustrative resuts based on experimenta trias and simuation are presented in Sections V and VI, respectivey. We concude in Section VII. Fig. 1: System framework of Wi-Dist based on Wi-Fi fingerprints. II. RELATED WORK Wi-Fi fingerprinting techniques, pioneered by Radar [], have been widey studied in recent years. The work by Horus [] estimates the target ocation using a probabiistic mode which refects the signa distribution in the site. Expectationmaximization [], compressive sensing [] and signa geometric patterns [] have been impemented for fingerprint-based indoor ocaization. The techniques above soey address Wi-Fi fingerprint issues. We study here fusing distance information with fingerprinting to achieve much better accuracy. Combining dead reckoning with fingerprints has been discussed in [1], [11], [12], [13]. These works assume that the waking path is conditionay independent. Therefore, the distance constraints over mutipe tempora Wi-Fi estimations have not been jointy considered. Furthermore, these works treat the outputs from dead reckoning and Wi-Fi fingerprint sequentiay [12]. Wi-Dist, on the other hand, formuates the ocaization probem as a singe joint convex-optimization probem. This greaty reduces the infuence of measurement noise and achieves higher accuracy. The work on Wi-Fi SLAM [13] impements robot odometer to determine the distance accuratey. Our work differs by considering step counter whose measurement may be noisy. There has been much work making use of Wi-Fi direct [14] and high-pitch sound [1], [1] to measure distance between devices. Some consider using a rigid graph constructed through rotation and transation [1], whie others consider using Bayesian approach to infer the device ocation [1]. It is found that higher accuracy can be achieved when each user is 3- connected [1], or the graph satisfies some specia requirements such as being pairwise connected [1]. In contrast, Wi- Dist considers jointy distance bounds and fingerprint noise, and formuates an optimization probem to estimate the target ocation. Therefore, Wi-Dist is a more versatie and reaistic framework accommodating measurement noises. In contrast to a the sensor fusion works above, Wi-Dist is a seamess generic framework appicabe to different sensor systems with tempora or spatia distance measurement. It may be extended to different appication scenarios with itte modification. Wi-Dist is an optimization-based approach for indoor ocaization, by jointy considering both Wi-Fi signa measurement noise and distance bounds. III. PROBLEM FORMULATION AND COMPLEXITY In this section, we show how Wi-Dist estimates user ocations by means of convex optimization. For concreteness, our discussion is in the context of Wi-Fi fingerprint signa (the extension to other signas is cear and straightforward. We first present the preiminaries of Wi-Dist in Section III-A. Then in Section III-B we present the objective function which is based on a nove difference metric for random Wi-Fi signas, and the probem formuation for ocation optimization. In Section III-C, we discuss the hardness of the probem. We show in Tabe I the symbos used in our probem formuation. 2

3 TABLE I: Major symbos in the probem formuation. Notations M Q x m X V Y I 2 r q R ω mq W Λ m Ω m L Ψ q ψ q σq Φ m φ m Δ (Φ m, Ψ q Γ(Φ m, Ψ q δ mn δ mn ˆδ mn Definitions Number of spatia or tempora targets Number of RPs in fingerprint database Estimated 2-D coordinate of target m M 2 matrix of a target ocations Index set of a targets to be ocated M M matrix for transformation in SDP 2 2 identity matrix 2-D coordinate of reference point (RP q 2 Q matrix of RPs Weight of RP q to estimate target m M Q matrix of weights at RPs Index set of targets to be estimated in V with distance measurements from m Set of distance bounds from m Number of Wi-Fi APs RSSI vector received at RP q Average RSSI of AP at q (dbm RSSI standard deviation of AP at q (db RSSI vector received at m RSSI of AP at m (dbm Expected signa difference of RSSI from AP between target m and RP q Overa expected signa difference between target m and RP q Distance between target m and n (m Lower distance bound between m and n (m Upper distance bound between m and n (m A. Preiminaries In the offine mode, a site survey is conducted with a tota of Q reference points (RPs. Let r q be the 2-D position of RP q, and R be a 2 Q matrix indicating the RP positions, i.e., R =[r 1, r 2,...,r Q ]. (1 Let L be the index set of the Wi-Fi access points (APs that cover the site, i.e., L = {1,...,L}. At each RP, time sampes of Wi-Fi RSSI readings are coected. Due to the random nature of radio signa, mutipe sampes are coected in order to reduce the uncertainty in the signa measurements. Denote the RSSI at RP q from AP at time t as {ψq(t,t= 1,...,S,S > 1}, with S being the tota number of sampes coected. Denote the average RSS readings from AP, L, at RP q as ψ q, and the unbiased estimate of the variance of the RSS time sampes for AP at RP q as ( 2. σq Then for each RP, the unbiased estimates for the mean RSSI and its corresponding standard deviation at RP q are computed as: ( ψ q = 1 S ψ S q(t, t=1 ( σq = 1 S (2 ( ψ S 1 q (t ψ q 2. t=1 Then the Wi-Fi RSSI vector at r q is Ψ q = [ ψ1 q, ψ q,..., 2 ψ q L ],q {1, 2,...,Q}, (3 where, by definition, ψ q =if AP is not detected at RP q. In the onine mode, et V be the set consisting of the indexes of the M target nodes (or simpy targets to be ocaized in the site, i.e., V = {1, 2,...,M}, and each of their 2-D ocations to be estimated is denoted as x m,m V. Note that these targets to be estimated can be either spatia or tempora. Let X be an M 2 matrix of a these points, i.e., X =[ x 1, x 2,..., x M ] T. (4 For each of the targets (spatia or tempora, et φ m be the RSSI vaue at target ocation x m for Wi-Fi AP, L. Simiar to the RP RSSI vector, we define the target m s samped RSSI vector as Φ m = [ φ 1 m,φ 2 m,...,φ L m],m V. ( where, by definition, φ m =if AP is not detected at target m. Given a target m, et Λ m be the set of its neighbors that have distance measurement with. Let δ mn be the distance (spatia or tempora between targets x m and x n for any m, n V, i.e., x m x n 2 = δmn, 2 n Λ m,n m. ( Based on statistica anaysis of distance (spatia or tempora, we can obtain a distance bound with high confidence eve. Given the distance measurement interva, we denote δ mn as the ower bound of the measurement and ˆδ mn as the upper bound. For each Wi-Fi target, Ω m stores the distance bounds, i.e., Ω m = {[δ mn, ˆδ mn ]}, n Λ m. Given the M targets to be ocaized in V, each of them contains the foowing information in the Wi-Dist probem: Π m {m, x m, Φ m, Λ m, Ω m },m V. ( B. Probem Formuation The RP positions R are used to estimate the ocation of the target. Let ω mq be the weight assigned to RP q to ocate m, so that Q x m = ω mq r q,m V, ( q=1 where the weights ω mq, m, satisfy Q ω mq =1,ω mq, q {1, 2,...,Q}. ( q=1 Let W be an M Q matrix of ω mq, r q R, i.e., ω ω 1Q W = ( ω M1... ω MQ Then the positions of a the targets in V given W are given by X = WR T, (11 The distance between two targets in Equation ( satisfies bound constraints, i.e., δ mn δ mn ˆδ mn, or equivaenty, δ 2 mn δ2 mn ˆδ 2 mn, [δ mn, ˆδ mn ] Ω m, m V. (12 2

4 Given the above, we present in the foowing the objective function for Wi-Dist probem. We first introduce a metric to evauate the difference between the target Wi-Fi sampes and the stored fingerprints under measurement noise. (Device heterogeneity in RSSI of onine and offine measurement is outside the scope of this paper; interested readers are referred to works ike [1] for more information on how to address it. We consider fingerprint noise at each RP. Define J mq as the shared APs between Wi-Fi measurement point m and RP q ( < J mq L. Given a target s Wi-Fi RSSI φ m (constant from AP J mq, the expected signa difference between RP q and the target m s RSSI in AP is derived as []: ( (φ Δ (Φ m, Ψ q E m ψq 2 (φ 2 =E( m 2φ m ψq + ( ψq 2 = ( φ 2 m 2φ m E ( ψq ( (ψ +E 2 q = ( φ 2 m 2φ m E ( ψq +E 2 ( ψq ( + σ 2 q = ( φ m ψ q 2 + ( σ q 2. (13 By definition, if either φ m = or ψ q = (or both, Δ (Φ m, Ψ q =. Thus the tota expected signa difference between the RP q and the target m s RSSI vector is given by Γ(Φ m, Ψ q 1 J mq Δ (Φ m, Ψ q. (14 J mq If J mq =(no shared APs between the target m and RP q, we have by definition Γ(Φ m, Ψ q =, i.e., RP q is essentiay excuded from the ater optimization formuation. Using Equation (14, we present in the foowing the objective function for Wi-Dist. To jointy measure the overa difference of a targets with the stored signa map, we find the weights which minimize a the targets weighted sum of expected signa difference as: M Q arg min Γ(Φ m, Ψ q ω mq, (1 W m=1 q=1 which jointy considers the signa difference and the physica distance constraints. To summarize, we are to find a matrix W so as to satisfy Objective: Equation (1, =1 subject to: Constraints (, (, (11 and (12. (1 C. Probem Hardness In this section we generaize the probem given by Formuation (1 and study its hardness. (In the next section we wi appy reaxation to sove it by semi-definite programming. Formuation (1 can be generaized into the foowing probem: Definition 1. Given the targets with RSSI measurements and the distance information between them, is there a set of ocations in the fingerprint database such that the tota difference between the measured RSSI vectors and the stored signa map is minimized whie their reative ocations satisfy the measured distance bounds? To prove the hardness in Definition 1, we introduce the subset sum probem (SSP, which is stated as foows: Definition 2. Given a set A of integer numbers and an integer number a, does there exist a subset of A such that the sum of its eements is equa to a? In reaity, soving the probem in Formuation (1 is chaenging. Here we are to prove that it is computationay hard by reduction from subset sum probem (SSP. Theorem 1. There is no efficient agorithm that soves the probem given by Definition 1 uness P = NP. Proof: We briefy describe our proof as foows. Suppose we have a poynomia-time agorithm that takes as input the distances between different targets as we as fingerprint measurement points to recover their origina positions. Therefore, we minimize the overa sum of target signa difference with the fingerprint map to obtain the candidate ocations, and then find those among them which satisfy the corresponding distance bounds. Then such an agorithm can be used to sove the SSP by appying it to an instance of the probem. After reaching its poynomia time bound, the agorithm wi either have returned a soution or not. In the first case, we can check if the soution with pairwise distances returned is consistent with the distance bounds. It is ike that in the SSP we check the sum of eements in poynomia time and accept if and ony if the check succeeds. In the second case, we can reject the instance. For both cases, we have returned the correct answer for SSP. Since SSP is aready NP-hard, our probem is as hard as the SSP. Thus, the probem in Definition 1 is NP-hard. IV. WI-DIST: SDP-BASED LOCATION OPTIMIZATION As the Wi-Dist probem is NP-hard, we use Semi-Definite Programming (SDP reaxation [1] to sove it. SDP has been appied in wireess communication [2] and sensor networks [21]. In this work, we impement SDP to fuse the noisy fingerprints and distance bounds for indoor ocaization. Given the distance bounds, we appy semi-definite reaxation [21] to reax the distance constraints. Let e mn be an M 1 coumn vector where the m-th eement is 1 and n-th eement is 1. The physica distance between node m and n can be therefore represented as δmn 2 = e T mnxx T e mn, [δ mn, ˆδ mn ] Ω m. (1 Denote an M M matrix Y for interna transformation, i.e., Y = XX T. (1 Finay the distance bound can be rewritten as δ 2 mn e T mnye mn ˆδ mn, 2 [δ mn, ˆδ mn ] Ω m. (1 Based on the distance bound constraints in Equation (1, we can rewrite Formuation (1 into Objective: Equation (1, (2 subject to: Constraints (, (11, (1 and (1. 2

5 Ceary, Constraint (1 is nonconvex due to δ 2 mn e T mnye mn. Given a symmetric matrix A, et A represent that A is a positive semidefinite matrix [1]. We can then reax this probem into a convex one by repacing the nonconvex equaity constraint, Y XX T =in Constraint (1, with a convex positive semi-definite constraint, i.e., Y XX T. (21 Constraint (21 is a noninear constraint, which can be further transformed into a inear matrix inequaity [1]. Then it can be soved efficienty by a convex optimization sover. The transformation is through a Schur compement: Definition 3. Let H be a matrix partitioned in four bocks, consisting of four matrices B,E,C and D, i.e., [ ] B E H =, (22 C D where B and D are symmetric and nonsinguar. The Schur compement of D in H, is defined as S = B ED 1 C. (23 If S, then H [1]. Thus, by using Schur compement, we can rewrite Constraint (21 as a matrix form, i.e., [ ] Y X X T. (24 I 2 Then Formuation (2 is finay transformed into an SDP probem [1]: Objective: Equation (1, (2 subject to Constraints (, (11, (1, (1 and (24. The formuation above can be directy appied in peerassisted ocaization. For dead-reckoning based ocaization, we may take m as the time stamp. We end by anayzing the computationa compexity of the soution. Given Q RPs and L APs, the compexity of signa difference cacuation is O(QL. Given M tempora or spatia target measurements (usuay M is sma, the computation of SDP reaxation is bounded by O(M 3 Q 3 [22]. Using some commercia SDP sover this probem can be soved efficienty [1]. Further computationa reduction can be achieved by AP fitering and RP custer mapping []. V. EXPERIMENTAL EVALUATION We have deveoped Wi-Dist based on Wi-Fi fingerprints in Android patforms and conducted experiments to study its performance. In this section, we first discuss the experimenta settings and performance metrics in Section V-A. Then we present iustrative experimenta resuts for dead reckoning and peer-assisted ocaization in Sections V-B and V-C, respectivey. A. Experimenta Settings and Performance Metrics We evauate Wi-Dist in the Hong Kong Internationa Airport (HKIA boarding area and HKUST campus atrium. In the airport we coect overa 1, 4 RPs in, m 2 area. On the campus we coect 34 RPs in, m 2 area. Figure 2(a and Figure 2(b show their corresponding foor pans. In the airport and the campus, we take overa Wi-Fi sampes at each RP using HTC One X+. A quarter of these sampes are coected when we are facing north, south, west and east respectivey. For a the appication scenarios, we use the foowing parameters as baseine: m survey grid size; 1 Wi-Fi RSSI sampe is used for each target; no Wi-Fi AP reduction is conducted over target RSSI vectors. We compare Wi-Dist with the foowing typica ocaization schemes in our experiments: Fingerprint-based ocaization (, the cassica agorithm such as [], [2], which evauates the Eucidean distance of target RSSI vector with each RP fingerprint and finds the interpoation of top k nearest neighbors for ocation estimation (k =1in our experiment. Sequentia Monte Caro ( ocaization, a typica fusion agorithm [12] based on Sequentia Monte Caro method (partice fiter which fuses INS data and Wi-Fi fingerprinting (. Through the propagation aong the tempora waking path, the partices transate from one ocation to the next. With map constraints, the spatia distribution of these partices gets corrected and resamped [11]. The fina estimation is based on the weighted average of partice ocations. Graph-based and fingerprint ocaization scheme (GB +, which uses graph construction and Wi-Fi fingerprinting for peer-assisted ocaization. With the pairwise spatia distances of peer targets, the server constructs the rigid graph consisting of a targets [14], [1], [23]. Then the system searches against the Wi-Fi signa map and finds a set of fingerprints to minimize the objective function Σ M m=1 Φ m Ψ q 2 through rotation and transation [1]. Let x m be target m s true ocation and x m be the estimated ocation. The performance metric in our experiment is the mean error (unit:m of the estimated target in set V: μ e = 1 x m x m. (2 V m V B. Dead Reckoning In this section we study how Wi-Dist performs for a mobie user with INS (step counter in his smartphone. Tempora waking distance of the target is estimated from the INS sensor which counts the steps of a waking target. Each step detection is based on the periodic changes in the vertica direction of the acceerometer readings [11]. Based on the number of steps, the distance traveed, or motion offset, can be estimated by mutipying the average stride ength of the target (which is reated to waking frequency as in [24]. As the target waks, the device aso coects the Wi-Fi RSSI vectors. Using the notation in Equation ( the index m 2

6 Fig. 2: (a Map of the HKIA boarding area. (b Map of the HKUST campus atrium. Bue points are the RPs ( m grid size. here represents the time stamp. Each target ocation x m now corresponds to a tempora measurement of a singe target. The most recent M tempora targets and M 1 distances between them form a siding window in time domain, and the estimation of the M-th target is returned as the current position. With the fast Wi-Fi scanning on smartphones [1], the sma curvature between two consecutive Wi-Fi sampes can be approximated as distance. Let β be the range of confidence interva for estimating the dispacement and σ mn be the statistica standard deviation based on experimenta resuts. Given a distance measurement δ mn at time m from the ast ocation with RSSI measurement (Λ m =[m 1], each of the distance bounds in Ω m is defined as δ mn βσ mn δ mn δ mn +βσ mn,n= m 1,m>1. (2 Based on Equation (, a Wi-Fi tempora target Π m at time m is defined as Π m {m, x m, Φ m, [m 1], Ω m }. (2 The distance bound for initia or the first target in the siding window is defined to be nu. Based on the empirica test, β is set to 2 in Equation (2. The size of siding window M is. 2 partices are used in the partice fiter of agorithm. Figure 3 pots the ocaization accuracy with respect to time for Wi-Dist and. The estimation error fuctuates as the user waks in the airport. Changes in wa partitions, crowded peope, user waking direction and smartphone hoding gesture introduce measurement noise in Wi-Fi and INS signas. sequentiay considers the fingerprints and INS measurements. It does not jointy consider the Wi-Fi fingerprints and the distances from the mutipe time periods. Therefore, arge error in ocation estimation happens. In contrast, Wi-Dist constrains its estimations through the distance bounds in a joint optimization formuation. Therefore, Wi-Dist can achieve ower ocaization errors and smaer estimation fuctuation. Figure 4 shows the mean dispacement measurement and corresponding standard deviation at each true waking distance. In the empirica studies, the major error of dispacement comes from misestimation in step counts and step ength. Meanwhie the device initiaization and waking curvature aso eads to additiona dispacement errors [1], [24]. Based on such empirica anaysis, we obtain the dispacement variance for each measured distance, which constitutes the distance bounds (Equation (2 in Wi-Dist tempora measurement. Figure shows the mean ocaization errors against the number of Wi-Fi tempora target measurements. We can see that the accuracy improves as we utiize more tempora sampes. It is because joint consideration of more periods further constrains the ocation estimations. When we further increase the number of measurements, the accuracy graduay converges, indicating that distance bounds aready provide sufficient constraints. Thus, to baance between ocaization accuracy and computationa compexity we choose severa tempora measurements (ike in our experiment in Wi-Dist. C. Peer-Assisted Locaization For some areas visited by many users, peer-assisted (PA ocaization may be used [1]. Peer ranging can be based on either RSS-distance mapping or sound ranging. In the experiment, we impement and test sound ranging under quiet and noisy campus environment. The mean peer ranging errors under these two conditions are. m and 2 m respectivey. Since distance constraint between two peers is asymmetric due to measurement uncertainty, we use the arger vaue in the distance measurements as the upper bound ˆδ mn and the smaer one as the ower bound δ mn. In the peer-assisted ocaization, targets are invoved in sound-based distance measurement. We do not excude the cases when was may partition peers during ocaization. Figure shows the mean ocaization errors against the proportion of APs removed at targets. We randomy remove some received APs of each target to evauate the infuence of AP reduction due to wa partitioning or crowds of peope. We can see that Wi-Dist and GB+ marginay rey on the number of received APs. It is because the mutipe users Wi- Fi sampes reduce the effect of sparse AP depoyment. To the contrary, reies on the APs to differentiate the RPs and therefore its estimation error increases as more APs are pruned. Figure shows that the ocation estimation errors against the number of Wi-Fi sampes at each target in PA ocaization. A the agorithms improve with more Wi-Fi sampes. It is because as the number of Wi-Fi sampes increases, noise from the random samping can be reduced []. Compared with GB+ and, Wi-Dist achieves higher ocaization accuracy because it further jointy considers the measurement noise in the optimization formuation and reduces the uncertainty. However, increasing the number of Wi-Fi sampes means that we need to wait for more sampes before fina estimation. A baance has to be made between accuracy and atency depending on appication scenarios. Figure shows the ocation errors against the survey grid size. As the minimum grid size is five meters, ines or rows of RPs are removed to form grid size with mutipes of 211

7 Locaization Error (m Time (sec Fig. 3: Locaization errors over time. Mean Measurement (m Step Counter Measurement True Dispacement (m Fig. 4: Waking distance measurements using step counter Number of Tempora Targets Fig. : Locaization errors vs. number of Wi-Fi tempora targets GB Reduced Ratio of Received APs Fig. : Locaization errors vs. removed proportion of received APs. GB Number of Sampes Fig. : Locaization errors vs. number of Wi-Fi sampes GB+ 1 2 Survey Grid Size (m Fig. : Locaization errors vs. survey grid size. five. Ceary, as the grid size increases, the accuracy of the three agorithms decreases. Though ess abor-intensive, arger survey grid size may more easiy ead to dispersed nearest neighbors under arge signa noise. Therefore, traditiona agorithms ike may not accuratey differentiate these RPs. Under different survey density, Wi-Dist and GB+ achieve more accurate ocation estimations with the constraints of peer-to-peer distances. However, the rigid graphs of targets in GB+ sti suffers from pairwise distance measurement noise. By fusing signa uncertainty and distance bounds, Wi- Dist achieves higher estimation accuracy under different grid sizes. Figure and Figure show the overa performance of Wi-Dist at different scenarios (INS and PA at baseine parameters in HKIA. Large indoor open space often eads to high uncertainty in Wi-Fi signas [2] and disperse nearest neighbors in signa space. Furthermore, the tempora and spatia distance measurement aso contains arge noise under the crowded scenarios. Compared with other state-of-the-art agorithms, Wi-Dist significanty reduces the estimation errors in HKIA. With distance constraints and joint optimization, Wi- Dist mitigates the effect of disperse nearest neighbors. Compared with the airport, the campus atrium is smaer with more buiding partitions, which may infuence the peerdistance measurement accuracy. We show the performance of Wi-Dist (INS and PA on HKUST campus in Figure 11 and Figure 12 respectivey. Wi-Dist achieves higher ocaization accuracy than the other state-of-the-art agorithms. As the resuts in HKUST are quaitativey simiar to those in HKIA, for brevity we do not repeat other experimenta resuts here. VI. ILLUSTRATIVE SIMULATION RESULTS To evauate more comprehensivey the performance of Wi- Dist in arge-scae indoor environment with many users, we have simuated Wi-Dist for the scenarios as mentioned in Section V. In this section, we first discuss the simuation setup (Section VI-A, foowed by the resuts for dead reckoning and peer-assisted ocaization (Sections VI-B and VI-C. A. Simuation Setup We simuate the Wi-Fi signa strength foowing the work in [2]. In the signa mode, the RSSI Φ (dbm from Wi-Fi AP at a distance D can be simuated as Φ=Φ TX L α og ( D D + ɛ, (2 where measurement noise is distributed as ɛ N(,σ 2 db. Uness otherwise stated, we use the foowing as our baseine parameters: the transmission power Φ TX =2dBm, the path oss exponent α =4., reference path oss L =3. db, reference distance D =1m, 2 m m survey site with m grid size; Wi-Fi signa noise σ db =db; APs are uniformy distributed in the survey area; a target takes a Wi-Fi sampe every 3 seconds. B. Dead Reckoning For dead reckoning, we use a random way-point mobiity mode with resting [2], and most recent Wi-Fi records are used for INS fusion. The step count error rate is distributed as N (,σr, 2 where σ r = 2%, and the stride ength error foows N (,σ 2, where σ =.2 m. Additiona waking 212

8 dispacement error is assumed to foow N (,σw, 2 where σ w =2m. Figure 13 shows the ocaization accuracy against the step count errors. Ceary, the performance of and Wi-Dist degrades with arger step count error. ocates the user based on the partice fiter, which sequentiay considers the Wi-Fi and INS measurements. Therefore, when step count accuracy degrades, the dispacement error increases and the partices become spatiay sparse, making it difficut for to converge to correct ocations. To the contrary, Wi-Dist ocaizes the target more accuratey because the joint consideration of Wi-Fi fingerprints and distance bounds of mutipe periods reduces the infuence of measurement uncertainty. Figure 14 shows the ocaization accuracy against the waking dispacement errors. We can see that as the dispacement error increases, the overa ocation accuracy decreases. Locaization error in increases because the partices converge sowy given arge distance errors and noisy Wi- Fi measurement. Different from, Wi-Dist achieves more accurate resuts because it utiizes the distance bounds instead of actua distance measurement. By constraining the target estimation within the intersection of these bounds, Wi-Dist is more robust to distance uncertainty. Figure 1 pots the ocaization errors versus the signa noise in Wi-Fi measurement (Equation (2. We can observe that the performance of both and Wi-Dist degrades when the random signa noise increases. It is because arger signa noise makes it more difficut to differentiate the fingerprints. Different from, Wi-Dist considers signa uncertainty through the expected signa difference. By minimizing the signa difference within constraints of distance bounds, Wi- Dist reduces the effect of disperse nearest neighbors and obtains better estimation resuts. C. Peer-Assisted Locaization We assume noisy peer-assisted distance error ɛ m N (,σm, 2 σ m = 3. m; neighborhood detection range is 1 m; four peers together initiate a peer-assisted ocaization; users are randomy distributed in the survey site. Figure 1 shows the ocaization errors versus the number of users. Ceary, more peer assistance provides more distance constraints over the invoved users and improves the ocaization accuracy. Different from GB+, Wi-Dist shows ess dependency on user connectivity. It is because Wi-Dist considers the measurement uncertainty in the optimization and jointy constrains a the users. Therefore, it does not have to invove many users to achieve high ocaization accuracy. Figure 1 shows the ocaization accuracy against the peer distance errors. We assume a Gaussian noise is added to the inter-device distance measurement. When the peer distance error is sma, both agorithms achieve high accuracy given ony Wi-Fi measurement noise in fingerprints. As distance error further increases, both agorithms degrade in ocaization accuracy. GB+ constructs a rigid graph to constrain reative positions of different users. However, the graph shape deforms under arge distance measurement errors. Wi-Dist, in contrast, shows more robustness by using joint optimization based on fingerprints and distance bounds. Without assuming a rigid graph, Wi-Dist can achieve more robust ocaization estimation. VII. CONCLUSION In this paper, we have proposed Wi-Dist, a nove and convex-optimization framework fusing wireess fingerprints with mutua distance information for indoor ocaization. The mutua distance can be tempora or spatia between different target measurements (as obtained from dead reckoning or peerassisted manner. Due to random signa fuctuation, fingerprints are noisy in nature and distance cannot be measured exacty. 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