Voronoi Tessellation Based Interpolation Method for

V oronoi Tessellation Based Interpolation Method for Wi-Fi Radio Map ConstructionMinkyu Lee and Dongsoo Han,Member,IEEEAbstract—Thefingerprint-based approach for positioning in WLAN has been drawing great attention these days.However, the approach usually requires tremendous time and efforts to collect locationfingerprints for the target area.In this paper,we propose an interpolation method based on Voronoi tessellation to significantly reduce such calibration efforts and to improve accuracy.Unlike other interpolation methods,our method refines the propagation model for each cell of the target area tessellated by a higher-order Voronoi diagram.Consequently,our method can take into account the signal fading caused by walls and obstacles more accurately.The proposed method significantly outperformed other interpolation methods in accuracy.Index Terms—Indoor environments,interpolation,radio prop-agation,wireless LAN.I.I NTRODUCTIONT HE proliferation of Wi-Fi infrastructures has facilitated numerous indoor positioning techniques using Wi-Fi lo-cationfingerprints.However,thefingerprint-based techniques usually require a lot of efforts to calibrate the target area. Several attempts to reduce the calibration efforts have been made.Some of them were developed based on the radio propagation model[1],[2],and others were based on inter-polation methods[3]–[6].However,in most cases,Reference Points(RPs),at whichfingerprints are to be collected,are not regularly distributed due to the structure of buildings,so interpolation methods for scattered data are widely used in practice.Radial Basis Function(RBF)[3],Linear[5],Inverse Distance Weighting(IDW)[6],and Kriging[4]are well-known methods for scattered data.Despite the numerous interpolation methods mentioned above,there is still potential to improve the conventional techniques in terms of increased accuracy and reduced ef-forts.In this paper,we propose a new interpolation method for scattered data based on higher-order V oronoi tessellation [8].Our method differs from conventional methods in two points.First,it adopts the Log-Distance Path Loss(LDPL) model which is widely used to estimate Signal Strength(SS) depending on the distance from the Access Point(AP).Other methods often use conventional models such as Gaussian, linear,or spline,but these models do not conform to signal fading patterns of Wi-Fi signals.Second,our method refines Manuscript received September26,2011.The associate editor coordinating the review of this letter and approving it for publication was S.Gupta.The authors are with the Department of Computer Science,Korea Advanced Institute of Science and Technology(KAIST),291Daehak-ro,Yuseong-gu, Daejeon,Korea(e-mail:{niklaus,dshan}@kaist.ac.kr).This research was supported by the MKE(The Ministry of Knowledge Economy),Korea,under the ITRC(Information Technology Research Center) support program supervised by the NIPA(National IT Industry Promotion Agency)(NIPA-2011-(C1090-1111-0013)).Digital Object Identifier10.1109/LCOMM.2012.020212.111992the propagation model for each cell of a target area tessellated by a higher-order V oronoi diagram in order to take into account signal fading caused by walls and obstacles more accurately.According to the experiments performed in two buildings,the proposed method outperformed other commonly used interpolation methods in accuracy.This indicates that the proposed method can significantly reduce the calibration efforts for localization.II.T HE P ROPOSED I NTERPOLATION M ETHODOur method consists of two phases:Thefirst phase esti-mates locations of all the APs.The second phase estimates signal fading parameters of the APs for each V oronoi cell in order to take into account walls and obstacles more accurately.A.First PhaseThe propagation model based on LDPL is defined by:P(d)=P(d0)−10θlog10(d/d0)(1) where P(d),θ,d0,P(d0)represents SS at distance d,path loss exponent,reference distance,and SS at the reference distance respectively.Here,we set d0=1(meter)and substitute Euclidean distance from the location of AP k(ˆx k,ˆy k)for d, then interpolation function is defined by:Z k(x,y)=ˆc k−10ˆθk log10(ˆx k−x)2+(ˆy k−y)2(2) where Z k(x,y)represents SS at location x,y from AP k and ˆc k=P(d0).In order to compute SS value at an arbitrary point,we need to determine the parametersˆx k,ˆy k,ˆθk andˆc k. Although the purpose of thefirst phase is to estimate locations of all the APs,(ˆx k,ˆy k),all four of the parameters should be estimated together because the parametersˆθk and ˆc k are also unknown.To estimate the four parameters,actual observations are required at the pre-designated RPs.Let N be the number of RPs(RP1,RP2,...,RP N),s k,i be the observed SS value at RP i of AP k,and(x i,y i)be the location of the RP i.The values of the parameters is obtained using the Nelder-Mead simplex algorithm[7],in which the sum of the weighted residual squares are minimized(3).(ˆx k,ˆy k,ˆθk,ˆc k)=arg minx k,y k,θk,c kNi=1w iNj=1w jγ2i(3)whereγi=s k,i−c k+10θlog10(x k−x i)2+(y k−y i)2(4)w i=1/|s k,i|p.(5)1089-7798/12$31.00c 2012IEEEFig.1:Example cases of interpolation.The top shows thefloor layout tessellated by the2nd-order V oronoi diagram where the squares denote RPs,thefilled square denotes AP and the crosses denote locations to be interpolated.The bottom shows the propagation modelsfitted for each cell of L1,L2,and L3.γi is a residual(i.e.the difference between s k,i and Z k(x i,y i)).Each residual is weighted by(5)in which a proper value for the power parameter p is essential in order to obtain the expected parameters.Since the Nelder-Mead algorithm is one of the direct search methods,a starting point for the parameters is essential.We use¯x,¯y,2.0,s k,max as a starting point forˆx k,ˆy k,ˆθk,ˆc k respectively,where¯x,¯y are the means of the x,y coordinates of RP i respectively,and s k,max is bigger than the strongest s k,i by one.Once the algorithm is successfully applied for all APs,we generate a radio map by(2)with the estimated parameters.In reality,however,SS attenuation does not follow this model exactly.A large gap still remains between the actual SS observation at the RP i and the SS value estimated by the model at the same RP i.This is because the parameters for the model are obtained by simply minimizing the weightedsquares.In the second phase,we reduce these gaps create a more accurate radio map.Second PhaseIn the second phase,we refine the parameterˆθk andˆc k each cell at the target area decomposed by a higher-order diagram.For a given set S of sites in a EuclideanV oronoi diagram is a partition of the space into regions cells).All the points in each region share the same closest in S.While a normal V oronoi cell considers only a single site,a higher-order V oronoi cell considers multiplesites.In our method,the RPs are used as the sites.1shows example cases of interpolating the three different L1,L2and L3,marked by crosses.While there is wall on the Line Of Sight(LOS)between L1&L3and marked by thefilled square,there are three walls on the between L2and the AP.We can easily guess that the SS of the L2case will be much more severe than the cases.If the target area is tessellated by Q th-order V oronoi each cell v has a set of closest RP v,j where ≤j≤Q.For example,for the cell including L2,RP3 RP6are the closest RPs when Q=2.As in thefirstwe also need observations for the refinement of each For each cell v,we can obtain a set of(s k,v,j,d k,v,j)where s k,v,j is the SS value of AP k at the RP v,jd k,v,j=(ˆx k−x v,j)2+(ˆy k−y v,j)2is the Euclidean between RP v,j and AP k.In addition,we can use onepair(s k,v,0=ˆc k,d k,v,0=1)becauseˆc k was estimated thefirst phase,which means the SS value is obtained at meter away from AP k.By using the Q+1pairs,the two can be refined further by(6)for each cell v and AP k.(ˆc k,v,ˆθk,v)=arg min(c k,v,θk,v)Qj=0(s k,v,j−c k,v+10θk,v log10(d k,v,j))2(6) Unlike in thefirst phase,the residues are not weighted in the second phase.With one or a small number of pairs,the sum of the residual squares is close to zero,and Q is very small in most cases;thus,the weighting effect is negligible.The Nelder-Mead algorithm is used to refine the parameters.The estimated values ofˆθk andˆc k obtained in thefirst phase are used as starting point values.We can get a more accurate radio map using the refined parametersˆθk,v andˆc k,v for(2).III.E XPERIMENTAL R ESULTSWe performed experiments in two buildings as illustrated in Fig.2.Thefirst one was COEX Mall,a large-scale shopping mall in Seoul,Korea,and the second one was N5building,a campus building at KAIST in Daejeon,Korea.The details of the two environments are summarized in Table I.We compared the proposed method(VORO)with the other two popular interpolation methods.Thefirst method is RBF described in[3]and the second one is IDW described in [6].The RBF method requiresσparameter,which denotes a standard deviation for the isotropic Gaussian used as a kernel function,and the IDW requiresλparameter,which denotesTABLE I:Summary of Experimental EnvironmentsCOEXN5Area (m 2)189000m 27650m 2#RPs 31115#Test Points 24273#Fingerprints/RP2060power to the weights.As described previously our method also requires two parameters p denoting power to the weights and Q denoting the number of the closest RPs for the refinement.Exactly the same KNN algorithm used in RADAR [1]is used for location estimation.Therefore,we need to determine theK parameter for the KNN algorithm.For a pair comparison,the parameters resulting in the best position accuracy were chosen by linear search.Half of the training dataset was used for training,and the remaining half was used for testing.For the shopping mall,σ=0.18,λ=3,p =4,Q =2and K =5were chosen,and for the campus building,σ=0.2,λ=1,p =1,Q =1and K =3were chosen.Interpolation was performed by 1meter interval only in the corridor area to reduce computing time.After the parameters were determined,we compared them in terms of mean distance error by varying the number of RPs for the two buildings as shown in Fig.3.RPs were randomly selected and then we took an average of the meandistanceFig.2:Floor plans of two buildings where the experiments were conducted.The top is COEX Mall (B1)and the bottom is N5Building at KAIST (2F).Rectangles denote RPs and crosses denote test points where test data were collected.Fig.3:A comparison of three interpolation methods with a varying number of RPs.errors of10times repetitions.In the case of the shopping mall, the mean error was8.79m for VORO,10.22m for RBF,and 9.73m for IDW when using all the311RPs.In the case of the campus building,the mean error was3.48m for VORO,4.69m for RBF,and4.88m for IDW.Even when we used a smaller number of RPs,VORO still outperformed other methods in both cases.IV.C ONCLUSIONThis paper presented a new interpolation method based on higher-order V oronoi tessellation.Unlike other interpolation methods,the proposed method adopted the LDPL model and took into account the signal fading caused by walls and obstacles.We confirmed that this method achieves better accu-racy than other conventional methods such as RBF and IDW through experiments with two infrastructures.In conclusion, due to the refined propagation model for each V oronoi cell, the proposed method can create more accurate Wi-Fi radio map from scattered 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