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Wireless sensor networks (WSNs) typically gather data at a discrete number of locations. However, it is desirable to be able to design applications and reason about the data in more abstract forms ...

Wireless sensor networks (WSNs) typically gather data at a discrete number of locations. However, it is desirable to be able to design applications and reason about the data in more abstract forms than in points of data. By bestowing the ability to predict inter-node values upon the network, it is proposed that it will become possible to build applications that are unaware of the concrete reality of sparse data. This interpolation capability is realised as a service of the network. In this paper, the ‘map’ style of presentation has been identified as a suitable sense data visualisation format. Although map generation is essentially a problem of interpolation between points, a new WSN service, called the map generation service, which is based on a Shepard interpolation method, is presented. A modified Shepard method that aims to deal with the special characteristics of WSNs is proposed. It requires small storage, can be localised and integrates the information about the application domain to further reduce the map generation cost and improve the mapping accuracy. Flood management application is considered to demonstrate how MGS-generated maps can be used in various applications. Empirical analysis has shown that the map generation service is an accurate, a flexible and an efficient method.

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- 1. WIRELESS COMMUNICATIONS AND MOBILE COMPUTING Wirel. Commun. Mob. Comput. 2010; 00:1–17 DOI: 10.1002/wcmRESEARCH ARTICLEInterpolation Techniques for Building a Continuous Map fromDiscrete Wireless Sensor Network DataMohammad Hammoudeh1∗ Robert Newman2 , Christopher Dennett2 , and Sarah Mount21 School of Computing, Manchester Metropolitan University, Manchester, UK2 School of Computing and IT, University of Wolverhampton, Wolverhampton, UKABSTRACTWireless Sensor Networks (WSNs) typically gather data at a discrete number of locations. However, it isdesirable to be able to design applications and reason about the data in more abstract forms than points ofdata. By bestowing the ability to predict inter-node values upon the network, it is proposed that it will becomepossible to build applications that are unaware of the concrete reality of sparse data. This interpolationcapability is realised as a service of the network. In this paper, the ‘map’ style of presentation has beenidentiﬁed as a suitable sense data visualisation format. While map generation is essentially a problem ofinterpolation between points, a new WSN service, called the Map Generation Service (MGS), which is basedon a Shepard Interpolation method, is presented. A modiﬁed Shepard method that aims to deal with thespecial characteristics of WSNs is proposed. It requires small storage, it can be localised, and it integratesthe information about the application domain to further reduce the map generation cost and improve the ©mapping accuracy. Empirical analysis has shown that the MGS is an accurate, ﬂexible and eﬃcient method.Copyright 2010 John Wiley & Sons, Ltd.KEYWORDSWireless Sensor Networks; Services; Visualisation; Information Extraction; Interpolation∗ CorrespondenceSchool of Computing, Manchester Metropolitan University, Manchester, UK. Email: m.hammoudeh@mmu.ac.uk1. INTRODUCTION content. The ability to interpolate point information is necessary for carrying out mapping tasks.With the increase in applications of WSNs, infor- The problem of map generation is essentially amation extraction and visualisation have become a problem of interpolation from sparse and irregularkey issue to develop and operate these networks. points. This interpolation capability is realised as aWSNs typically gather data at a discrete number of service of the network. In this paper, one particularlocations. By bestowing the ability to predict inter- interpolation approach, Shepard interpolation [1],node values upon the network, it is proposed that it is examined and shown to be suitable for thewill become possible to build applications that are constraints imposed by the nature of WSNs.unaware of the concrete reality of sparse data. Visual aspects, sensitivity to parameters, and timing Not all information that is collected from a requirements were used to test the characteristics ofWSN comes ready to use. Often, WSNs ﬁeld data this methodcollection takes the form of single points that need to The rest of the paper is organised as follows.be processed to get a continuous data presentation. Section 2 explains why map is a suitable discreteInterpolation describes this process of taking many data visualisation format. Sections 3 and 4 providesingle points and building a complete surface, the a brief description of map generation algorithmsinter-node gaps being ﬁlled based on the spatial and mapping applications in the literature. Sectionstatistics of the observation points. Interpolating 5 deﬁnes the problem on map generation. Sectionthese points will produce more useful information for 6 deﬁnes Shepard interpolation method. Sections 7the end user such as maps related to water chemical and 8 describe the modiﬁed Shepard map generation.Copyright © 2010 John Wiley & Sons, Ltd. 1Prepared using wcmauth.cls [Version: 2010/07/01 v2.00]
- 2. A map generation service for WSNs M. Hammoudeh et al.Evaluation of the MGS is presented in Section 9. The 3. A SURVEY ON ALGORITHMS FORpaper concludes in Section 10. MAP GENERATION The following section provides a brief on related work in map generation methods. Map generation techniques have previously been explored in the context of WSNs [3, 4]. Chang et al. [4] implemented an algorithm to estimate sensor nodes faulty behaviour on top of a cluster-based network. This approach is based on Bayesian Belief Networks (BBNs) which make it problematic to compute all2. SENSE DATA VISUALISATION: the probabilities and the revised probabilities once a MAPS new sensor reading is received. In dense multi-modal WSNs, the number of dependencies increases rapidlyThe integration of data visualisation tools and the and probabilities computation becomes an NP-hardraw data sent by the WSN makes the sensor network problem. This approach also lacks precision whensystem useful to diﬀerent potential users. Visual updating the fault rate table since it is based on aformats, such as maps, can be easily understood predeﬁned threshold value.by people possibly from diﬀerent communities, Event detection based on matching the contourthus allowing them to derive conclusions based maps of in-network data distribution has been shownon substantial understanding of the available data. eﬀective for event detection in WSNs [3]. MapMaps are eﬀective to understand the spatial construction starts from each node generating adistribution of environmental features since humans partial map of its own. When a node forwardscan use their natural interpretation capabilities to data for its neighbours, it adds each contour regionunderstand colours, patterns, and spatial relevance. in these partial maps with its own. This processA map is a visual representation of an area, although is repeated until the ﬁnal map is generated. Thismost commonly used to depict geography, maps may approach works well with grid network topologiesrepresent any space without regard to context or and less well with random topologies. When a gridscale such as weather data mapping [2]. However, is overlaid on top of a random topology some cellsa map could be overlaid over a geographic map to in the grid may be empty. These empty cells willenable observation of the data in a real-world map. not participate in the ﬁnal map construction. Hence, In a WSN, a map may be used as an information the ﬁnal map will not cover the entire network area.representation and extraction tool in which visual This makes the scheme sensitive and unsuitable forfeatures such as symbols and colours are used to random WSNs deployments. Furthermore, the loss ofcode diﬀerent attributes of the data to provide any partial maps will result in an incomplete networkthe information for end users to analyse and map.examine. These unique visualisation and analysis In both [3] and [4] the sink node is required tobeneﬁts oﬀered by maps make them more visually know the location and the ID of all nodes in thecommunicative, they imply the distributions and network. Furthermore, the work in both papers isstates; provide information about spatial patterns; application-dependent and requires major lower leveland imply the association of diverse phenomena. modiﬁcations if the application is to change. In [3], Maps can be either static or dynamic and the assumptions made on the network topologyallow data representation on 2D or 3D space. and the way the grid is formed are not eﬃcientThey allow the user to infer the actual sizes and and may dissipate the energy savings achieved bydistance between objects. The users can zoom the in-network map construction. In [4], it is notin or zoom out respectively meaning showing clear how the hierarchy is built. Besides, it is onlymore or less details. Furthermore, maps allow suitable for small size networks due to the single hopthe extraction of information that can not be communication scheme.obtained by looking at sensor readings separately DIMENSIONS [5] made the case for a large-scaleand are more eﬃcient to compute in both time distributed multi-resolution storage system thatand energy. For instance, maps may capture trends provides a uniﬁed view of data handling in WSNsor correlations among sense data. Where there incorporating long-term storage, multi-resolutionis no operating sensor, predictions can be made data access and spatio-temporal correlations inusing these spatial and temporal correlations among sensor data. This work is related to ours, butsensor readings. Finally, a map provides a higher- diﬀerent in focus at both the system architecturelevel information-rich representation which can be and coding level. It outlines an approach forsuitable for informing other network services and the relatively power-rich devices, focused on encodingdelivery of ﬁeld information visualisation.2 Wirel. Commun. Mob. Comput. 2010; 00:1–17 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm Prepared using wcmauth.cls
- 3. M. Hammoudeh et al. A map generation service for WSNsregularly-gridded, spatial wavelets over time series. limited to particular applications and constrainedBy contrast, we focus on highly resource constrained with unreliable assumptions. The grid alignment ofdevices, and integrate diﬀerent network services with sensors in [10], for example, is one such assumption.the MGS. Our work is also focused on spatially In the wider literature, mapping was soughtdeployed networks and is independent on a particular as a useful tool in respect to network diagnosisrouting algorithm. and monitoring [6], power management [11], and In centralised map generation approaches, deliv- jammed-area detections [12]. For instance, contourering all network sensory data back to the sink maps were found to be an eﬀective solution toincurs heavy transmission traﬃc. Several aggregation the pattern matching problem that works forbased map generation methods have been proposed limited resource networks [3]. These are examples ofto address this problem [6, 3, 7, 8]. However, aggre- speciﬁc instances of the mapping problem and, asgation based methods can not further improve the such, motivate the development of a generic MGS,scalability of the network as all sensors are required furthering the area of research by moving beyondto report to the sink. Moreover, the aggregation the limitations of the centralised approaches.process increases the computation overhead on the A service oriented approach has special properties.intermediate nodes. To address the inherent limita- It is made up of components and interconnectionstions of aggregation based methods, [9] proposed a that stress interoperability and transparency. Ser-method called Iso-Map that intelligently selects a vices and service-oriented approaches address de-small portion of the nodes, isoline nodes, to generate signing and building systems using heterogeneousand report mapping data to reduce the network network software components. This allows the devel-traﬃc and computation overhead. Partial utilisation opment of a MGS that works with existing networkof the network information leads to a decrease in components, e.g. routing protocols, and resourcesthe mapping ﬁdelity and isoline nodes will suﬀer without adding extra overhead on the network.from heavy computation and communication load.Furthermore, the location of mapping nodes can alsoaﬀect the directions of traﬃc ﬂow and thereby have asigniﬁcant impact of the network lifetime. Finally, insparsely deployed low density networks it is diﬃcult 5. UNDERSTANDING THE PROBLEMto construct contour maps based only on isoline OF MAP GENERATION FROMnodes. The positions of isoline nodes provide only SPARSE DATAdiscrete iso-positions, which does not deﬁne how todeduce how the isolines pass through these positions. Given a set of known data points representing the To conclude, mapping is often employed in WSN nodes’ perception of a given measurable parameterapplications but as yet there is no clear deﬁnition of the phenomenon, what is the most likely complete(or published work towards) a localised MGS that and continuous map of that parameter? In thewould aid the development of more sophisticated ﬁeld of computer graphics, this problem is knownapplications. The development and analysis of such as an unorganised points problem, or a cloud ofa service is the key novel contribution of the work points problem. That is, since the position of theproposed here. points in xy is assumed to be known, the third parameter can be thought of as height and surface reconstruction algorithms can be applied. Simple algorithms use the point cloud as vertices in the4. MAPPING APPLICATIONS IN THE reconstructed surface. These are not diﬃcult to LITERATURE calculate, but can be ineﬃcient if the point cloud is not evenly distributed, or is dense in areas of littleWithin the WSN ﬁeld, mapping applications found geometric variation.in the literature are ultimately concerned with the Approximation, or iterative ﬁtting algorithmsproblem of mapping measurements onto a model deﬁne a new surface that is iteratively shaped to ﬁtof the environment. Estrin et al.[10] proposed the the point cloud. Although approximation algorithmsconstruction of isobar maps in sensor networks can be more complex, the positions of vertices areand showed how in-network merging of isobars not bound to the positions of points from the cloud.could help reduce the amount of communication. For applications in WSNs, this means that we canFurthermore, [6] proposed an eﬃcient data-collection deﬁne a mesh density diﬀerent to the number ofscheme, and the building of contour maps, for sensor nodes, and produce a mesh that makes moreevent monitoring and network-wide diagnosis, in eﬃcient use of the vertices. Self organising maps arecentralised networks. Solutions such as distributed one of the algorithms that can be used for surfacemapping have been proposed to the general reconstruction [13]. This method uses a ﬁxed numbermapping domain. However, many solutions are of vertices that move towards the known data.Wirel. Commun. Mob. Comput. 2010; 00:1–17 © 2010 John Wiley & Sons, Ltd. 3DOI: 10.1002/wcmPrepared using wcmauth.cls
- 4. A map generation service for WSNs M. Hammoudeh et al. Note that surface reconstruction on typical non- Shepard’s expression for globally modelling aoverlapping terrains is equivalent to sparse-data surface is:interpolation. This kind of geometric parameter interpolation has been shown to work well for N (di )−u × zi reconstructing underlying geography when the entire i=1network has been queried [14]. However, It does N if di = 0 ∀Di (u > 0) f1 (P ) =not extend well to variable surfaces or overlapping (di ) −ulocal mapping, since it requires a complete data i=1set to deﬁne the surface. A more general method z i if di = 0is interpolation by inverse distance and, speciﬁcally, (1)Shepard interpolation [1] which improves on it. where di is the distance from P to D numbered i in the N known points set and zi is the known value at Di . The exponent u is used to control the smoothness of the interpolation. As P approaches a data point Di , di tends to zero and the ith terms in both the numerator and denominator exceeds all bounds while other terms remain bounded. Therefore, the limP →Di f1 (P ) = zi is as desired and the function f1 (P ) is continuously diﬀerentiable even at the6. SHEPARD INTERPOLATION junctions of local functions.Shepard Interpolation is an inverse distance weighted 6.1. Global Shepard Algorithm Shortcomings andscattered data interpolation algorithm. It is widely Solutionsused in practise and has also been shown towork well with noisy data [15]. Shepard deﬁned a Shepard’s interpolation suﬀers from several short-continuous function where the weighted average of comings imposed by the fact that each sample pointdata is inversely proportional to the distance from has a radially symmetric inﬂuence despite the naturethe interpolated location. The algorithm explicitly of the underlying data [21]. Among the well knownimplies that the further away a point is from an artifacts are cusps, corners, and ﬂat spots at the datainterpolated location, P , the less eﬀect it will have points, as well as the excessive inﬂuence of pointson the interpolated value. that are far away [22]. Further shortcomings include Known points, Di , are weighted during interpo- that the global function necessitates all weights tolation relative to their distance from P . Weighting be recomputed if any points are added, removed,is assigned to data through the use of a weighting or modiﬁed. In WSNs this is impractical due topower, which controls how the weighting factors drop the network dynamics such as: node failures, nodeoﬀ as the distance from P increases. The greater the mobility, or deployment of new nodes. Shepard hasweighting power, the less eﬀect far points have on the identiﬁed three main shortcomings of his method andinterpolation result. As the power increases, Shep- proposed modiﬁcations to deal with them as follows:ard interpolation approaches the nearest neighbourinterpolation method [16] where the interpolated Building the Support Setvalue simply takes on the value of the closest sample We deﬁne the support set as the set containingpoint [16, 15]. all points used to calculate P . The global method Many modiﬁcations to the original Shepard has a linear running-time O(N), which makesalgorithm have been proposed in the literature [17, it impractical and ineﬃcient especially when the18, 19, 20], however, most of these methods are number of nodes is large. To overcome this, adesigned for computer graphics and image processing local Shepard algorithm was deﬁned. This algorithmﬁelds. These algorithms usually trade accuracy eliminates distant points from the calculation of anywith computation complexity. Nevertheless, when interpolated value since only nearby data points haveapplying interpolation to WSN applications it is signiﬁcant inﬂuence. To select nearby nodes, Sheparddesirable to keep the interpolation simple to reduce deﬁned two criteria:the amount of processing as well as communicatedinformation across the network. Therefore, we shall 1. Arbitrary distance criterion: All data pointsuse the original method with the modiﬁcations within radius r of the point P are includedproposed by Shepard that further reduce the amount in computation. This is computationally easyof processing and data communication to achieve but allows the possibility that there are nomore energy savings using the limited available data points or a suﬃciently large number ofbandwidth. These modiﬁcations are described in data points within the radius r. A collection ofsubsection 6.1. points, Cp , within a search radius r is deﬁned4 Wirel. Commun. Mob. Comput. 2010; 00:1–17 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm Prepared using wcmauth.cls
- 5. M. Hammoudeh et al. A map generation service for WSNs as Cp = {Di |di ≤ r} and n(Cp ) is the total interpolation function f2 (P ) is given by: number of data points in Cp . 2. Arbitrary number criterion: Only the closest n (si )2 zi data points are considered in the computation D ∈C i if di = 0 ∀Di of any interpolated value. This approach f2 (P ) = (si )2 (4) ignores the relative location and spacing of D ∈C i the points and requires deep searching and zi if di = 0 complex ranking procedure for data points. In addition, it assumes that a single number, n, The function f2 (P ) requires less physical resources, of interpolating points was optimal. If N is in terms of both computation and memory, than the total number of data points, then, a the previous functions. The running-time is reduced n new collection of data points, CP , is deﬁned to O(CP ). Consequently, the amount of memory used n as CP = {Di1 , Di2 ..., Din } where (n ≤ N ) and by the algorithm on data set of size N is reduced the subscripts ij are deﬁned such that 0 ≤ to inputs of CP . This reduction in computation and di1 ≤ di2 ≤ ... ≤ diN . memory cost is invaluable in large scale WSNs whichShepard has chosen a mix of the two criteria, which must be capable of in-network processing at all levels,combined their advantages. An initial radius r is including the application level.deﬁned depending on the overall density of data Since topologies in WSNs changes frequently,points such that seven data points are included on the MGS should be topology-independent andaverage in a circle of radius r. r is written as follows: decentralised. That is, each node or subsystem, e.g. cluster, uses only local information when making 7A mapping decisions. MGS can be implemented on πr2 = (2) N each node with varying the size of the support setwere A is the area of the largest polygon enclosed by and the way it is used. For instance, each node canthe data points. use the Cp to build a set of neighbours to collaborateA function si = s(di ) is deﬁned to guarantee the with in building a local map for their vicinity. Thislocal behaviour of the interpolating algorithm by local map can be used to respond to user quires,calculating a surface model for any d ≤ r, and which to update the global map maintained at a clusterweights the points at r ≤ r more heavily: head (partial) or a sink (global), used as a local 3 accuracy model, etc. The local map can be used to calculate whether a new reading should be forwarded 1/d if 0 < d ≤ r 3 to upper nodes in the hierarchy by calculating the s(d) = 4r2 ( r − 1)2 27 d if r < d ≤ r (3) 3 impact of the new reading on the local map. At 0 if r < d cluster heads and the sink, the support set contains all nodes within the cluster and all nodes in thewhere r is a radius of inﬂuence about P chosen network respectively. With hierarchical approaches,large enough to include n points and deﬁned as the global map can be built and updated by cluster n / nr (Cp ) = min{dij |Dij ∈ CP } = din+1 . In order that heads. MGS deals eﬃciently with the addition orthe interpolation algorithm works realistically, if deletion of nodes due to the local mapping, i.e. anythe data points were girded, a minimum of four topological changes will be dealt with locally withoutdata points was chosen. A maximum of ten was recalculating the global map.established to limit the complexity and amount ofcomputation required [1]. Thus Cp and rp are deﬁned Including Directionas follow: The current method ignores the direction factor 4 in computing the weightings. To make the Cp if 0 ≤ n(Cp ) ≤ 4 method intuitively reasonable, Shepard included the Cp = Cp if 4 < n(Cp ) ≤ 10 direction in computing interpolated values. A new 10 Cp if 10 < n(Cp ) directional weighting for each data point Di close to P is deﬁned by:and 4 r (Cp ) if n(Cp ) ≤ 4 ti = sj [1 − cos(Di P Dj )]/ sj (5) rp = r if 4 < n(Cp ) ≤ 10 Dj ∈C Dj ∈C 10 r (Cp ) if 10 < n(Cp ) were the cos(Di P Dj ) is deﬁned as: [(x − xi )(x −The resulting function f2 (P ), has similar behaviour xj ) + (y − yi )(y − yj )]/di dj . The appropriateness ofto the original function but it is capable of the cosine function and computation ease makeshandling much larger data sets and it is much it a good measure of direction. The function sjmore suitable for parallel implementations. The is included in the new function to preserve theWirel. Commun. Mob. Comput. 2010; 00:1–17 © 2010 John Wiley & Sons, Ltd. 5DOI: 10.1002/wcmPrepared using wcmauth.cls
- 6. A map generation service for WSNs M. Hammoudeh et al. A new parameter v is deﬁned with the distance dimension to bound the maximum eﬀect the slope terms may have on the ﬁnal interpolated value. For a contour mapping application, v can be deﬁned as: 0.1[max{zi } − min{zi }] v= (9) max{(A2 + Bi )} i 2 Ai (x−xi )+Bi (y−yi ) An increment ∆zi = v is computed v+di for each Di ∈ CP as a function of P to include the Figure 1. Fire spreading and wind direction. eﬀect of the slope in interpolating values at P . Thus the latest version of the interpolation function is:original weighting assumption. Within the direction wi (zi + ∆zi )considered, a new weighting function wi = (si )2 × D ∈C i(1 + ti ) is deﬁned and the ﬁnal interpolation function if di = 0 ∀Di ∈ C f4 (P ) = wiis deﬁned as: Di ∈C zi if di = 0 w i zi / wi if di = 0 ∀Di f3 (P ) = Di ∈C Di ∈C (10) zi if di = 0 In function f3 (P ), the interpolated surface has a (6) zero gradient at every Di . This modiﬁcation is This modiﬁcation is useful for mapping modalities valuable to WSNs applications since it reduces noisewhere the direction is vital, for instance wind and redundant small details in the image that aredirection in forest ﬁre monitoring applications. The perceptually unnoticeable to human eyes.two conﬁgurations in Figure 1, for example, wouldyield identical interpolated values at location 2.However, if the algorithm to be intuitively practical, 7. MAPPING IN HIGHERthe value at location 2 in the top conﬁguration DIMINESIONAL-SPACEshould be closer to the value at sensor 1 than in thelower conﬁguration, because wind direction should This section deﬁnes a new metric for distance,be expected to screen the eﬀect of more distant point. suitable for higher dimensions (multi-modal sensing), in which the concept of closeness is described inDetermining Slope terms of relationships between sets rather than inThe arbitrary and undesirable zero gradient at every terms of the Euclidean distance between points.point Di still exists on the f3 (P ), generated surface. Using this distance metric, a new generalisedIf di is very small, si will equal d−1 and wi will vary i mapping function f , that is suitable for an arbitraryas d−2 . To correct this, weighted averages of divided i number of sensed modalities, is deﬁned.diﬀerences of zi about Di , Ai and Bi , were added In higher diminsional-space mapping every set Sito suﬃciently nearby data points to achieve partial corresponds to an input variable i.e. a sense modality,derivatives at Di . Constants Ai and Bi represent the called i, and referred to as a dimension. The powerslope in the x and y directions at each data point Di , of such a generalisation can be seen when we includeAi and Bi are deﬁned as: the time variable as one dimension. The spatial map generation problem can be stated as follows: (zj − zi )(xj − xi ) Given a set of randomly distributed data points wj (d[Dj , Di ])2 Dj ∈Ci Ai = (7) xi ∈ Ω, i ∈ [1, N ] , Ω ⊂ Rn (11) wj Dj ∈Ci with function values yi ∈ R, and i ∈ [1, N ] we require a continuous function f : Ω −→ R to interpolateand unknown intermediate points such that (zj − zi )(yj − yi ) f (xi ) = yi where i ∈ [1, N ] (12) wj (d[Dj , Di ])2 Dj ∈Ci Bi = (8) We refer to xi as the observation points. The wj integer n is the number of dimensions and Ω Dj ∈Ci is a suitable domain containing the observation points. When rewriting this deﬁnition in terms ofwhere Ci = CDi − {Di }. relationships between sets we get the following:6 Wirel. Commun. Mob. Comput. 2010; 00:1–17 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm Prepared using wcmauth.cls
- 7. M. Hammoudeh et al. A map generation service for WSNsLemma 7.1 Given N ordered pairs of separated sets by using Geometric Algebra (GA) in a special waySi ⊂ Ω with continuous functions while using a point to set distance metric. Burley et. al [25] discuss the usefulness of GA for adapting fi : Si −→ R, i ∈ [1, n] (13) uni-variate numerical methods to multivariate datawe require a multivariate continuous function f : using no additional mathematical derivation. TheirΩ −→ R, deﬁned in the domain Ω = S1 ∪ S2 ∪ ... ∪ work was motivated by the fact that it is possible toSn−1 ∪ Sn of the n-dimensional Euclidean space deﬁne GAs over an arbitrary number of geometricwhere dimensions and that it is therefore theoretically possible to work with any number of dimensions. f (xi ) = fi (xi ) ∀xi ∈ Si where i ∈ [1, n] (14) This is done simply by replacing the algebra of theProof of Lemma 7.1 The existence of the global real numbers by that of the GA. We apply the ideascontinuous function f can be veriﬁed as follows. in [25] to ﬁnd a multivariate analogue of uni-variateFirst, the data set is deﬁned as interpolation functions. To show how this approach (0) works, an example of Shepard interpolation of this (1) (n ) v1 , v1 , · · · , v1 form is given below: (0) v , v (1) , · · · , v (n ) Given a set of n distinct points X = 2 2 2 S= . . . (15) {x0 , x1 , ..., xn } ⊂ Rs , the classical Shepard’s . . . . . . interpolation function is deﬁned by (0) (1) (n ) vn , vn , · · · , vn n owhere n ≤ N and vi = (xi , ri ) , i ∈ [1, N ] and ri is Sn,µ f (x) = wk (x) f (xk ) (18)a reading value of some distinctive modality (e.g. k=0temperature). Let Φ be a topological space on S and andthere exists open subsets Si , i ∈ [1, n] |x − xk |−µ wk (x) = n (19) (0) (1) (n ) S1 = v1 , v1 , · · · , v1 |x − xk |−µ (0) (1) (n ) k=0 S2 = v2 , v2 , · · · , v2 . (16) where |.| denotes the Euclidean norm in Rs . . . 0 In the uni-variate case (s = 1) and Sn,2 f . The (0) (1) (n ) 0 Sn = vn , vn , · · · , vn basic properties of Sn,µ f are: 0which are topological subspaces of Φ such that 1. Sn,µ f (xi ) = f (xi ) , i = 0, ..., n; 0 2. doe Sn,µ f = 0, where doe is an abbreviation ΦSi = {Si ∩ U |U ∈ Φ} (17) of degree of exactness.Also deﬁne Ψ as a topological space on the co-domain R of function f . Then there exists afunction, f , that has the following properties: 8. APPLICATION-BASED LOCAL MAP GENERATION 1. Let f : S1 ∪ S2 ∪ ... ∪ Sn−1 ∪ Sn be a mapping deﬁned on the union of subsets Si , i ∈ [1, N ] Because the accuracy level of generated maps such that the restriction mappings f|Si are may vary signiﬁcantly depending on the speciﬁc continuous. If subsets Si are open subspaces application, e.g. existence of barriers, in this section of S or weakly separated, then there exist a we modify the distance metric to include the function f that is continuous over S (proved knowledge known about the application domain. by [23]). The proposed metric attempts to balance the size 2. If f : Ω → Ψ is continuous, then the restriction of the support set with the interpolation algorithm to Si , i ∈ [1, N ] is continuous (property, computation complexity as well as interpolation see [24]). The restriction of a continuous global accuracy. mapping function to a smaller local set, Si , We deﬁne the term scale for determining the is still continuous. The local set follows since weight of every given dimension with respect to P open sets in the subspace topology are formed based on a combined Euclidean distance criteria from open sets in the topology of the whole as well as information already known about the space. application domain a priori to network deployment.Using the point to set distance generalisation, the While the term weight is reserved for the relevancefunction f can be determined as a natural generali- of a data site by calculating the Euclidean distancesation of methods developed for approximating uni- between P and Di .variate functions. Well-known uni-variate interpola- We deﬁne a new scale-based weighting metric, mP ,tion formulas are extended to the multivariate case which includes application domain information. TheWirel. Commun. Mob. Comput. 2010; 00:1–17 © 2010 John Wiley & Sons, Ltd. 7DOI: 10.1002/wcmPrepared using wcmauth.cls
- 8. A map generation service for WSNs M. Hammoudeh et al.support set is Ci , where Ci ⊆ Si , for each dimension Table I. Linux-class sensor node hardware platforms.contains the nearest points for P using mP .Symbolically, Ci is calculated as MicroServer Gumsense CPU au1550 MarvellPXA270 Ci = L (d (P, Ej ) , δ (Si )) ∀Ej ∈ Si (20) Clock speed 400MHz 100 to 600MHzwhere i ∈ [1, n], L is a local model that selects the CPU power consumption 0.5W 72µWsupport set for calculating P , d is an Euclideandistance function, Ej is an observation point in the Memory 128MB 128MBdimension Si , and δ(Si ) a set of parameters for Flash ROM 128MB 32MBdimension Si . These parameters are usually a set ofrelationships between diﬀerent dimensions or other discontinuous as P crosses a barrier which resultapplication domain characteristics such as obstacles. discontinuous interpolated surface at an obstacle.In uni-dimensional distance weighting methods, the The inclusion of barriers in the interpolation willweight, ω can be calculated as follows result in the selection of a diﬀerent set of nearby ω = d (P, Ej ) , Ej ∈ Si (21) data points, weightings, and slopes.This function can be extended to multi-dimensionaldistance weighting systems as follows 9. SHEPARD INTERPOLATION ANALYSIS ω = K (P, Si ) , i ∈ [0, n] (22) In this section the eﬀectiveness of the Shepard inter-where K (P, Si ) is the distance from P to data set Si polation algorithm is veriﬁed and its characteristicsand n is the number of dimensions in the system. are studied quantitatively and qualitatively. TheEquation 22 can now be extended to include the Shepard interpolation performance was compareddomain model parameters of arbitrary dimensional with that of Triangulation with Linear Interpolationsystem. Then the dimension-based scaling metric can algorithm (TLI) [16].be deﬁned as TLI was chosen for comparison because it is an exact interpolator which uses the optimal Delaunay mP = L (K(P, Si ), δ (Si )) i ∈ [0, n] & Si = CP triangulation. Delaunay triangulation is used exten- i sively in the ﬁeld of WSN. Uses include: adaptable (23) network deployment [26], network coverage [27],where CP is the dimension containing P . locating and bypassing routing holes [28], distributed area computation [29], position-aware routing [30],8.1. Example: Mapping Surfaces with Barriers and spatial clustering [31]. Furthermore, TLI is widely referred to in the literature including in theShepard interpolation is based on the intuitive image processing ﬁeld [32] and reported to be oneassumption that there is a logical relationship of the simplest and most eﬃcient algorithms with abetween adjacent points. This assumption is, good running time [16, 33].however, violated if some barrier, such as a river,ruptures the continuity of the surface. The eﬀect 9.1. Hardware Requirementsof physical barriers can be simulated easily due tothe distance-dependent interpolation by including The following experiments target WSNs built fromvirtual barriers. The user may specify discontinuities Linux-class devices that have higher storage andin the metric space in which di is calculated using processing capabilities. The choice of less constraineda diﬀerent selection set of nearby data points and hardware platform was for two reasons:diﬀerent weightings and slopes are being calculated. 1. Distributed mapping is desirable but intro-Given a detour of length b[P, Di] perpendicular to duces a considerable storage and computationthe line between P and Di , Shepard interpolation complexity on sensing devices when consider-deﬁnes the eﬀective distance to travel between the ing current sensor node capabilities.two points as: 2. In-network visualisation has requirements typ- 1 ical of any non-trivial processing. For example, di = {(d[P, Di ])2 + (b[P, Di ])2 } 2 (24) the MICA/MICA2 mote [34] microcontroller has no support for ﬂoating point arithmetic orwhere b[P, Di] is the strength of the barrier. When integer multiplications.a barrier exists, di replace di in all calculations.Whereas, if there is no barrier between Di and P , The Gumsense [35] and EmStar MicroServers [36,di = di and b[P, Di] = 0. The eﬀective distance is 37], amongst other Linux boxes are example8 Wirel. Commun. Mob. Comput. 2010; 00:1–17 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm Prepared using wcmauth.cls
- 9. M. Hammoudeh et al. A map generation service for WSNshardware platforms that are available in the market method. The 2D maps depict the height where theand are capable of running the MGS. Table I shows pixel intensities depict depth values.the speciﬁcations of these hardware platforms. In an extreme situation, assume that there is 9.4. Experimental Setupa sensor node that store 500 observation points.Assuming mapping data are represented as tripletsof 32-bit ﬂoats, the data alone requires 3.9KBof memory. Let Id be the number of instructionsrequired to estimate the value at a location. Knowingthe clock speed of the processors allows making asimple estimate of the execution time. Combinedwith the 600MHz clock speed, execution time tocalculate a partial map is estimated at 1.4583s.What is deﬁned as an acceptable execution time isdependent on the application requirements. Figure 2. Grand Canyon height map9.2. TLI Algorithm Details These experiments made use of the Grand CanyonThis method connects data points to form triangles height map [39]. The studied region is 15360m2 ,that do not intersect with each other. The result with heights ranging from 165m to 284m aboveof this process is a patchwork of triangular faces sea level. This map was sampled to 65536over the extent of the grid. The slope and elevation points. The sensor nodes were randomly distributedof the triangle is determined by the original data over the sensing ﬁeld, i.e. the height map, atpoints deﬁning the triangle and all nodes within position (xi , yj ), where the pixel intensities depictthe triangular plane are deﬁned by the triangular altitude values. The height map was chosen becausesurface. Since the triangles are determined by the using numerous wide-distributed height points hasoriginal data, the data must be sampled at a high been an important topic in the ﬁeld of spatialrate. TLI is fast with all data sets but it is not information [16]. Furthermore, the height is a staticeﬀective with few points [16]. One advantage of measure which makes it suitable for the evaluationtriangulation is that, with enough data, triangulation of various interpolation algorithms. The primarycan preserve break lines deﬁned in a data ﬁle. For purpose of these experiments is to take spatialexample, if a fault is delimited by enough data points interpolation to calculate the unknown heights byon both sides of the fault line, the surface generated using the information of neighbouring points andby triangulation will show the discontinuity [16]. to report results. Shepard Algorithm with all three modiﬁcations is implemented and used in all of the9.3. Comparison Metrics following experiements. Using the same data set, the diﬀerence in quality and accuracy of generated mapsTo determine the accuracy of the interpolation is determined by the interpolation method used.quantitatively, the skewness and kurtosis of ahigh resolution source data and the result of the 9.5. Experiment 1: The Eﬀect of Network Densityinterpolation using a subset of that data has beenchosen as a measure of the surface deviation. Aim: The eﬀect of network density on the recon-The kurtosis is a measure of the peakedness of a struction quality of both interpolation algorithms isreal-valued random variable where a high kurtosis studied.distribution has a sharper peak and fatter tails and Procedure: Interpolation methods are run withlow kurtosis distribution has a more rounded peak diﬀerent network densities and results are recorded.with wider shoulders [38]. Skewness is a measure Results and discussion: Figures 3 and 4 showof the asymmetry of the probability distribution of how the network density and choice of interpolationa real-valued random variable [38]. A distribution algorithm aﬀect the reconstruction results. It iscould have two kinds of skewness; positive skew or observed that higher network densities increase thenegative skew, where the mass of the distribution is smoothness in the re-constructed maps. For instance,concentrated on the left of the ﬁgure or the right of contour maps made from high network densitythe ﬁgure respectively. Further qualitative accuracy are visibly smoother due to shorter line segmentsassessments are done using empirical peak proﬁling between data points.to obtain peak information for studying the two Compared to the actual height map, Figure 2,algorithms local behaviour. it is visually evident that both interpolation Visually, we use 2D and 3D height maps to algorithms produced acceptable quality 2D anddetermine the global accuracy of the interpolation 3D maps. However, the reconstruction quality ofWirel. Commun. Mob. Comput. 2010; 00:1–17 © 2010 John Wiley & Sons, Ltd. 9DOI: 10.1002/wcmPrepared using wcmauth.cls
- 10. A map generation service for WSNs M. Hammoudeh et al.TLI with the absence of suﬃcient data density (N ) Shepard TLI(e.g. 50, and 100 node) is largely ﬁctitious andunconstrained especially on the map boundaries.TLI requires the number of boundaries of theobserved area and higher density of sensing nodeson these locations. As the network density increasesthe reconstruction quality for both algorithms isimproved. TLI performed better than Shepard with 50the reconstruction of the right hand side portion ofthe map due to the smoothness of its surface. Thisresult is due to the assumption that TLI makes, thatthe height is changing at constant rate, which wasthe case in that portion of the map. Nevertheless,total map produced by Shepard interpolation wereequal to or better than that of the TLI algorithmdespite the little geometric variation in that part 100of the map. Shepard interpolation captured smallerfeatures of the surface and reﬂected more details thanTLI. However, the cost (in terms of computation andcommunication, i.e. the size of the support set) ofinterpolation in Shepard was much less than thatwhen using TLI.Conclusion: Shepard interpolation resulted inequal or better reconstruction results than TLI. The 200Shepard algorithm proved to produce more accurateresults especially on the boundaries and at lownetwork density. Also, Shepard has also capturedsmaller features and reﬂected more details of thesurface than TLI.9.6. Experiment 2: Interpolation Local Behaviour 300Aim: In this experiment the local performance ofShepard and TLI algorithms is to be evaluatedthrough application of image processing approaches.Procedure: Peak proﬁling and statistical measuresare used to quantitatively characterise and comparelocal features extraction capabilities of bothalgorithms at various network densities. The highest 500peak in the Grand Canyon height data, labelled inFigure 2, was selected as the local feature that isquantised from maps produced by each algorithm.Results and discussion: Figures 5 to 9 show theproﬁling results of the selected peak from the originalmap and from maps produced by the Shepard andTLI interpolation algorithms. It is observed from 1000the ﬁgures that Shepard interpolation appears tobe more visually plausible and has always rendered Figure 3. (1): 2D maps produced by Shepard and TLI at variousa smoother surface than TLI. This is because that network densities (N ).TLI surface passes through all points whose valuesare known. Shepard algorithm maintained the localshape properties of the nodal functions because thereis a mild decrease in a point’s inﬂuence as it gets to sample points within the neighbourhood reducedfarther from the prediction location. While in TLI the eﬀect of distant points and produced a ﬁnalcurves, all locations within the relevant triangle surface that is much closer to the original forget the same weight regardless of how far they some features. At low network densities (e.g. 50)are from the prediction location. In local Shepard Shepard algorithm yields a surface which is muchinterpolation, the enforced restriction of support set more representative of the original surface than that10 Wirel. Commun. Mob. Comput. 2010; 00:1–17 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm Prepared using wcmauth.cls
- 11. M. Hammoudeh et al. A map generation service for WSNs (N ) Shepard TLI 50 100 Figure 5. Peak proﬁling with 100 nodes network density 200 300 Figure 6. Peak proﬁling with 200 nodes network density 500 1000Figure 4. (2): 3D maps produced by Shepard and TLI at various network densities (N ).yielded by TLI. This is because TLI requires a Figure 7. Peak proﬁling with 300 nodes network densitymedium-to-large number of data points to generateacceptable results. With a highly variable surface Table II. Peak proﬁling statistical measures, where N is thesuch as this, 50 data points are insuﬃcient for TLI number of nodes.to re-create the source data, despite the relativelysmall size of the region in question. At all network N Skewness Kurtusisdensities, TLI suﬀer from edge eﬀects because data Shepard TLI Shepard TLIsets that contain sparse areas result in distinct 10 -0.340 -0.129 2.072 2.085triangular facets on a surface plot or contour map. 50 -0.314 -0.105 2.088 1.782At slightly higher network densities (100 and 200), 100 -1.698 -0.610 5.372 2.086TLI was less representative of the original data range 200 -1.065 -1.069 2.862 3.380than Shepard because it tends to capture broad 300 -1.635 -1.613 4.669 4.486regional trends in the surface. TLI does not provide 500 -1.117 -1.186 4.211 3.465the ‘ﬂatness’ on the edges we would hope for. As 1000 -1.249 -0.964 4.211 3.092the network density increases, both interpolationalgorithms give almost equal results with betterperformance from the Shepard algorithm on the basis Table II presents a summary of statistics forof adherence to the original surface. peak proﬁling results at various network densitiesWirel. Commun. Mob. Comput. 2010; 00:1–17 © 2010 John Wiley & Sons, Ltd. 11DOI: 10.1002/wcmPrepared using wcmauth.cls
- 12. A map generation service for WSNs M. Hammoudeh et al. Original 10 50 100 200 300 500 1000 Figure 10. Contour maps drawn on maps produced by Shepard Figure 8. Peak proﬁling with 500 nodes network density interpolation 9.7. Experiment 3: Acceptable Level of Data Presentation Aim: In this experiment the question of what is an acceptable level of data presentation needed for a particular application was investigated. Procedure: The required accuracy level of inter- polated maps may vary signiﬁcantly depending on the speciﬁc application. Contour map was chosen as an application to determine the network density required to reﬂect some terrain characteristics with Figure 9. Peak proﬁling with 1000 nodes network density particular levels of accuracy and details. In this experiment the eﬀect of the network density on the quality of the contour maps is to be studied. We restrict the data representation quality experi- ments to Shepard’s algorithm because Experiment 1 and Experiment 2 proved that it is more suitable for spatial data interpolation at lower times and processing complexities than TLI.using Shepard and TLI interpolated maps. The Results and discussion: Figure 10, shows askewness and kurtosis values measured from the number of contour maps overlaying the height maporiginal map are −1.419 and 4.423 respectively. The generated using the Shepard interpolation algorithmvalues recorded in table II shows that as the network at various network densities. By comparing contourdensity increases, the quality of the produced maps maps constructed using low (10, 50, and 100),increase. The skewness measurements in table II medium (200 and 300), and high (500 and 100)conﬁrm the results found in the previous experiment network densities, it is noticed that the reconstructedthat Shepard interpolation produced a more accurate maps are very similar to the original one. Thepresentation of the interpolated surface at smaller lowest network density at which the selected peakdata sets. However, with bigger data sets (200 was successfully captured is 100, however it didand 300) the peak deviation diﬀerence of Shepard not precisely identify the size of the peak. At 200and TLI interpolated surfaces from the original nodes network density both the size and the heightsurface is minimised. Looking at the kurtusis of the peak were represented correctly on themeasures, Shepard gives more accurate results of how contour map. With higher network densities, contourpeaked a distribution is. This success of Shepard was maps exactness increased rapidly and the diﬀerencedue to the use of a subset of the observation points between the contour maps generated using 200, 300,which is more related to the interpolation location 500 and 1000 is insigniﬁcant. Thus a network densityand ignores the eﬀect of distant points. of 200 is enough to give an acceptable presentationConclusion: The results of these experiments of that desired feature.showed that Shepard interpolation was more capable Conclusion: From the contour map and theof extracting local features of the interpolated terrain peak proﬁling results, it can be seen that mostthan TLI. This result makes the Shepard method of the topographic variations of the terrain weremore suitable for implementing the localised MGS represented with accuracy levels enough to supplyin large WSNs. information on the topography of the land surface12 Wirel. Commun. Mob. Comput. 2010; 00:1–17 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm Prepared using wcmauth.cls
- 13. M. Hammoudeh et al. A map generation service for WSNsat a 200 nodes network density. In current real-lifeWSN deployments, this network density is achievablewhich proves the appropriateness and eﬃciency ofShepard interpolation when applied in WSNs.9.8. Practical Analysis of Modiﬁed Shepard-based Map Generation Figure 11. Heat diﬀusion map taken by IR camera.Aim: This experiment aims to study the eﬀect ofintegrating the knowledge given by the applicationdomain into the multi-modal MGS.Procedure: While no single domain of scientiﬁcendeavour can serve as a basis for designinga general framework, an appropriate choice ofspeciﬁc application domain is important in providingsigniﬁcant insights relating to requirements of such a Figure 12. Heat map generated by the Shepard-mappingmapping service. Therefore, to illustrate the beneﬁts method.of exploiting the domain model in map generationwe consider heat diﬀusion in metals model.A FLIR ThermaCAM P65 Infrared (IR) camera [40],is used to take sharp thermal images. A heat sourcewas placed on the middle of one edge of the brasssheet with the segment hole excavation. Brass (analloy of copper and zinc) sheet was chosen becauseit is a good thermal conductor and allows imaging Figure 13. Heat map generated by the modiﬁed Shepard- mapping method.within the temperature range of the available IRcamera with less reﬂection than other metals suchas Aluminium and Steel. After applying heat for 30seconds, a thermal image was taken for the sheet. reconstructed and has caused hard edges around theThis map has been randomly down-sampled to 1000 location of heat source. This is due to attenuationpoints, that is 1.5% of the total 455 × 147 to be between adjacent points and the fact that some areasused by the MGS to re-generate the total heat map. contain many sensor readings with almost the sameThe mapping service integrates all the knowledge elevation.given by the application domain. Particularly, the Figure 13 shows the map generated by modiﬁedpresence of the obstacle, its position, length, and Shepard-mapping method with knowledge about thestrength. It is assumed here that the obstacle (hole application model. A better approximation to theexcavation) is continuous and the existence of this real surface near the obstacle is observed. Theobstacle between two directly communicating nodes new details included in the domain model removedwill break the wireless links between them. This artifacts from both ends of the obstacle. This is duemeans that the nearest neighbour triangulation RF to the inclusion of the obstacle width in weightingconnectivity map is used as a dimension by the MGS. sensor readings when calculating P which furtherResults and discussion: The nearest neighbour reduces the eﬀect of geographically nearby sensorstriangulation In this experiment, the RF connectiv- that are disconnected from P by the obstacle.ity map is used as one dimension to predict the Conclusion: This experiment shows that theheat map. Figure 11 shows the heat diﬀusion map incorporation of the application domain informationcaptured by the FLIR ThermaCAM P65 IR camera. in the MGS signiﬁcantly improves the mapGiven that the heat is applied at the middle of the production quality.top edge of the brass sheet and the location of theobstacle, by comparing the left side and right sideareas around the heat source, this ﬁgure shows that 10. EXAMPLE APPLICATION OF THEthe existence of the obstacle has strongly reduced the MGStemperature rise in the area on its right side. Figure 12 shows the map generated by the In this subsection, ﬂood management applicationShepard-mapping. Compared with Figure 11, the is considered to demonstrates how MGS generatedobtained map conserves perfectly the global maps can be used in various applications. In this ap-appearance and many of the details of the original plication an elevation data, Figure 14−(a), acquiredmap with 98.5% less data. However, the area by the Shuttle Radar Topography (SRTM) [41] iscontaining the obstacle has not been correctly used. The map is 42.2 × 40.4 kilometres where theWirel. Commun. Mob. Comput. 2010; 00:1–17 © 2010 John Wiley & Sons, Ltd. 13DOI: 10.1002/wcmPrepared using wcmauth.cls
- 14. A map generation service for WSNs M. Hammoudeh et al. height is depicted as brightness. This map clariﬁes the continuity of the drainage network, which can be used for ﬂoodplain zoning (a procedure used to identify areas of varying ﬂood hazard). Consider a WSN that is deployed for ﬂood monitoring in which nodes equipped with sensors to measure the river and weather conditions. Measured information can be integrated into maps that can be overlaid over each other to provide more meaning of the mapped data. The MGS can be used to generate high-risk ﬂoodplain map (Figure 14−(b)) that can be used to forecast, notify, plan, and manage ﬂoods. Such maps can be used to answer questions related to the above tasks; for instance, what is the deepest river channel (a)Gotel Mountains, height as brightness. with the fasted water ﬂow? The response generated by the MGS based on simulated data (Figure 14−(c)) can be used to identify locations of where to install portable inﬂatable tubes, e.g. at sharp corners, and to inform emergency services. To measure the cost of generating the ﬂood hazard map we used the same experimental setup as above with MuMHR [42] as a commmunication protocol. We instantiate unit transmission cost on a communication link between two nodes using the ﬁrst order radio model values presented in [43]. The typical energy consumption per bit on the transmitter and receiver circuit is set to 40nJ/bit. We simulate 16 diﬀerent network (b) Areas of ﬂood hazard. topologies with various node densities because the network topologies and nodes density will aﬀect the behaviour of diﬀerent map generation algorithms. We also performed the same experiment with TLI, the generated maps are similar and omitted here. Figure 14−(d) shows that the MGS with Shepard interpolation expends less energy in map generation than with TLI. This can be attributed to the localised behaviour of the underlying Shepard method, which reduces energy consumed by the MGS for propagating mapping information to the central location. The MGS can be used to inform other network services such as routing or calibration services. For (c) The deepest river channel identiﬁed by the example, a node can estimate its reading from MGS. its neighbours readings using the MGS. If the 1.1 diﬀerence between the estimated value and the 1 actual reading exceeds a certain threshold then the 0.9 0.8 node initiates the calibration service and indicates that the sensor readings are erroneous. This example Cost (mJ) 0.7 0.6 can be generalised to detect anomalies in the 0.5 network. Another example is when the MGS is used 0.4 TLI MGS by the routing service to decide whether to forward 0.3 0.2 a reading or not by examining the impact of the new 25 100 175 250 325 400 reading on the local map. Number of nodes (d) Cost of generating map in mJ Figure 14. MGS application.14 Wirel. Commun. Mob. Comput. 2010; 00:1–17 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm Prepared using wcmauth.cls
- 15. M. Hammoudeh et al. A map generation service for WSNs11. CONCLUSIONThis paper presented a new WSN service, theMGS. Mapping was deﬁned as a problem ofinterpolation from sparse and irregular points.Shepard interpolation method was identiﬁed andemppirically proved to work well with the constriantsimposed by WSNs. Shepard method is intuitivelyunderstandable and provides a large variety ofpossible customisations to suit particular purposes.Also, Shepard was found to be easily modiﬁed toincorporate diﬀerent external conditions that mighthave an impact on the mapping results, such asbarriers. Furthermore, this method is simple toimplement with fast computation and modellingtime [44, 45] and it is easy to generalise to more thantwo independent sensed modalities. This method canbe localised, which is an advantage for large andfrequently changing data sets, making it suitable forWSNs applications. Local map generation reducesdata communication across the network and evadesthe computation of the complete network map whenone or more observations are changed. Finally,there are few parameter decisions and it makesonly one assumption which gives it the advantageover other methods [44]. Shepard was modiﬁed toutilise the special characteristics of the applicationdomain to render visualisations in a map formatthat are a precise reﬂection of the concrete reality.This modiﬁed service is suitable for visualising anarbitrary number of sense modalities. It is capableof visualising from multiple independent types ofthe sense data to overcome the limitations ofgenerating visualisations from a single type of a sensemodality. Experimental evaluation demonstrates theusefulness of the modiﬁed Shepard mapping service.Future work will investigate how this higher-levelinformation-rich representations can be used forinforming other network services besides the deliveryof ﬁeld information visualisations.Wirel. Commun. Mob. Comput. 2010; 00:1–17 © 2010 John Wiley & Sons, Ltd. 15DOI: 10.1002/wcmPrepared using wcmauth.cls
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