Interpolation Techniques for Building a Continuous Map from Discrete Wireless Sensor Network Data

WIRELESS COMMUNICATIONS AND MOBILE COMPUTING

Wirel. Commun. Mob. Comput. 2010; 00:1–17

DOI: 10.1002/wcm

RESEARCH ARTICLE

Interpolation Techniques for Building a Continuous Map fromDiscrete Wireless Sensor Network DataMohammad Hammoudeh1∗ Robert Newman2, Christopher Dennett2, and Sarah Mount2

1School of Computing, Manchester Metropolitan University, Manchester, UK2School of Computing and IT, University of Wolverhampton, Wolverhampton, UK

ABSTRACT

Wireless 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 themapping accuracy. Empirical analysis has shown that the MGS is an accurate, flexible and efficient method.Copyright© 2010 John Wiley & Sons, Ltd.

KEYWORDS

Wireless Sensor Networks; Services; Visualisation; Information Extraction; Interpolation

∗Correspondence

School of Computing, Manchester Metropolitan University, Manchester, UK. Email: [email protected]

1. INTRODUCTION

With the increase in applications of WSNs, infor-mation extraction and visualisation have become akey issue to develop and operate these networks.WSNs typically gather data at a discrete number oflocations. By bestowing the ability to predict inter-node values upon the network, it is proposed that itwill become possible to build applications that areunaware of the concrete reality of sparse data.

Not all information that is collected from aWSN comes ready to use. Often, WSNs field datacollection takes the form of single points that need tobe processed to get a continuous data presentation.Interpolation describes this process of taking manysingle points and building a complete surface, theinter-node gaps being filled based on the spatialstatistics of the observation points. Interpolatingthese points will produce more useful information forthe end user such as maps related to water chemical

content. The ability to interpolate point informationis necessary for carrying out mapping tasks.

The problem of map generation is essentially aproblem of interpolation from sparse and irregularpoints. This interpolation capability is realised as aservice of the network. In this paper, one particularinterpolation approach, Shepard interpolation [1],is examined and shown to be suitable for theconstraints imposed by the nature of WSNs.Visual aspects, sensitivity to parameters, and timingrequirements were used to test the characteristics ofthis method

The rest of the paper is organised as follows.Section 2 explains why map is a suitable discretedata visualisation format. Sections 3 and 4 providea brief description of map generation algorithmsand mapping applications in the literature. Section5 defines the problem on map generation. Section6 defines Shepard interpolation method. Sections 7and 8 describe the modified Shepard map generation.

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A map generation service for WSNs M. Hammoudeh et al.

Evaluation of the MGS is presented in Section 9. Thepaper concludes in Section 10.

2. SENSE DATA VISUALISATION:MAPS

The integration of data visualisation tools and theraw data sent by the WSN makes the sensor networksystem useful to different potential users. Visualformats, such as maps, can be easily understoodby people possibly from different communities,thus allowing them to derive conclusions basedon substantial understanding of the available data.Maps are effective to understand the spatialdistribution of environmental features since humanscan use their natural interpretation capabilities tounderstand colours, patterns, and spatial relevance.A map is a visual representation of an area, althoughmost commonly used to depict geography, maps mayrepresent any space without regard to context orscale such as weather data mapping [2]. However,a map could be overlaid over a geographic map toenable observation of the data in a real-world map.

In a WSN, a map may be used as an informationrepresentation and extraction tool in which visualfeatures such as symbols and colours are used tocode different attributes of the data to providethe information for end users to analyse andexamine. These unique visualisation and analysisbenefits offered by maps make them more visuallycommunicative, they imply the distributions andstates; provide information about spatial patterns;and imply the association of diverse phenomena.

Maps can be either static or dynamic andallow data representation on 2D or 3D space.They allow the user to infer the actual sizes anddistance between objects. The users can zoomin or zoom out respectively meaning showingmore or less details. Furthermore, maps allowthe extraction of information that can not beobtained by looking at sensor readings separatelyand are more efficient to compute in both timeand energy. For instance, maps may capture trendsor correlations among sense data. Where thereis no operating sensor, predictions can be madeusing these spatial and temporal correlations amongsensor readings. Finally, a map provides a higher-level information-rich representation which can besuitable for informing other network services and thedelivery of field information visualisation.

3. A SURVEY ON ALGORITHMS FORMAP GENERATION

The following section provides a brief on relatedwork in map generation methods. Map generationtechniques have previously been explored in thecontext of WSNs [3, 4]. Chang et al. [4] implementedan algorithm to estimate sensor nodes faultybehaviour on top of a cluster-based network. Thisapproach is based on Bayesian Belief Networks(BBNs) which make it problematic to compute allthe probabilities and the revised probabilities once anew sensor reading is received. In dense multi-modalWSNs, the number of dependencies increases rapidlyand probabilities computation becomes an NP-hardproblem. This approach also lacks precision whenupdating the fault rate table since it is based on apredefined threshold value.

Event detection based on matching the contourmaps of in-network data distribution has been showneffective for event detection in WSNs [3]. Mapconstruction starts from each node generating apartial map of its own. When a node forwardsdata for its neighbours, it adds each contour regionin these partial maps with its own. This processis repeated until the final map is generated. Thisapproach works well with grid network topologiesand less well with random topologies. When a gridis overlaid on top of a random topology some cellsin the grid may be empty. These empty cells willnot participate in the final map construction. Hence,the final map will not cover the entire network area.This makes the scheme sensitive and unsuitable forrandom WSNs deployments. Furthermore, the loss ofany partial maps will result in an incomplete networkmap.

In both [3] and [4] the sink node is required toknow the location and the ID of all nodes in thenetwork. Furthermore, the work in both papers isapplication-dependent and requires major lower levelmodifications if the application is to change. In [3],the assumptions made on the network topologyand the way the grid is formed are not efficientand may dissipate the energy savings achieved bythe in-network map construction. In [4], it is notclear how the hierarchy is built. Besides, it is onlysuitable for small size networks due to the single hopcommunication scheme.

DIMENSIONS [5] made the case for a large-scaledistributed multi-resolution storage system thatprovides a unified view of data handling in WSNsincorporating long-term storage, multi-resolutiondata access and spatio-temporal correlations insensor data. This work is related to ours, butdifferent in focus at both the system architectureand coding level. It outlines an approach forrelatively power-rich devices, focused on encoding

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DOI: 10.1002/wcm

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M. Hammoudeh et al. A map generation service for WSNs

regularly-gridded, spatial wavelets over time series.By contrast, we focus on highly resource constraineddevices, and integrate different network services withthe MGS. Our work is also focused on spatiallydeployed networks and is independent on a particularrouting algorithm.

In centralised map generation approaches, deliv-ering all network sensory data back to the sinkincurs heavy transmission traffic. Several aggregationbased map generation methods have been proposedto address this problem [6, 3, 7, 8]. However, aggre-gation based methods can not further improve thescalability of the network as all sensors are requiredto report to the sink. Moreover, the aggregationprocess increases the computation overhead on theintermediate nodes. To address the inherent limita-tions of aggregation based methods, [9] proposed amethod called Iso-Map that intelligently selects asmall portion of the nodes, isoline nodes, to generateand report mapping data to reduce the networktraffic and computation overhead. Partial utilisationof the network information leads to a decrease inthe mapping fidelity and isoline nodes will sufferfrom 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 difficultto construct contour maps based only on isolinenodes. The positions of isoline nodes provide onlydiscrete iso-positions, which does not define how todeduce how the isolines pass through these positions.

To conclude, mapping is often employed in WSNapplications but as yet there is no clear definition(or published work towards) a localised MGS thatwould aid the development of more sophisticatedapplications. The development and analysis of sucha service is the key novel contribution of the workproposed here.

4. MAPPING APPLICATIONS IN THELITERATURE

Within the WSN field, mapping applications foundin the literature are ultimately concerned with theproblem of mapping measurements onto a modelof the environment. Estrin et al.[10] proposed theconstruction of isobar maps in sensor networksand showed how in-network merging of isobarscould help reduce the amount of communication.Furthermore, [6] proposed an efficient data-collectionscheme, and the building of contour maps, forevent monitoring and network-wide diagnosis, incentralised networks. Solutions such as distributedmapping have been proposed to the generalmapping domain. However, many solutions are

limited to particular applications and constrainedwith unreliable assumptions. The grid alignment ofsensors in [10], for example, is one such assumption.

In the wider literature, mapping was soughtas a useful tool in respect to network diagnosisand monitoring [6], power management [11], andjammed-area detections [12]. For instance, contourmaps were found to be an effective solution tothe pattern matching problem that works forlimited resource networks [3]. These are examples ofspecific instances of the mapping problem and, assuch, motivate the development of a generic MGS,furthering the area of research by moving beyondthe limitations of the centralised approaches.

A service oriented approach has special properties.It is made up of components and interconnectionsthat stress interoperability and transparency. Ser-vices and service-oriented approaches address de-signing and building systems using heterogeneousnetwork software components. This allows the devel-opment of a MGS that works with existing networkcomponents, e.g. routing protocols, and resourceswithout adding extra overhead on the network.

5. UNDERSTANDING THE PROBLEMOF MAP GENERATION FROMSPARSE DATA

Given a set of known data points representing thenodes’ perception of a given measurable parameterof the phenomenon, what is the most likely completeand continuous map of that parameter? In thefield of computer graphics, this problem is knownas an unorganised points problem, or a cloud ofpoints problem. That is, since the position of thepoints in xy is assumed to be known, the thirdparameter can be thought of as height and surfacereconstruction algorithms can be applied. Simplealgorithms use the point cloud as vertices in thereconstructed surface. These are not difficult tocalculate, but can be inefficient if the point cloudis not evenly distributed, or is dense in areas of littlegeometric variation.

Approximation, or iterative fitting algorithmsdefine a new surface that is iteratively shaped to fitthe point cloud. Although approximation algorithmscan be more complex, the positions of vertices arenot bound to the positions of points from the cloud.For applications in WSNs, this means that we candefine a mesh density different to the number ofsensor nodes, and produce a mesh that makes moreefficient use of the vertices. Self organising maps areone of the algorithms that can be used for surfacereconstruction [13]. This method uses a fixed numberof vertices that move towards the known data.

Wirel. Commun. Mob. Comput. 2010; 00:1–17© 2010 John Wiley & Sons, Ltd. 3DOI: 10.1002/wcm



Note that surface reconstruction on typical non-overlapping terrains is equivalent to sparse-datainterpolation. This kind of geometric parameterinterpolation has been shown to work well forreconstructing underlying geography when the entirenetwork has been queried [14]. However, It doesnot extend well to variable surfaces or overlappinglocal mapping, since it requires a complete dataset to define the surface. A more general methodis interpolation by inverse distance and, specifically,Shepard interpolation [1] which improves on it.

6. SHEPARD INTERPOLATION

Shepard Interpolation is an inverse distance weightedscattered data interpolation algorithm. It is widelyused in practise and has also been shown towork well with noisy data [15]. Shepard defined acontinuous function where the weighted average ofdata is inversely proportional to the distance fromthe interpolated location. The algorithm explicitlyimplies that the further away a point is from aninterpolated location, P , the less effect it will haveon the interpolated value.

Known points, Di, are weighted during interpo-lation relative to their distance from P . Weightingis assigned to data through the use of a weightingpower, which controls how the weighting factors dropoff as the distance from P increases. The greater theweighting power, the less effect far points have on theinterpolation result. As the power increases, Shep-ard interpolation approaches the nearest neighbourinterpolation method [16] where the interpolatedvalue simply takes on the value of the closest samplepoint [16, 15].

Many modifications to the original Shepardalgorithm have been proposed in the literature [17,18, 19, 20], however, most of these methods aredesigned for computer graphics and image processingfields. These algorithms usually trade accuracywith computation complexity. Nevertheless, whenapplying interpolation to WSN applications it isdesirable to keep the interpolation simple to reducethe amount of processing as well as communicatedinformation across the network. Therefore, we shalluse the original method with the modificationsproposed by Shepard that further reduce the amountof processing and data communication to achievemore energy savings using the limited availablebandwidth. These modifications are described insubsection 6.1.

Shepard’s expression for globally modelling asurface is:

f1(P ) =

N∑i=1

[(di)

−u × zi]

N∑i=1

(di)−u

if di 6= 0 ∀Di(u > 0)

zi if di = 0

(1)where di is the distance from P to D numbered i inthe N known points set and zi is the known value atDi. The exponent u is used to control the smoothnessof the interpolation. As P approaches a data pointDi, di tends to zero and the ith terms in boththe numerator and denominator exceeds all boundswhile other terms remain bounded. Therefore, thelimP→Dif1(P ) = zi is as desired and the functionf1(P ) is continuously differentiable even at thejunctions of local functions.

6.1. Global Shepard Algorithm Shortcomings andSolutions

Shepard’s interpolation suffers from several short-comings imposed by the fact that each sample pointhas a radially symmetric influence despite the natureof the underlying data [21]. Among the well knownartifacts are cusps, corners, and flat spots at the datapoints, as well as the excessive influence of pointsthat are far away [22]. Further shortcomings includethat the global function necessitates all weights tobe recomputed if any points are added, removed,or modified. In WSNs this is impractical due tothe network dynamics such as: node failures, nodemobility, or deployment of new nodes. Shepard hasidentified three main shortcomings of his method andproposed modifications to deal with them as follows:

Building the Support SetWe define the support set as the set containingall points used to calculate P . The global methodhas a linear running-time O(N), which makesit impractical and inefficient especially when thenumber of nodes is large. To overcome this, alocal Shepard algorithm was defined. This algorithmeliminates distant points from the calculation of anyinterpolated value since only nearby data points havesignificant influence. To select nearby nodes, Sheparddefined two criteria:

1. Arbitrary distance criterion: All data pointswithin radius r of the point P are includedin computation. This is computationally easybut allows the possibility that there are nodata points or a sufficiently large number ofdata points within the radius r. A collection ofpoints, Cp, within a search radius r is defined


DOI: 10.1002/wcm



as Cp = Di|di ≤ r and n(Cp) is the totalnumber of data points in Cp.

2. Arbitrary number criterion: Only the closest ndata points are considered in the computationof any interpolated value. This approachignores the relative location and spacing ofthe points and requires deep searching andcomplex ranking procedure for data points. Inaddition, it assumes that a single number, n,of interpolating points was optimal. If N isthe total number of data points, then, anew collection of data points, CnP , is definedas CnP = Di1 , Di2 ..., Din where (n ≤ N) andthe subscripts ij are defined such that 0 ≤di1 ≤ di2 ≤ ... ≤ diN .

Shepard has chosen a mix of the two criteria, whichcombined their advantages. An initial radius r isdefined depending on the overall density of datapoints such that seven data points are included onaverage in a circle of radius r. r is written as follows:

πr2 =7A

N(2)

were A is the area of the largest polygon enclosed bythe data points.A function si = s(di) is defined to guarantee thelocal behaviour of the interpolating algorithm bycalculating a surface model for any d ≤ r, and whichweights the points at r ≤ r

3more heavily:

s(d) =

1/d if 0 < d ≤ r′

3274r2

( dr′ − 1)2 if r′

3< d ≤ r′

0 if r′ < d

(3)

where r′ is a radius of influence about P chosenlarge enough to include n points and defined asr′(Cnp ) = mindij |Dij /∈ CnP = din+1 . In order thatthe interpolation algorithm works realistically, ifthe data points were girded, a minimum of fourdata points was chosen. A maximum of ten wasestablished to limit the complexity and amount ofcomputation required [1]. Thus C′p and r′p are definedas follow:

C′p =

C4p if 0 ≤ n(Cp) ≤ 4

Cp if 4 < n(Cp) ≤ 10

C10p if 10 < n(Cp)

and

r′p =

r′(C4

p) if n(Cp) ≤ 4

r if 4 < n(Cp) ≤ 10

r′(C10p ) if 10 < n(Cp)

The resulting function f2(P ), has similar behaviourto the original function but it is capable ofhandling much larger data sets and it is muchmore suitable for parallel implementations. The

interpolation function f2(P ) is given by:

f2(P ) =

∑Di∈C′

(si)2zi∑

Di∈C′(si)

2if di 6= 0 ∀Di

zi if di = 0

(4)

The function f2(P ) requires less physical resources,in terms of both computation and memory, thanthe previous functions. The running-time is reducedto O(C′P). Consequently, the amount of memory usedby the algorithm on data set of size N is reducedto inputs of C′P. This reduction in computation andmemory cost is invaluable in large scale WSNs whichmust be capable of in-network processing at all levels,including the application level.

Since topologies in WSNs changes frequently,the MGS should be topology-independent anddecentralised. That is, each node or subsystem, e.g.cluster, uses only local information when makingmapping decisions. MGS can be implemented oneach node with varying the size of the support setand the way it is used. For instance, each node canuse the Cp to build a set of neighbours to collaboratewith in building a local map for their vicinity. Thislocal map can be used to respond to user quires,to update the global map maintained at a clusterhead (partial) or a sink (global), used as a localaccuracy model, etc. The local map can be used tocalculate whether a new reading should be forwardedto upper nodes in the hierarchy by calculating theimpact of the new reading on the local map. Atcluster heads and the sink, the support set containsall nodes within the cluster and all nodes in thenetwork respectively. With hierarchical approaches,the global map can be built and updated by clusterheads. MGS deals efficiently with the addition ordeletion of nodes due to the local mapping, i.e. anytopological changes will be dealt with locally withoutrecalculating the global map.

Including DirectionThe current method ignores the direction factorin computing the weightings. To make themethod intuitively reasonable, Shepard included thedirection in computing interpolated values. A newdirectional weighting for each data point Di closeto P is defined by:

ti =∑Dj∈C′

sj [1− cos(DiPDj)]/∑Dj∈C′

sj (5)

were the cos(DiPDj) is defined as: [(x− xi)(x−xj) + (y − yi)(y − yj)]/didj . The appropriateness ofthe cosine function and computation ease makesit a good measure of direction. The function sjis included in the new function to preserve the




Figure 1. Fire spreading and wind direction.

original weighting assumption. Within the directionconsidered, a new weighting function wi = (si)

2 ×(1 + ti) is defined and the final interpolation functionis defined as:

f3(P ) =

∑Di∈C′

wizi/∑Di∈C′

wi if di 6= 0 ∀Di

zi if di = 0

(6)This modification is useful for mapping modalities

where the direction is vital, for instance winddirection in forest fire monitoring applications. Thetwo configurations in Figure 1, for example, wouldyield identical interpolated values at location 2.However, if the algorithm to be intuitively practical,the value at location 2 in the top configurationshould be closer to the value at sensor 1 than in thelower configuration, because wind direction shouldbe expected to screen the effect of more distant point.

Determining SlopeThe arbitrary and undesirable zero gradient at everypoint Di still exists on the f3(P ), generated surface.If di is very small, si will equal d−1

i and wi will varyas d−2

i . To correct this, weighted averages of divideddifferences of zi about Di, Ai and Bi, were addedto sufficiently nearby data points to achieve partialderivatives at Di. Constants Ai and Bi represent theslope in the x and y directions at each data point Di,Ai and Bi are defined as:

Ai =

∑Dj∈C

′′i

wj(zj − zi)(xj − xi)

(d[Dj , Di])2∑Dj∈C

′′i

wj(7)

and

Bi =

∑Dj∈C

′′i

wj(zj − zi)(yj − yi)

(d[Dj , Di])2∑Dj∈C

′′i

wj(8)

where C′′i = C′Di

− Di.

A new parameter v is defined with the distancedimension to bound the maximum effect the slopeterms may have on the final interpolated value. Fora contour mapping application, v can be defined as:

v =0.1[maxzi −minzi]√

max(A2i +B2

i )(9)

An increment ∆zi = Ai(x−xi)+Bi(y−yi)v

v+di

is computed

for each Di ∈ C′P as a function of P to include theeffect of the slope in interpolating values at P . Thusthe latest version of the interpolation function is:

f4(P ) =

∑Di∈C′

wi(zi + ∆zi)∑Di∈C′

wiif di 6= 0 ∀Di ∈ C′

zi if di = 0

(10)In function f3(P ), the interpolated surface has azero gradient at every Di. This modification isvaluable to WSNs applications since it reduces noiseand redundant small details in the image that areperceptually unnoticeable to human eyes.

7. MAPPING IN HIGHERDIMINESIONAL-SPACE

This section defines a new metric for distance,suitable for higher dimensions (multi-modal sensing),in which the concept of closeness is described interms of relationships between sets rather than interms of the Euclidean distance between points.Using this distance metric, a new generalisedmapping function f , that is suitable for an arbitrarynumber of sensed modalities, is defined.

In higher diminsional-space mapping every set Sicorresponds to an input variable i.e. a sense modality,called i, and referred to as a dimension. The powerof such a generalisation can be seen when we includethe time variable as one dimension. The spatial mapgeneration problem can be stated as follows:Given a set of randomly distributed data points

xi ∈ Ω, i ∈ [1, N ] , Ω ⊂ Rn (11)

with function values yi ∈ R, and i ∈ [1, N ] we requirea continuous function f : Ω −→ R to interpolateunknown intermediate points such that

f (xi) = yi where i ∈ [1, N ] (12)

We refer to xi as the observation points. Theinteger n is the number of dimensions and Ωis a suitable domain containing the observationpoints. When rewriting this definition in terms ofrelationships between sets we get the following:


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Lemma 7.1 Given N ordered pairs of separated setsSi ⊂ Ω with continuous functions

fi : Si −→ R, i ∈ [1, n] (13)

we require a multivariate continuous function f :Ω −→ R, defined in the domain Ω = S1 ∪ S2 ∪ ... ∪Sn−1 ∪ Sn of the n-dimensional Euclidean spacewhere

f (xi) = fi (xi) ∀xi ∈ Si where i ∈ [1, n] (14)

Proof of Lemma 7.1 The existence of the globalcontinuous function f can be verified as follows.First, the data set is defined as

S =

v(0)1 , v

(1)1 , · · · , v

(n′)1

v(0)2 , v

(1)2 , · · · , v

(n′)2

......

...

v(0)n , v

(1)n , · · · , v

(n′)n

(15)

where n′ ≤ N and vi = (xi, ri) , i ∈ [1, N ] and ri isa reading value of some distinctive modality (e.g.temperature). Let Φ be a topological space on S andthere exists open subsets Si, i ∈ [1, n]

S1 =v(0)1 , v

(1)1 , · · · , v(n

′)1

S2 =

v(0)2 , v

(1)2 , · · · , v(n

′)2

...

Sn =v(0)n , v

(1)n , · · · , v(n

′)n

(16)

which are topological subspaces of Φ such that

ΦSi = Si ∩ U |U ∈ Φ (17)

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:

1. Let f : S1 ∪ S2 ∪ ... ∪ Sn−1 ∪ Sn be a mappingdefined on the union of subsets Si, i ∈ [1, N ]such that the restriction mappings f|Si

arecontinuous. If subsets Si are open subspacesof S or weakly separated, then there exist afunction f that is continuous over S (provedby [23]).

2. If f : Ω→ Ψ is continuous, then the restrictionto Si, i ∈ [1, N ] is continuous (property,see [24]). The restriction of a continuous globalmapping function to a smaller local set, Si,is still continuous. The local set follows sinceopen sets in the subspace topology are formedfrom open sets in the topology of the wholespace.

Using the point to set distance generalisation, thefunction f can be determined as a natural generali-sation of methods developed for approximating uni-variate functions. Well-known uni-variate interpola-tion formulas are extended to the multivariate case

by using Geometric Algebra (GA) in a special waywhile using a point to set distance metric. Burleyet. al [25] discuss the usefulness of GA for adaptinguni-variate numerical methods to multivariate datausing no additional mathematical derivation. Theirwork was motivated by the fact that it is possible todefine GAs over an arbitrary number of geometricdimensions and that it is therefore theoreticallypossible to work with any number of dimensions.This is done simply by replacing the algebra of thereal numbers by that of the GA. We apply the ideasin [25] to find a multivariate analogue of uni-variateinterpolation functions. To show how this approachworks, an example of Shepard interpolation of thisform is given below:

Given a set of n distinct points X =x0, x1, ..., xn ⊂ Rs, the classical Shepard’sinterpolation function is defined by

(Son,µf

)(x) =

n∑k=0

wk (x) f (xk) (18)

and

wk (x) =|x− xk|−µn∑k=0

|x− xk|−µ(19)

where |.| denotes the Euclidean norm in Rs.In the uni-variate case (s = 1) and S0

n,2f . Thebasic properties of S0

n,µf are:

1.(S0n,µf

)(xi) = f (xi) , i = 0, ..., n;

2. doe(S0n,µf

)= 0, where doe is an abbreviation

of degree of exactness.

8. APPLICATION-BASED LOCAL MAPGENERATION

Because the accuracy level of generated mapsmay vary significantly depending on the specificapplication, e.g. existence of barriers, in this sectionwe modify the distance metric to include theknowledge known about the application domain.The proposed metric attempts to balance the sizeof the support set with the interpolation algorithmcomputation complexity as well as interpolationaccuracy.

We define the term scale for determining theweight of every given dimension with respect to Pbased on a combined Euclidean distance criteriaas well as information already known about theapplication domain a priori to network deployment.While the term weight is reserved for the relevanceof a data site by calculating the Euclidean distancebetween P and Di.

We define a new scale-based weighting metric,mP ,which includes application domain information. The




support set is Ci, where Ci ⊆ Si, for each dimensioncontains the nearest points for P using mP .Symbolically, Ci is calculated as

Ci = L (d (P,Ej) , δ (Si)) ∀Ej ∈ Si (20)

where i ∈ [1, n], L is a local model that selects thesupport set for calculating P , d is an Euclideandistance function, Ej is an observation point in thedimension Si, and δ(Si) a set of parameters fordimension Si. These parameters are usually a set ofrelationships between different dimensions or otherapplication domain characteristics such as obstacles.In uni-dimensional distance weighting methods, theweight, ω can be calculated as follows

ω = d (P,Ej) , Ej ∈ Si (21)

This function can be extended to multi-dimensionaldistance weighting systems as follows

ω = K (P, Si) , i ∈ [0, n] (22)

where K (P, Si) is the distance from P to data set Siand n is the number of dimensions in the system.Equation 22 can now be extended to include thedomain model parameters of arbitrary dimensionalsystem. Then the dimension-based scaling metric canbe defined as

mP =∑i

L (K(P, Si), δ (Si)) i ∈ [0, n] & Si 6= CP

(23)where CP is the dimension containing P .

8.1. Example: Mapping Surfaces with Barriers

Shepard interpolation is based on the intuitiveassumption that there is a logical relationshipbetween adjacent points. This assumption is,however, violated if some barrier, such as a river,ruptures the continuity of the surface. The effectof physical barriers can be simulated easily due tothe distance-dependent interpolation by includingvirtual barriers. The user may specify discontinuitiesin the metric space in which di is calculated usinga different selection set of nearby data points anddifferent weightings and slopes are being calculated.Given a detour of length b[P,Di] perpendicular tothe line between P and Di, Shepard interpolationdefines the effective distance to travel between thetwo points as:

d′i = (d[P,Di])2 + (b[P,Di])

212 (24)

where b[P,Di] is the strength of the barrier. Whena barrier exists, d′i replace di in all calculations.Whereas, if there is no barrier between Di and P ,di = di and b[P,Di] = 0. The effective distance is

Table I. Linux-class sensor node hardware platforms.

MicroServer Gumsense

CPU au1550 MarvellPXA270

Clock speed 400MHz 100 to 600MHz

CPU powerconsumption 0.5W 72µW

Memory 128MB 128MB

Flash ROM 128MB 32MB

discontinuous as P crosses a barrier which resultdiscontinuous interpolated surface at an obstacle.The inclusion of barriers in the interpolation willresult in the selection of a different set of nearbydata points, weightings, and slopes.

9. SHEPARD INTERPOLATIONANALYSIS

In this section the effectiveness of the Shepard inter-polation algorithm is verified and its characteristicsare studied quantitatively and qualitatively. TheShepard interpolation performance was comparedwith that of Triangulation with Linear Interpolationalgorithm (TLI) [16].

TLI was chosen for comparison because it is anexact interpolator which uses the optimal Delaunaytriangulation. Delaunay triangulation is used exten-sively in the field of WSN. Uses include: adaptablenetwork deployment [26], network coverage [27],locating and bypassing routing holes [28], distributedarea computation [29], position-aware routing [30],and spatial clustering [31]. Furthermore, TLI iswidely referred to in the literature including in theimage processing field [32] and reported to be oneof the simplest and most efficient algorithms with agood running time [16, 33].

9.1. Hardware Requirements

The following experiments target WSNs built fromLinux-class devices that have higher storage andprocessing capabilities. The choice of less constrainedhardware platform was for two reasons:

1. Distributed mapping is desirable but intro-duces a considerable storage and computationcomplexity on sensing devices when consider-ing current sensor node capabilities.

2. In-network visualisation has requirements typ-ical of any non-trivial processing. For example,the MICA/MICA2 mote [34] microcontrollerhas no support for floating point arithmetic orinteger multiplications.

The Gumsense [35] and EmStar MicroServers [36,37], amongst other Linux boxes are example


DOI: 10.1002/wcm



hardware platforms that are available in the marketand are capable of running the MGS. Table I showsthe specifications of these hardware platforms.

In an extreme situation, assume that there isa 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.

9.2. TLI Algorithm Details

This method connects data points to form trianglesthat do not intersect with each other. The resultof this process is a patchwork of triangular facesover the extent of the grid. The slope and elevationof the triangle is determined by the original datapoints defining the triangle and all nodes withinthe triangular plane are defined by the triangularsurface. Since the triangles are determined by theoriginal data, the data must be sampled at a highrate. TLI is fast with all data sets but it is noteffective with few points [16]. One advantage oftriangulation is that, with enough data, triangulationcan preserve break lines defined in a data file. Forexample, if a fault is delimited by enough data pointson both sides of the fault line, the surface generatedby triangulation will show the discontinuity [16].

9.3. Comparison Metrics

To determine the accuracy of the interpolationquantitatively, the skewness and kurtosis of ahigh resolution source data and the result of theinterpolation using a subset of that data has beenchosen as a measure of the surface deviation.The kurtosis is a measure of the peakedness of areal-valued random variable where a high kurtosisdistribution has a sharper peak and fatter tails andlow kurtosis distribution has a more rounded peakwith wider shoulders [38]. Skewness is a measureof the asymmetry of the probability distribution ofa real-valued random variable [38]. A distributioncould have two kinds of skewness; positive skew ornegative skew, where the mass of the distribution isconcentrated on the left of the figure or the right ofthe figure respectively. Further qualitative accuracyassessments are done using empirical peak profilingto obtain peak information for studying the twoalgorithms local behaviour.

Visually, we use 2D and 3D height maps todetermine the global accuracy of the interpolation

method. The 2D maps depict the height where thepixel intensities depict depth values.

9.4. Experimental Setup

Figure 2. Grand Canyon height map

These experiments made use of the Grand Canyonheight map [39]. The studied region is 15360m2,with heights ranging from 165m to 284m abovesea level. This map was sampled to 65536points. The sensor nodes were randomly distributedover the sensing field, i.e. the height map, atposition (xi, yj), where the pixel intensities depictaltitude values. The height map was chosen becauseusing numerous wide-distributed height points hasbeen an important topic in the field of spatialinformation [16]. Furthermore, the height is a staticmeasure which makes it suitable for the evaluationof various interpolation algorithms. The primarypurpose of these experiments is to take spatialinterpolation to calculate the unknown heights byusing the information of neighbouring points andto report results. Shepard Algorithm with all threemodifications is implemented and used in all of thefollowing experiements. Using the same data set, thedifference in quality and accuracy of generated mapsis determined by the interpolation method used.

9.5. Experiment 1: The Effect of Network Density

Aim: The effect of network density on the recon-struction quality of both interpolation algorithms isstudied.Procedure: Interpolation methods are run withdifferent network densities and results are recorded.Results and discussion: Figures 3 and 4 showhow the network density and choice of interpolationalgorithm affect the reconstruction results. It isobserved that higher network densities increase thesmoothness in the re-constructed maps. For instance,contour maps made from high network densityare visibly smoother due to shorter line segmentsbetween data points.

Compared to the actual height map, Figure 2,it is visually evident that both interpolationalgorithms produced acceptable quality 2D and3D maps. However, the reconstruction quality of




TLI with the absence of sufficient data density(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 withthe 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 partof 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. TheShepard 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

Aim: 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 highestpeak 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 fromthe figures that Shepard interpolation appears tobe more visually plausible and has always rendereda smoother surface than TLI. This is because thatTLI 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 getsfarther from the prediction location. While in TLIcurves, all locations within the relevant triangleget the same weight regardless of how far theyare from the prediction location. In local Shepardinterpolation, the enforced restriction of support set

(N) Shepard TLI

50

100

200

300

500

1000

Figure 3. (1): 2D maps produced by Shepard and TLI at various

network densities (N).

to sample points within the neighbourhood reducedthe effect of distant points and produced a finalsurface that is much closer to the original forsome features. At low network densities (e.g. 50)Shepard algorithm yields a surface which is muchmore representative of the original surface than that


DOI: 10.1002/wcm



(N) Shepard TLI

50

100

200

300

500

1000

Figure 4. (2): 3D maps produced by Shepard and TLI at various

network densities (N).

yielded by TLI. This is because TLI requires amedium-to-large number of data points to generateacceptable results. With a highly variable surfacesuch as this, 50 data points are insufficient for TLIto re-create the source data, despite the relativelysmall size of the region in question. At all networkdensities, TLI suffer from edge effects because datasets that contain sparse areas result in distincttriangular facets on a surface plot or contour map.At slightly higher network densities (100 and 200),TLI was less representative of the original data rangethan Shepard because it tends to capture broadregional trends in the surface. TLI does not providethe ‘flatness’ on the edges we would hope for. Asthe network density increases, both interpolationalgorithms give almost equal results with betterperformance from the Shepard algorithm on the basisof adherence to the original surface.

Figure 5. Peak profiling with 100 nodes network density



Table II. Peak profiling statistical measures, where N is the

number of nodes.

N Skewness Kurtusis

Shepard TLI Shepard TLI

10 -0.340 -0.129 2.072 2.085

50 -0.314 -0.105 2.088 1.782

100 -1.698 -0.610 5.372 2.086

200 -1.065 -1.069 2.862 3.380

300 -1.635 -1.613 4.669 4.486

500 -1.117 -1.186 4.211 3.465

1000 -1.249 -0.964 4.211 3.092

Table II presents a summary of statistics forpeak profiling results at various network densities






using Shepard and TLI interpolated maps. Theskewness and kurtosis values measured from theoriginal map are −1.419 and 4.423 respectively. Thevalues recorded in table II shows that as the networkdensity increases, the quality of the produced mapsincrease. The skewness measurements in table IIconfirm the results found in the previous experimentthat Shepard interpolation produced a more accuratepresentation of the interpolated surface at smallerdata sets. However, with bigger data sets (200and 300) the peak deviation difference of Shepardand TLI interpolated surfaces from the originalsurface is minimised. Looking at the kurtusismeasures, Shepard gives more accurate results of howpeaked a distribution is. This success of Shepard wasdue to the use of a subset of the observation pointswhich is more related to the interpolation locationand ignores the effect of distant points.Conclusion: The results of these experimentsshowed that Shepard interpolation was more capableof extracting local features of the interpolated terrainthan TLI. This result makes the Shepard methodmore suitable for implementing the localised MGSin large WSNs.

Original 10 50 100

200 300 500 1000

Figure 10. Contour maps drawn on maps produced by Shepard

interpolation

9.7. Experiment 3: Acceptable Level of DataPresentation

Aim: In this experiment the question of what is anacceptable level of data presentation needed for aparticular application was investigated.Procedure: The required accuracy level of inter-polated maps may vary significantly depending onthe specific application. Contour map was chosenas an application to determine the network densityrequired to reflect some terrain characteristics withparticular levels of accuracy and details. In thisexperiment the effect of the network density on thequality of the contour maps is to be studied.

We restrict the data representation quality experi-ments to Shepard’s algorithm because Experiment 1and Experiment 2 proved that it is more suitablefor spatial data interpolation at lower times andprocessing complexities than TLI.Results and discussion: Figure 10, shows anumber of contour maps overlaying the height mapgenerated using the Shepard interpolation algorithmat various network densities. By comparing contourmaps constructed using low (10, 50, and 100),medium (200 and 300), and high (500 and 100)network densities, it is noticed that the reconstructedmaps are very similar to the original one. Thelowest network density at which the selected peakwas successfully captured is 100, however it didnot precisely identify the size of the peak. At 200nodes network density both the size and the heightof the peak were represented correctly on thecontour map. With higher network densities, contourmaps exactness increased rapidly and the differencebetween the contour maps generated using 200, 300,500 and 1000 is insignificant. Thus a network densityof 200 is enough to give an acceptable presentationof that desired feature.Conclusion: From the contour map and thepeak profiling results, it can be seen that mostof the topographic variations of the terrain wererepresented with accuracy levels enough to supplyinformation on the topography of the land surface


DOI: 10.1002/wcm



at 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-basedMap Generation

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 amapping service. Therefore, to illustrate the benefitsof 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 imagingwithin 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.This map has been randomly down-sampled to 1000points, that is 1.5% of the total 455× 147 to beused by the MGS to re-generate the total heat map.The mapping service integrates all the knowledgegiven by the application domain. Particularly, thepresence of the obstacle, its position, length, andstrength. It is assumed here that the obstacle (holeexcavation) is continuous and the existence of thisobstacle between two directly communicating nodeswill break the wireless links between them. Thismeans that the nearest neighbour triangulation RFconnectivity map is used as a dimension by the MGS.Results and discussion: The nearest neighbourtriangulation In this experiment, the RF connectiv-ity map is used as one dimension to predict theheat map. Figure 11 shows the heat diffusion mapcaptured by the FLIR ThermaCAM P65 IR camera.Given that the heat is applied at the middle of thetop 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 thatthe existence of the obstacle has strongly reduced thetemperature rise in the area on its right side.

Figure 12 shows the map generated by theShepard-mapping. Compared with Figure 11, theobtained map conserves perfectly the globalappearance and many of the details of the originalmap with 98.5% less data. However, the areacontaining the obstacle has not been correctly

Figure 11. Heat diffusion map taken by IR camera.

Figure 12. Heat map generated by the Shepard-mapping

method.

Figure 13. Heat map generated by the modified Shepard-

mapping method.

reconstructed and has caused hard edges around thelocation of heat source. This is due to attenuationbetween adjacent points and the fact that some areascontain many sensor readings with almost the sameelevation.

Figure 13 shows the map generated by modifiedShepard-mapping method with knowledge about theapplication model. A better approximation to thereal surface near the obstacle is observed. Thenew details included in the domain model removedartifacts from both ends of the obstacle. This is dueto the inclusion of the obstacle width in weightingsensor readings when calculating P which furtherreduces the effect of geographically nearby sensorsthat are disconnected from P by the obstacle.Conclusion: This experiment shows that theincorporation of the application domain informationin the MGS significantly improves the mapproduction quality.

10. EXAMPLE APPLICATION OF THEMGS

In this subsection, flood management applicationis considered to demonstrates how MGS generatedmaps can be used in various applications. In this ap-plication an elevation data, Figure 14−(a), acquiredby the Shuttle Radar Topography (SRTM) [41] isused. The map is 42.2× 40.4 kilometres where the




(a)Gotel Mountains, height as brightness.

(b) Areas of flood hazard.

(c) The deepest river channel identified by theMGS.

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

25 100 175 250 325 400

Cos

t (m

J)

Number of nodes

TLIMGS

(d) Cost of generating map in mJ

Figure 14. MGS application.

height is depicted as brightness. This map clarifiesthe continuity of the drainage network, which canbe used for floodplain zoning (a procedure used toidentify areas of varying flood hazard). Consider aWSN that is deployed for flood monitoring in whichnodes equipped with sensors to measure the riverand weather conditions. Measured information canbe integrated into maps that can be overlaid overeach other to provide more meaning of the mappeddata. The MGS can be used to generate high-riskfloodplain map (Figure 14−(b)) that can be used toforecast, notify, plan, and manage floods. Such mapscan be used to answer questions related to the abovetasks; for instance, what is the deepest river channelwith the fasted water flow? The response generatedby the MGS based on simulated data (Figure 14−(c))can be used to identify locations of where to installportable inflatable tubes, e.g. at sharp corners, andto inform emergency services.

To measure the cost of generating the floodhazard map we used the same experimental setupas above with MuMHR [42] as a commmunicationprotocol. We instantiate unit transmission coston a communication link between two nodesusing the first order radio model values presentedin [43]. The typical energy consumption per biton the transmitter and receiver circuit is setto 40nJ/bit. We simulate 16 different networktopologies with various node densities because thenetwork topologies and nodes density will affect thebehaviour of different map generation algorithms.We also performed the same experiment withTLI, the generated maps are similar and omittedhere. Figure 14−(d) shows that the MGS withShepard interpolation expends less energy in mapgeneration than with TLI. This can be attributed tothe localised behaviour of the underlying Shepardmethod, which reduces energy consumed by theMGS for propagating mapping information to thecentral location.

The MGS can be used to inform other networkservices such as routing or calibration services. Forexample, a node can estimate its reading fromits neighbours readings using the MGS. If thedifference between the estimated value and theactual reading exceeds a certain threshold then thenode initiates the calibration service and indicatesthat the sensor readings are erroneous. This examplecan be generalised to detect anomalies in thenetwork. Another example is when the MGS is usedby the routing service to decide whether to forwarda reading or not by examining the impact of the newreading on the local map.


DOI: 10.1002/wcm



11. CONCLUSION

This 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.




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Interpolation Techniques for Building a Continuous Map from Discrete Wireless Sensor Network Data

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