Interpolation techniques for building a continuous map from discrete ...

WIRELESS COMMUNICATIONS AND MOBILE COMPUTING Wirel. Commun. Mob. Comput. (2011) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/wcm.1139

RESEARCH ARTICLE

Interpolation techniques for building a continuous map from discrete wireless sensor network data Mohammad Hammoudeh1* , Robert Newman2 , Christopher Dennett2 and Sarah Mount2 1 2

Department of Computing and Mathematics, Manchester Metropolitan University, Manchester, U.K. School of Computing and Information Technology, University of Wolverhampton, Wolverhampton, U.K.

ABSTRACT 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. Empirical analysis has shown that the map generation service is an accurate, a flexible and an efficient method. Copyright © 2011 John Wiley & Sons, Ltd. KEYWORDS wireless sensor networks; services; visualisation; information extraction; interpolation *Correspondence Mohammad Hammoudeh, Department of Computing and Mathematics, Manchester Metropolitan University, Manchester, U.K. E-mail: [email protected]

1. INTRODUCTION With the increase in applications of wireless sensor networks (WSNs), information extraction and visualisation have become a key issue to develop and to operate in these networks. WSNs typically gather data at a discrete number of locations. By bestowing the ability to predict internode 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. Not all information that is collected from a WSN comes ready to use. Often, WSNs field data collection takes the form of single points that need to be processed to have a continuous data presentation. Interpolation describes this process of taking many single points and building a complete surface, the inter-node gaps being filled based on the spatial statistics of the observation points. Interpolating these points will produce more useful information such as maps related to water chemical content for the end-user. The ability to interpolate

Copyright © 2011 John Wiley & Sons, Ltd.

point information is necessary for carrying out mapping tasks. The problem of map generation is essentially a problem of interpolation from sparse and irregular points. This interpolation capability is realised as a service of the network. In this paper, one particular interpolation approach, Shepard interpolation [1], is examined and shown to be suitable for the constraints imposed by the nature of WSNs. Visual aspects, sensitivity to parameters and timing requirements were used to test the characteristics of this method. The rest of the paper is organised as follows. Section 2 explains why map is a suitable discrete data visualisation format. Sections 3 and 4 provide a brief description of map generation algorithms and mapping applications in the literature, respectively. Section 5 defines the problem on map generation. Section 6 defines Shepard interpolation method. Sections 7 and 8 describe the modified Shepard map generation. Evaluation of the map generation service (MGS) is presented in Section 9. Example application of

A map generation service for WSNs

MGS is discussed in Section 10. The paper concludes in Section 11.

2. SENSE DATA VISUALISATION: MAPS The integration of data visualisation tools and the raw data sent by the WSN make the sensor network system useful to different potential users. Visual formats, such as maps, can be easily understood by people possibly from different communities, thus allowing them to derive conclusions based on substantial understanding of the available data. Maps are effective to understand the spatial distribution of environmental features because humans can use their natural interpretation capabilities to understand colours, patterns and spatial relevance. A map is a visual representation of an area. Although most commonly used to depict geography, maps may represent any space without regard to context or scale such as weather data mapping [2]. However, a map could be overlaid over a geographic map to enable observation of the data in a real world map. In a WSN, a map may be used as an information representation and extraction tool in which visual features such as symbols and colours are used to code different attributes of the data to provide the information for end-users to analyse and examine. These unique visualisation and analysis benefits offered by maps make them more visually communicative. They imply the distributions and states, provide information about spatial patterns, and imply the association of diverse phenomena. Maps can be either static or dynamic and allow data representation on a 2D or 3D space. They allow the user to infer the actual sizes and distance between objects. The users can zoom in or zoom out respectively meaning showing more or less details. Furthermore, maps allow the extraction of information that cannot be obtained by looking at sensor readings separately and are more efficient to compute in both time and energy. For instance, maps may capture trends or correlations among sense data. Where there is no operating sensor, predictions can be made using these spatial and temporal correlations among sensor readings. Finally, a map provides a higher-level information-rich representation that can be suitable for informing other network services and for delivering field information visualisation.

3. A SURVEY ON ALGORITHMS FOR MAP GENERATION This section provides a brief discussion on related works 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

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estimate sensor nodes faulty behaviour on top of a clusterbased network. This approach is based on Bayesian belief networks, which make it problematic to compute all the probabilities and the revised probabilities once a new sensor reading is received. In dense multi-modal WSNs, the number of dependence increases rapidly, and probabilities computation becomes a NP-hard problem. This approach also lacks precision when updating the fault rate table because it is based on a predefined threshold value. Event detection based on matching the contour maps of in-network data distribution has been shown effective for event detection in WSNs [3]. Map construction starts from each node generating a partial map of its own. When a node forwards data to its neighbours, it adds each contour region in these partial maps with its own. This process is repeated until the final map is generated. This approach works well with grid network topologies and less well with random topologies. When a grid is overlaid on top of a random topology, some cells in the grid may be empty. These empty cells will not participate in the final map construction. Hence, the final map will not cover the entire network area. This makes the scheme sensitive and unsuitable for random WSNs deployments. Furthermore, the loss of any partial maps will result in an incomplete network map. In both [3] and [4], the sink node is required to know the location and the ID of all nodes in the network. Furthermore, the work in both papers is an applicationdependent and requires major lower level modifications if the application is to change. In [3], the assumptions made on the network topology and the way the grid is formed are not efficient and may dissipate the energy savings achieved by the in-network map construction. In [4], it is not clear how the hierarchy is built. Besides, it is only suitable for small-size networks due to the single-hop communication scheme. DIMENSIONS [5] made the case for a large-scale distributed multi-resolution storage system that provides a unified view of data handling in WSNs incorporating long-term storage, multi-resolution data access and spatiotemporal correlations in sensor data. This work is related to ours but different in focus at both the system architecture and coding level. It outlines an approach for relatively power-rich devices, focused on encoding regularly gridded, spatial wavelets over time series. By contrast, we focus on highly resource-constrained devices and integrate different network services with the MGS. Our work is also focused on spatially deployed networks and is independent on a particular routing algorithm. In centralised map generation approaches, delivering all network sensory data back to the sink incurs heavy transmission traffic. Several aggregation-based map generation methods have been proposed to address this problem [3,6–8]. However, aggregation-based methods cannot further improve the scalability of the network as all sensors are required to report to the sink. Moreover, the aggregation process increases the computation overhead on the Wirel. Commun. Mob. Comput. (2011) © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

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intermediate nodes. To address the inherent limitations of aggregation-based methods, [9] proposes a method called Iso-Map that intelligently selects a small portion of the nodes, isoline nodes, to generate and report mapping data to reduce the network traffic and computation overhead. Partial utilisation of the network information leads to a decrease in the mapping fidelity, and isoline nodes will suffer from heavy computation and communication load. Furthermore, the location of mapping nodes can also affect the directions of traffic flow and thereby have a significant impact on the network lifetime. Finally, in sparsely deployed low-density networks, it is difficult to construct contour maps based only on isoline nodes. The positions of isoline nodes provide only discrete iso-positions, which does not define how to deduce and how the isolines pass through these positions. To conclude, mapping is often employed in WSN applications, but as yet, there is no clear definition (or published work) towards a localised MGS that would aid the development of more sophisticated applications. The development and analysis of such a service is the key novel contribution of the work proposed here.

4. MAPPING APPLICATIONS IN THE LITERATURE Within the WSN field, mapping applications found in the literature are ultimately concerned with the problem of mapping measurements onto a model of the environment. Estrin et al.[10] proposed the construction of isobar maps in sensor networks and showed how in-network merging of isobars could help reduce the amount of communication. Furthermore, [6] proposes an efficient data collection scheme, and the building of contour maps, for event monitoring and network-wide diagnosis in centralised networks. Solutions such as distributed mapping have been proposed to the general mapping domain. However, many solutions are limited to particular applications and constrained with unreliable assumptions. The grid alignment of sensors in [10], for example, is one such assumption. In the wider literature, mapping was sought as a useful tool in respect to network diagnosis and monitoring [6], power management [11], and jammed-area detections [12]. For instance, contour maps were found to be an effective solution to the pattern matching problem that works for limited resource networks [3]. These are examples of specific instances of the mapping problem and, as such, motivate the development of a generic MGS, furthering the area of research by moving beyond the limitations of the centralised approaches. A service-oriented approach has special properties. It is made up of components and interconnections that stress interoperability and transparency. Services and serviceoriented approaches address designing and building systems using heterogeneous network software components. This allows the development of an MGS that works with Wirel. Commun. Mob. Comput. (2011) © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/wcm


existing network components, for example, routing protocols, and resources without adding extra overhead on the network.

5. UNDERSTANDING THE PROBLEM OF MAP GENERATION FROM SPARSE DATA Given a set of known data points representing the nodes’ perception of a given measurable parameter of the phenomenon, what is the most likely complete and continuous map of that parameter? In the field of computer graphics, this problem is known as an unorganised points problem, or a cloud of points problem. That is, because the position of the points in xy is assumed to be known, the third parameter can be thought of as a height and surface reconstruction algorithms can be applied. Simple algorithms use the point cloud as vertices in the reconstructed surface. These are not difficult to calculate but can be inefficient if the point cloud is not evenly distributed, or is dense in areas of little geometric variation. Approximation, or iterative fitting algorithms, define a new surface that is iteratively shaped to fit the point cloud. Although approximation algorithms can be more complex, the positions of vertices are not bound to the positions of points from the cloud. For applications in WSNs, this means that we can define a mesh density different to the number of sensor nodes and produce a mesh that makes a more efficient use of the vertices. Self-organising maps are one of the algorithms that can be used for surface reconstruction [13]. This method uses a fixed number of vertices that moves towards the known data. Note that surface reconstruction on typical nonoverlapping terrains is equivalent to sparse data interpolation. This kind of geometric parameter interpolation has been shown to work well for reconstructing underlying geography when the entire network has been queried [14]. However, it does not extend well to variable surfaces or overlapping local mapping because it requires a complete data set to define the surface. A more general method is the interpolation by inverse distance and, specifically, the Shepard interpolation [1] that improved from it.

6. SHEPARD INTERPOLATION Shepard interpolation is an inverse distance weighted scattered data interpolation algorithm. It is widely used in practise and has also been shown to work well with noisy data [15]. Shepard defined a continuous function where the weighted average of data is inversely proportional to the distance from the interpolated location. The algorithm explicitly implies that the farther away a point is from an interpolated location, P , the less effect it will have on the interpolated value.

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Known points, Di , are weighted during interpolation relative to their distance from P . Weighting is assigned to data through the use of a weighting power that controls how the weighting factors drop off as the distance from P increases. The greater the weighting power, the less effect far points have on the interpolation result. As the power increases, Shepard interpolation approaches the nearest neighbour interpolation method [16] where the interpolated value simply takes on the value of the closest sample point [15,16]. Many modifications to the original Shepard algorithm have been proposed in the literature [17–20]; however, most of these methods are designed for computer graphics and image processing fields. These algorithms usually trade accuracy with computational complexity. Nevertheless, when applying an interpolation to WSN applications, it is desirable to keep the interpolation simple to reduce the amount of processing as well as communicated information across the network. Therefore, we shall use the original method with the modifications proposed by Shepard that further reduce the amount of processing and data communication to achieve more energy savings using the limited available bandwidth. These modifications are described in Section 6.1. Shepard’s expression for globally modelling a surface is: 8 N X ˆ ˆ ˆ ˆ .di /u zi ˆ ˆ ˆ < iD1 N f1 .P / D X ˆ ˆ .di /u ˆ ˆ ˆ ˆ iD1 ˆ : zi

if

di ¤ 0 8Di .u > 0/

if

di D 0 (1)

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 i th terms in both the numerator and the denominator exceed all bounds, whereas other terms remain bounded. Therefore, the limP !Di f1 .P / D zi is as desired, and the function f1 .P / is continuously differentiable even at the junctions of local functions. 6.1. Global Shepard algorithm shortcomings and solutions Shepard’s interpolation suffers from several shortcomings imposed by the fact that each sample point has a radially symmetric influence despite the nature of the underlying data [21]. Among the well-known artifacts are cusps, corners and flat spots at the data points, as well as the excessive influence of points that are far away [22]. Further shortcomings include that the global function necessitates all weights to be recomputed if any points are added, removed or modified. In WSNs, this is impractical because of the network dynamics such as node failures, node mobility or new nodes deployment. Shepard has

identified three main shortcomings of his method and proposed modifications to deal with them as follows. 6.1.1. Building the support set. We define the support set as the set containing all points used to calculate P . The global method has a linear running-time O(N), which makes it impractical and inefficient especially when the number of nodes is large. To overcome this, a local Shepard algorithm was defined. This algorithm eliminates distant points from the calculation of any interpolated value because only nearby data points have significant influence. To select nearby nodes, Shepard defined two criteria: (1) Arbitrary distance criterion: All data points within radius r of the point P are included in the computation. This is computationally easy but allows the possibility that there are no data points or there is a sufficiently large number of data points within the radius r. A collection of points, Cp , within a search radius r is defined as Cp D fDi jdi rg, and n.Cp / is the total number of data points in Cp . (2) Arbitrary number criterion: Only the closest n data points are considered in the computation of any interpolated value. This approach ignores the relative location and spacing of the points and requires deep searching and complex ranking procedure for data points. In addition, it assumes that a single number, n, of interpolating points was optimal. If N is the total number of data points, then a new collect˚ ion of data points, CPn , is defined as CPn D Di1 ; Di2 : : : ; Din , where .n N /, and the subscripts ij are defined such that 0 di1 di2 : : : diN . Shepard has chosen a mix of the two criteria and has combined their advantages. An initial radius r is defined depending on the overall density of data points such that seven data points are included on average in a circle of radius r. r is written as follows: r 2 D

7A N

(2)

where A is the area of the largest polygon enclosed by the data points. A function si D s.di /, which weights the points at r 3r more heavily, is defined to guarantee the local behaviour of the interpolating algorithm by calculating a surface model for any d r:

s.d / D

8 ˆ < 1=d ˆ :

27 4r 2

0

d r0

2 1

if if if

0 = > > > > ;

(15)

where n0 N and vi D.xi ; ri / ; i 2 Œ1; N , and ri is a reading value of some distinctive modality (e.g. temperature). Let ˆ be a topological space on S , and there exists open subsets Si ; i 2 Œ1; n o n .0/ .1/ .n0 / S1 D v1 ; v1 ; ; v1 o n .0/ .1/ .n0 / S2 D v2 ; v2 ; ; v2 :: : o n .0/ .1/ .n0 / Sn D vn ; vn ; ; vn

(16)

which are topological subspaces of ˆ such that: ˆSi D fSi \ U jU 2 ˆg

(17)

Wirel. Commun. Mob. Comput. (2011) © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

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Also define ‰ as a topological space on the co-domain R of function f . Then there exists a function, f , that has the following properties: (1) Let f W S1 [ S2 [ : : : [ Sn1 [ Sn be a mapping defined on the union of subsets Si ; i 2 Œ1; N such that the restriction mappings fjSi are continuous. If subsets Si are open subspaces of S or weakly separated, then there exist a function f that is continuous over S (proved by [23]). (2) If f W ! ‰ is continuous, then the restriction to Si ; i 2 Œ1; N is continuous (property, see [24]). The restriction of a continuous global mapping function to a smaller local set, Si , is still continuous. The local set follows since open sets in the subspace topology are formed from open sets in the topology of the whole space. Using the point to set distance generalisation, the function f can be determined as a natural generalisation of methods developed for approximating uni-variate functions. Wellknown uni-variate interpolation formulas are extended to the multivariate case by using Geometric Algebra (GA) in a special way while using a point to set distance metric. Burley et. al. [25] discussed the usefulness of GA for adapting uni-variate numerical methods to multivariate data using no additional mathematical derivation. Their work was motivated by the fact that it is possible to define GAs over an arbitrary number of geometric dimensions and that it is therefore theoretically possible to work with any number of dimensions. This is carried out simply by replacing the algebra of the real numbers by that of the GA. We apply the ideas in [25] to find a multivariate analogue of uni-variate interpolation functions. To show how this approach works, an example of Shepard interpolation of this form is given below: Given a set of n distinct points X D fx0 ; x1 ; : : : ; xn g Rs , the classical Shepard’s interpolation function is defined by n X o f .x/ D wk .x/ f .xk / Sn;

and jx xk j n X

jx xk j

Ci D L.d .P ; Ej /; ı.Si //

(19)

where j:j denotes the Euclidean norm in Rs . 0 f . The basic In the uni-variate case .s D 1/ and Sn;2 0 properties of Sn; f are: 0 f .x / D f .x / ; i D 0; : : : ; nI Sn; i i 0 (2) doe Sn; f D 0, where doe is an abbreviation of degree of exactness.


8Ej 2 Si

(20)

where i 2 Œ1; n, L is a local model that selects the support set for calculating P , d is a Euclidean distance function, Ej is an observation point in the dimension Si and ı.Si / a set of parameters for dimension Si . These parameters are usually a set of relationships between different dimensions or other application domain characteristics such as obstacles. In uni-dimensional distance weighting methods, the weight, !, can be calculated as follows: ! D d P ; Ej ; Ej 2 Si

(21)

This function can be extended to multi-dimensional distance weighting systems as follows: ! D K .P ; Si / ;

i 2 Œ0; n

(22)

where K .P ; Si / is the distance from P to data set Si and n is the number of dimensions in the system. Equation 22 can now be extended to include the domain model parameters of arbitrary dimensional system. Then the dimension-based scaling metric can be defined as:

kD0

(1)

Because the accuracy level of generated maps may vary significantly depending on the specific application, for example, the existence of barriers, in this section, we modify the distance metric to include the knowledge known about the application domain. The proposed metric attempts to balance the size of the support set with the interpolation algorithm computational complexity as well as with the interpolation accuracy. We define the term scale for determining the weight of every given dimension with respect to P based on a combined Euclidean distance criteria as well as the information already known about the application domain a priori to network deployment. Whereas, the term weight is reserved for the relevance of a data site by calculating the Euclidean distance between 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 dimension contains the nearest points for P using mP . Symbolically, Ci is calculated as

(18)

kD0

wk .x/ D

8. APPLICATION-BASED LOCAL MAP GENERATION

mP D

X

L.K.P ; Si /; ı.Si //

i 2 Œ0; n & Si ¤ CP

i

(23) where CP is the dimension containing P . 8.1. Example: Mapping surfaces with barriers Shepard interpolation is based on the intuitive assumption that there is a logical relationship between adjacent points.

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This assumption is, however, violated if some barrier, such as a river, ruptures the continuity of the surface. The effect of physical barriers can be simulated easily because of the distance-dependent interpolation by including virtual barriers. The user may specify discontinuities in the metric space in which di is calculated using a different selection set of nearby data points and in which different weightings and slopes are also being calculated. Given a detour of length bŒP ; Di perpendicular to the line between P and Di , Shepard interpolation defines the effective distance to travel between the two points as: 1

di0 D f.d ŒP ; Di /2 C .bŒP ; Di /2 g 2

(24)

where bŒP ; Di is the strength of the barrier. When a barrier exists, di0 replaces di in all calculations. Whereas, if there is no barrier between Di and P , di D di and bŒP ; Di D 0. The effective distance is discontinuous as P crosses a barrier that results in a discontinuous interpolated surface at an obstacle. The inclusion of barriers in the interpolation will result in the selection of a different set of nearby data points, weightings and slopes.

Milpitas, CA, USA) microcontroller has no support for floating point arithmetic or integer multiplications. The Gumstix (Gumstix Inc., Portola Valley, CA, USA) [35] and EmStar MicroServers (MicroServer, Newark, NJ, USA) [36], amongst other Linux boxes, are example hardware platforms that are available in the market and are capable of running the MGS. Table I shows the specifications of these hardware platforms. In an extreme situation, assume that there is a sensor node that stores 500 observation points. Assuming mapping data are represented as triplets of 32-bit floats, the data alone require 3.9 KB of memory. Let Id be the number of instructions required to estimate the value at a location. Knowing the clock speed of the processors allows the making of a simple estimate of the execution time. Combined with the 600 MHz clock speed, the execution time to calculate a partial map is estimated at 1:4583 s. What is defined as an acceptable execution time is dependent on the application requirements. 9.2. TLI algorithm details

9. SHEPARD INTERPOLATION ANALYSIS In this section, the effectiveness of the Shepard interpolation algorithm is verified, and its characteristics are studied quantitatively and qualitatively. The Shepard interpolation performance was compared with that of the triangulation with linear interpolation (TLI) algorithm [16]. The TLI was chosen for comparison because it is an exact interpolator that uses the optimal Delaunay triangulation. The Delaunay triangulation is used extensively in the field of WSN. Its uses include adaptable network deployment [26], network coverage [27], locating and bypassing routing holes [28], distributed area computation [29], position-aware routing [30] and spatial clustering [31]. Furthermore, TLI is widely referred to in the literature including in the image processing field [32] and is reported to be one of the simplest and most efficient algorithms with a good running time [16,33]. 9.1. Hardware requirements The following experiments target WSNs built from Linuxclass devices that have higher storage and processing capabilities. The choice of a less-constrained hardware platform was for two reasons:

This method connects data points to form triangles that do not intersect with each other. The result of this process is a patchwork of triangular faces over the extent of the grid. The slope and elevation of the triangle is determined by the original data points defining the triangle, and all the nodes within the triangular plane are defined by the triangular surface. Because the triangles are determined by the original data, the data must be sampled at a high rate. TLI is fast with all data sets, but it is not effective with few points [16]. One advantage of triangulation is that, with enough data, triangulation can preserve break lines defined in a data file. For example, if a fault is delimited by enough data points on both sides of the fault line, the surface generated by triangulation will show the discontinuity [16]. 9.3. Comparison metrics To determine the accuracy of the interpolation quantitatively, the skewness and kurtosis of a high-resolution source data and the result of the interpolation using a subset of that data has been chosen as a measure of the surface deviation. The kurtosis is a measure of the peakedness of a real-valued random variable, where a high-kurtosis

Table I. Linux-class sensor node hardware platforms.

(1) Distributed mapping is desirable but introduces a considerable storage and computational complexity on sensing devices when considering current sensor node capabilities. (2) In-network visualisation has requirements typical of any non-trivial processing. For example, the MICA/ MICA2 mote [34] (Crossbow Technologies, Inc.,

MicroServer CPU Clock speed CPU power consumption Memory Flash ROM

au1550 400 MHz 0.5 W 128 MB 128 MB

Gumsense MarvellPXA270 100–600 MHz 72 W 128 MB 32 MB


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distribution has a sharper peak and fatter tails and a lowkurtosis distribution has a more rounded peak with wider shoulders [37]. The skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable [37]. A distribution could have two kinds of skewness, positive skew or negative skew, where the mass of the distribution is concentrated on the left or on the right of the figure, respectively. Further qualitative accuracy assessments are performed using an empirical peak profiling to obtain peak information for studying the two algorithms local behaviour. Visually, we use 2D and 3D height maps to determine the global accuracy of the interpolation method. The 2D maps depict the height where the pixel intensities depict depth values.


(N)

Shepard

TLI

50

100

9.4. Experimental Setup These experiments made use of the Grand Canyon height map [38]. The studied region is 15 360 m2 , with heights ranging from 165 to 284 m above sea level. This map was sampled to 65 536 points. The sensor nodes were randomly distributed over the sensing field, that is, the height map, at position .xi ; yj /, where the pixel intensities depict altitude values. The height map was chosen because using numerous wide-distributed height points has been an important topic in the field of spatial information [16]. Furthermore, the height is a static measure, which makes it suitable for the evaluation of various interpolation algorithms. The primary purpose of these experiments is to take spatial interpolation to calculate the unknown heights by using the information of neighbouring points and to report results. Shepard algorithm with all three modifications is implemented and used in all of the following experiments. Using the same data set, the difference in quality and accuracy of generated maps is determined by the interpolation method used.

200

300

500

9.5. Experiment 1: the effect of network density Aim: The effect of network density on the reconstruction quality of both interpolation algorithms is studied. Procedure: Interpolation methods are run with different network densities, and results are recorded. Results and discussion: Figures 2 and 3 show how the network density and choice of interpolation algorithm affect the reconstruction results. It is observed that higher network densities increase the smoothness in the reconstructed maps. For instance, contour maps made from high network density are visibly smoother because of shorter line segments between data points. Compared with the actual height map, Figure 4, it is visually evident that both interpolation algorithms produced an acceptable quality of 2D and 3D maps. However, the reconstruction quality of TLI with the absence of sufficient data density (e.g. 50 and 100 nodes) is largely Wirel. Commun. Mob. Comput. (2011) © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

1000 Figure 2. 2D maps produced by Shepard and triangulation with linear interpolation (TLI) at various network densities (N ).

fictitious and unconstrained especially on the map boundaries. TLI requires the number of boundaries of the observed area and a higher density of sensing nodes on these locations. As the network density increases, the reconstruction quality for both algorithms is improved. TLI performed better than Shepard with the reconstruction of

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(N )

Shepard

TLI

100

100

200

300

500

1000 Figure 3. 3D maps produced by Shepard and triangulation with linear interpolation (TLI) at various network densities (N ).


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Figure 4. Grand Canyon height map.

the right-hand side portion of the map due to the smoothness of its surface. This result is due to the assumption made by TLI that the height is changing at constant rate, which was the case in that portion of the map. Nevertheless, the total map produced by Shepard interpolation were equal to or better than that of the TLI algorithm despite the little geometric variation in that part of the map. Shepard interpolation captured smaller features of the surface and reflected more details than TLI. However, the cost (in terms of computation and communication, i.e. the size of the support set) of interpolation in Shepard was much less than that when using TLI. Conclusion: Shepard interpolation resulted in equal or better reconstruction results than TLI. The Shepard algorithm proved to produce more accurate results especially on the boundaries and at low network density. Also, Shepard has also captured smaller features and reflected more details of the surface than TLI.

9.6. Experiment 2: interpolation local behaviour Aim: In this experiment, the local performance of Shepard and TLI algorithms is to be evaluated through the application of image processing approaches. Procedure: Peak profiling and statistical measures were used to quantitatively characterise and compare local features extraction capabilities of both algorithms at various network densities. The highest peak in the Grand Canyon height data, labelled in Figure 2, was selected as the local feature that was quantised from maps produced by each algorithm. Results and discussion: Figures 5 to 9 show the profiling results of the selected peak from the original map and from maps produced by the Shepard and TLI interpolation algorithms. It is observed from the figures that the Shepard interpolation appears to be more visually

Figure 5. Peak profiling with 100-node network density.


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plausible and has always rendered a smoother surface than the TLI. This is because the TLI surface passes through all points whose values are known. Shepard algorithm maintained the local shape properties of the nodal functions because there is a mild decrease in a point’s influence as it goes farther from the prediction location. Whereas in TLI curves, all locations within the relevant triangle obtain the same weight regardless of how far they are from the prediction location. In local Shepard interpolation, the enforced restriction of support set to sample points within the neighbourhood reduce the effect of distant points and produce a final surface that is much closer to the original for some features. At low network densities (e.g. 50), Shepard algorithm yields a surface that is much more a

representative of the original surface than that yielded by TLI. This is because TLI requires a medium-to-large number of data points to generate acceptable results. With a highly variable surface such as this, 50 data points are insufficient for TLI to recreate the source data, despite the relatively small size of the region in question. At all network densities, TLI suffer from edge effects because data sets that contain sparse areas result in distinct triangular facets on a surface plot or contour map. At slightly higher network densities (100 and 200), TLI was less a representative of the original data range than Shepard because it tends to capture broad regional trends in the surface. TLI does not provide the ‘flatness’ on the edges we would hope for. As the network density increases, both interpolation Wirel. Commun. Mob. Comput. (2011) © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

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algorithms give almost equal results with better performance from the Shepard algorithm on the basis of adherence to the original surface. Table II presents a summary of statistics for peak profiling results at various network densities using Shepard and TLI interpolated maps. The skewness and kurtosis values measured from the original map are 1:419 and 4.423, respectively. The values recorded in Table II show that as the network density increases, the quality of the produced maps increases. The skewness measurements in Table II confirm the results found in the previous experiment that Shepard interpolation produced a more accurate presentation of the interpolated surface at smaller data sets. Wirel. Commun. Mob. Comput. (2011) © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

However, with bigger data sets (200 and 300), the peak deviation difference of Shepard and TLI interpolated surfaces from the original surface is minimised. Looking at the kurtosis measures, Shepard gives more accurate results of how peaked a distribution is. This success of Shepard was due to the use of a subset of the observation points, which is more related to the interpolation location and which ignores the effect of distant points. Conclusion: The results of these experiments showed that Shepard interpolation was more capable of extracting local features of the interpolated terrain than TLI. This result makes the Shepard method more suitable for implementing the localised MGS in large WSNs.

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Table II. Peak profiling statistical measures.

N

10 50 100 200 300 500 1000

Skewness

Kurtosis

Shepard

TLI

Shepard

TLI

0.340 0.314 1.698 1.065 1.635 1.117 1.249

0:129 0:105 0:610 1:069 1:613 1:186 0:964

2:072 2:088 5:372 2:862 4:669 4:211 4:211

2:085 1:782 2:086 3:380 4:486 3:465 3:092

N, number of nodes; TLI, triangulation with linear interpolation.

9.7. Experiment 3: acceptable level of data presentation Aim: In this experiment, the question of what acceptable level of data presentation is needed for a particular application was investigated. Procedure: The required accuracy level of interpolated 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 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 experiments to the Shepard’s algorithm because Experiment 1 and Experiment 2 proved that it is more suitable for spatial data interpolation at lower times and for processing complexities than the TLI.

Results and discussion: Figure 10 shows a number of contour maps overlaying the height map generated using the Shepard interpolation algorithm at various network densities. By comparing contour maps constructed using low (10, 50 and 100), medium (200 and 300) and high (500 and 1000) network densities, it is noticed that the reconstructed maps are very similar to the original one. The lowest network density at which the selected peak was successfully captured is 100; however, it did not precisely identify the size of the peak. At the 200-node network density, both the size and the height of the peak were represented correctly on the contour map. With higher network densities, contour maps exactness increased rapidly, and the difference between the contour maps generated using 200, 300, 500 and 1000 is insignificant. Thus, a network density of 200 is enough to give an acceptable presentation of that desired feature. Conclusion: From the contour map and the peak profiling results, it can be seen that most of the topographic variations of the terrain were represented with accuracy levels enough to supply information on the topography of the land surface at a 200-node network density. In current real-life WSN deployments, this network density is achievable, which proves the appropriateness and efficiency of Shepard interpolation when applied in WSNs. 9.8. Practical analysis of modified Shepard-based map generation Aim: This experiment aims to study the effect of integrating the knowledge given by the application domain into the multi-modal MGS.

Original

10

50

100

200

300

500

1000

Figure 10. Contour maps drawn on maps produced by Shepard interpolation.


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Procedure: Although no single domain of scientific endeavour can serve as a basis for designing a general framework, an appropriate choice of specific application domain is important in providing significant insights relating to requirements of such a mapping service. Therefore, to illustrate the benefits of exploiting the domain model in map generation, we considered heat diffusion in metals model. A FLIR ThermaCAM P65 (FLIR Systems Inc., Portland, OR, USA) infrared (IR) camera [39], was used to take sharp thermal images. A heat source was placed at the middle of one edge of the brass sheet with the segment hole excavation. Brass (an alloy of copper and zinc) sheet was chosen because it is a good thermal conductor and allows imaging within the temperature range of the available IR camera with less reflection than other metals such as aluminium and steel. After applying heat for 30 s, a thermal image was taken for the sheet. This map had been randomly down-sampled to 1000 points, that is, 1:5% of the total 455 147 to be used by the MGS to regenerate the total heat map. The mapping service integrated all the knowledge given by the application domain, particularly, the presence, position, length and strength of the obstacle. It is assumed here that the obstacle (hole excavation) is continuous, and the existence of this obstacle between two directly communicating nodes will break the wireless links between them. This means that the nearest neighbour triangulation radio frequency (RF) connectivity map is used as a dimension by the MGS. Results and discussion: In this experiment, the nearest neighbour triangulation the RF connectivity map was used as one dimension to predict the heat map. Figure 11 shows the heat diffusion map captured by the FLIR ThermaCAM P65 IR camera. Given that the heat was applied at the middle of the top edge of the brass sheet and at the location of the obstacle, by comparing the left side and right side areas around the heat source, this figure shows that the existence of the obstacle has strongly reduced the temperature rise in the area on its right side. Figure 12 shows the map generated by the Shepard mapping. Compared with Figure 11, the obtained map conserves perfectly the global appearance and many of the details of the original map with 98:5% less data. However, the area containing the obstacle has not been correctly reconstructed and has caused hard edges around the location of heat source. This is due to attenuation between adjacent points and the fact that some areas contain many sensor readings with almost the same elevation.

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 Shepardmapping method.

Figure 13 shows the map generated by modified Shepard-mapping method with the knowledge about the application model. A better approximation to the real surface near the obstacle is observed. The new details included in the domain model removed artifacts from both ends of the obstacle. This is due to the inclusion of the obstacle width in weighting sensor readings when calculating P , which further reduces the effect of geographically nearby sensors that are disconnected from P by the obstacle. Conclusion: This experiment shows that the incorporation of the application domain information in the MGS significantly improves the map production quality.

10. EXAMPLE APPLICATION OF THE MAP GENERATION SERVICE In this section, flood management application is considered to demonstrate how MGS-generated maps can be used in various applications. In this application, an elevation data, Figure 14(a), acquired by the Shuttle Radar Topography [40], is used. The map is 42:2 40:4 km where the 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 are 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 with the fasted water flow? The response generated by the MGS based on simulated data

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(a)

(c)

1.1 1

Cost (mJ)

0.9 0.8 0.7 0.6 0.5 TLI MGS

0.4 0.3 0.2 25

(b)

100

175 250 Number of nodes

325

400

(d)

Figure 14. Map generation service (MGS) application in (a) Gotel Mountains, where height is depicted as brightness, in (b) areas of flood hazard, and in (c) the deepest river channel identified by the MGS, and (d) cost of generating a map in mJ. TLI, triangulation with linear interpolation.

(Figure 14(c)) can be used to identify locations of where to install portable inflatable tubes, for example, 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 mentioned with multi-path, multi-hop hierarchical routing [41] as a communication protocol. We instantiated unit transmission cost on a communication link between two nodes using the first-order radio model values presented in [42]. The typical energy consumption per bit on the transmitter and receiver circuit is set to 40 nJ=bit. We simulated 16 different network 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 were 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 example, a node can estimate its reading from its neighbours’ readings using the MGS. If the difference between the estimated value and the actual reading exceeds a certain threshold, then the node initiates the calibration service and indicates that the sensor readings are erroneous. This example can be generalised to detect anomalies in the network. Another example is when the MGS is used by the routing service to decide whether to forward a reading or not by examining the impact of the new reading on the local map.

11. CONCLUSION This paper presented a new WSN service, the MGS. Mapping was defined as a problem of interpolation from sparse Wirel. Commun. Mob. Comput. (2011) © 2011 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

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and irregular points. Shepard interpolation method was identified and empirically proved to work well with the constraints imposed by WSNs. Shepard method is intuitively understandable and provides a large variety of possible customisations to suit particular purposes. Also, Shepard was found to be easily modified to incorporate different external conditions that might have an impact on the mapping results, such as barriers. Furthermore, this method is simple to implement with fast computation and modelling time [43,44], and it is easy to generalise to more than two independent sense modalities. This method can be localised, which is an advantage for large and frequently changing data sets, making it suitable for WSNs applications. Local map generation reduces data communication across the network and evades the computation of the complete network map when one or more observations are changed. Finally, there are few parameter decisions, and it makes only one assumption, which gives it the advantage over other methods [43]. Shepard was modified to utilise the special characteristics of the application domain to render visualisations in a map format that are a precise reflection of the concrete reality. This modified service is suitable for visualising an arbitrary number of sense modalities. It is capable of visualising from multiple independent types of the sense data to overcome the limitations of generating visualisations from a single type of sense modality. Experimental evaluation demonstrates the usefulness of the modified Shepard mapping service. Future work will investigate how this higher-level information-rich representations can be used for informing other network services besides for the delivery of field information visualisations.


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AUTHORS’ BIOGRAPHIES Mohammad Hammoudeh is a lecturer in computer networks and security in the School of Computing, Mathematics and Digital Technology at the Manchester Metropolitan University. He received his PhD in Computer Science from the University of Wolverhampton. His research interests include distributed systems, wireless networks, information extraction and visualisation. Robert Newman received his BSc in Physics from the University of Birmingham in 1976 and his PhD in Computer Science from Coventry University in 1998. He took up his current position as Associate Dean


with the responsibility for research and academic development at the School of Computing and Information Technology, University of Wolverhampton in September 2007. Previously, he was at Coventry University, where he served in five roles for four different schools, culminating in his appointment as Head of Computer Science in 2002. His work in the field of design systems, that is, to develop IT systems in order to promote information flows and track, required transformations between disparate communities within development processes in the engineering sector and attracted collaboration with such major industry players as Ford, Rolls Royce, Volkswagen, BMW, BAE Systems and EADS. Latterly, he has moved his research focus to pervasive computing, specialising in the system design of distributed intelligent systems and their application within very large wireless intelligent sensor networks. Christopher Dennett is a senior lecturer in the School of Technology at the University of Wolverhampton. His doctoral thesis studies the use of error control codes in mobile communications. His current research interests are in the design of embedded devices, wireless sensor networks and context aware systems. Sarah Mount received a BA with Honours in Computer Science in 2000 and her MA 3 years later from Robinson College, Cambridge. She joined Coventry University’s teaching staff in 2000 as a tutorial assistant and later served as a research assistant, a lecturer and a senior lecturer. In 2007, she took up her current position as a senior lecturer at the School of Computing and Information Technology, University of Wolverhampton. Her current research interests include pervasive computing and the Internet of things. Sarah co-authored Python for Rookies (now Cengage Learning), which was published by Thomson Learning in February 2008.