The document discusses the development of a map generation service (MGS) for wireless sensor networks (WSNs) that allows for effective data visualization and interpolation from sparse sensor data. It proposes a modified Shepard interpolation method tailored to the unique constraints of WSNs, emphasizing low storage requirements and improved mapping accuracy. Through empirical analysis, the MGS is shown to be accurate, flexible, and efficient, offering significant advantages for applications relying on continuous data representation.

1. 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 from
Discrete Wireless Sensor Network Data
Mohammad Hammoudeh1∗ Robert Newman2 , Christopher Dennett2 , and Sarah Mount2
1
School of Computing, Manchester Metropolitan University, Manchester, UK
2
School 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 is
desirable to be able to design applications and reason about the data in more abstract forms than 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. While map generation is essentially a problem of
interpolation between points, a new WSN service, called the Map Generation Service (MGS), 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, it can be localised, and it 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 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: m.hammoudeh@mmu.ac.uk
1. INTRODUCTION content. The ability to interpolate point information
is necessary for carrying out mapping tasks.
With the increase in applications of WSNs, infor- The problem of map generation is essentially a
mation extraction and visualisation have become a problem of interpolation from sparse and irregular
key issue to develop and operate these networks. points. This interpolation capability is realised as a
WSNs typically gather data at a discrete number of service of the network. In this paper, one particular
locations. By bestowing the ability to predict inter- interpolation approach, Shepard interpolation [1],
node values upon the network, it is proposed that it is examined and shown to be suitable for the
will become possible to build applications that are constraints imposed by the nature of WSNs.
unaware of the concrete reality of sparse data. Visual aspects, sensitivity to parameters, and timing
Not all information that is collected from a requirements were used to test the characteristics of
WSN comes ready to use. Often, WSNs field data this method
collection takes the form of single points that need to The rest of the paper is organised as follows.
be processed to get a continuous data presentation. Section 2 explains why map is a suitable discrete
Interpolation describes this process of taking many data visualisation format. Sections 3 and 4 provide
single points and building a complete surface, the a brief description of map generation algorithms
inter-node gaps being filled based on the spatial and mapping applications in the literature. Section
statistics of the observation points. Interpolating 5 defines the problem on map generation. Section
these points will produce more useful information for 6 defines Shepard interpolation method. Sections 7
the end user such as maps related to water chemical and 8 describe the modified Shepard map generation.
Copyright © 2010 John Wiley & Sons, Ltd. 1
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2. A map generation service for WSNs M. Hammoudeh et al.
Evaluation of the MGS is presented in Section 9. The 3. A SURVEY ON ALGORITHMS FOR
paper concludes in Section 10. MAP GENERATION
The following section provides a brief on related
work in map generation methods. Map generation
techniques have previously been explored in the
context of WSNs [3, 4]. Chang et al. [4] implemented
an algorithm to estimate sensor nodes faulty
behaviour on top of a cluster-based network. This
approach is based on Bayesian Belief Networks
(BBNs) which make it problematic to compute all
2. SENSE DATA VISUALISATION: the probabilities and the revised probabilities once a
MAPS new sensor reading is received. In dense multi-modal
WSNs, the number of dependencies increases rapidly
The integration of data visualisation tools and the and probabilities computation becomes an NP-hard
raw data sent by the WSN makes the sensor network problem. This approach also lacks precision when
system useful to different potential users. Visual updating the fault rate table since it is based on a
formats, such as maps, can be easily understood predefined threshold value.
by people possibly from different communities, Event detection based on matching the contour
thus allowing them to derive conclusions based maps of in-network data distribution has been shown
on substantial understanding of the available data. effective for event detection in WSNs [3]. Map
Maps are effective to understand the spatial construction starts from each node generating a
distribution of environmental features since humans partial map of its own. When a node forwards
can use their natural interpretation capabilities to data for its neighbours, it adds each contour region
understand colours, patterns, and spatial relevance. in these partial maps with its own. This process
A map is a visual representation of an area, although is repeated until the final map is generated. This
most commonly used to depict geography, maps may approach works well with grid network topologies
represent any space without regard to context or and less well with random topologies. When a grid
scale such as weather data mapping [2]. However, is overlaid on top of a random topology some cells
a map could be overlaid over a geographic map to in the grid may be empty. These empty cells will
enable observation of the data in a real-world map. not participate in the final map construction. Hence,
In a WSN, a map may be used as an information the final map will not cover the entire network area.
representation and extraction tool in which visual This makes the scheme sensitive and unsuitable for
features such as symbols and colours are used to random WSNs deployments. Furthermore, the loss of
code different attributes of the data to provide any partial maps will result in an incomplete network
the information for end users to analyse and map.
examine. These unique visualisation and analysis In both [3] and [4] the sink node is required to
benefits offered by maps make them more visually know the location and the ID of all nodes in the
communicative, they imply the distributions and network. Furthermore, the work in both papers is
states; provide information about spatial patterns; application-dependent and requires major lower level
and imply the association of diverse phenomena. modifications if the application is to change. In [3],
Maps can be either static or dynamic and the assumptions made on the network topology
allow data representation on 2D or 3D space. and the way the grid is formed are not efficient
They allow the user to infer the actual sizes and and may dissipate the energy savings achieved by
distance between objects. The users can zoom the in-network map construction. In [4], it is not
in or zoom out respectively meaning showing clear how the hierarchy is built. Besides, it is only
more or less details. Furthermore, maps allow suitable for small size networks due to the single hop
the extraction of information that can not be communication scheme.
obtained by looking at sensor readings separately DIMENSIONS [5] made the case for a large-scale
and are more efficient to compute in both time distributed multi-resolution storage system that
and energy. For instance, maps may capture trends provides a unified view of data handling in WSNs
or correlations among sense data. Where there incorporating long-term storage, multi-resolution
is no operating sensor, predictions can be made data access and spatio-temporal correlations in
using these spatial and temporal correlations among sensor data. This work is related to ours, but
sensor readings. Finally, a map provides a higher- different in focus at both the system architecture
level information-rich representation which can be and coding level. It outlines an approach for
suitable for informing other network services and the relatively power-rich devices, focused on encoding
delivery of field information visualisation.
2 Wirel. Commun. Mob. Comput. 2010; 00:1–17 © 2010 John Wiley & Sons, Ltd.
DOI: 10.1002/wcm
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3. M. Hammoudeh et al. A map generation service for WSNs
regularly-gridded, spatial wavelets over time series. limited to particular applications and constrained
By contrast, we focus on highly resource constrained with unreliable assumptions. The grid alignment of
devices, and integrate different network services with sensors in [10], for example, is one such assumption.
the MGS. Our work is also focused on spatially In the wider literature, mapping was sought
deployed networks and is independent on a particular as a useful tool in respect to network diagnosis
routing algorithm. and monitoring [6], power management [11], and
In centralised map generation approaches, deliv- jammed-area detections [12]. For instance, contour
ering all network sensory data back to the sink maps were found to be an effective solution to
incurs heavy transmission traffic. Several aggregation the pattern matching problem that works for
based map generation methods have been proposed limited resource networks [3]. These are examples of
to address this problem [6, 3, 7, 8]. However, aggre- specific instances of the mapping problem and, as
gation based methods can not further improve the such, motivate the development of a generic MGS,
scalability of the network as all sensors are required furthering the area of research by moving beyond
to report to the sink. Moreover, the aggregation the limitations of the centralised approaches.
process increases the computation overhead on the A service oriented approach has special properties.
intermediate nodes. To address the inherent limita- It is made up of components and interconnections
tions of aggregation based methods, [9] proposed a that stress interoperability and transparency. Ser-
method called Iso-Map that intelligently selects a vices and service-oriented approaches address de-
small portion of the nodes, isoline nodes, to generate signing and building systems using heterogeneous
and report mapping data to reduce the network network software components. This allows the devel-
traffic and computation overhead. Partial utilisation opment of a MGS that works with existing network
of the network information leads to a decrease in components, e.g. routing protocols, and resources
the mapping fidelity and isoline nodes will suffer without adding extra overhead on the network.
from heavy computation and communication load.
Furthermore, the location of mapping nodes can also
affect the directions of traffic flow and thereby have a
significant impact of the network lifetime. Finally, in
sparsely deployed low density networks it is difficult
5. UNDERSTANDING THE PROBLEM
to construct contour maps based only on isoline OF MAP GENERATION FROM
nodes. The positions of isoline nodes provide only SPARSE DATA
discrete iso-positions, which does not define how to
deduce how the isolines pass through these positions. Given a set of known data points representing the
To conclude, mapping is often employed in WSN nodes’ perception of a given measurable parameter
applications but as yet there is no clear definition of the phenomenon, what is the most likely complete
(or published work towards) a localised MGS that and continuous map of that parameter? In the
would aid the development of more sophisticated field of computer graphics, this problem is known
applications. The development and analysis of such as an unorganised points problem, or a cloud of
a service is the key novel contribution of the work points problem. That is, since the position of the
proposed here. points in xy is assumed to be known, the third
parameter can be thought of as height and surface
reconstruction algorithms can be applied. Simple
algorithms use the point cloud as vertices in the
4. MAPPING APPLICATIONS IN THE reconstructed surface. These are not difficult to
LITERATURE calculate, but can be inefficient if the point cloud
is not evenly distributed, or is dense in areas of little
Within the WSN field, mapping applications found geometric variation.
in the literature are ultimately concerned with the Approximation, or iterative fitting algorithms
problem of mapping measurements onto a model define a new surface that is iteratively shaped to fit
of the environment. Estrin et al.[10] proposed the the point cloud. Although approximation algorithms
construction of isobar maps in sensor networks can be more complex, the positions of vertices are
and showed how in-network merging of isobars not bound to the positions of points from the cloud.
could help reduce the amount of communication. For applications in WSNs, this means that we can
Furthermore, [6] proposed an efficient data-collection define a mesh density different to the number of
scheme, and the building of contour maps, for sensor nodes, and produce a mesh that makes more
event monitoring and network-wide diagnosis, in efficient use of the vertices. Self organising maps are
centralised networks. Solutions such as distributed one of the algorithms that can be used for surface
mapping have been proposed to the general reconstruction [13]. This method uses a fixed number
mapping domain. However, many solutions are of vertices that move towards the known data.
Wirel. Commun. Mob. Comput. 2010; 00:1–17 © 2010 John Wiley & Sons, Ltd. 3
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4. A map generation service for WSNs M. Hammoudeh et al.
Note that surface reconstruction on typical non- Shepard’s expression for globally modelling a
overlapping terrains is equivalent to sparse-data surface is:
interpolation. This kind of geometric parameter 
interpolation has been shown to work well for  N
(di )−u × zi


reconstructing underlying geography when the entire 


 i=1
network has been queried [14]. However, It does 
N
if di = 0 ∀Di (u > 0)
f1 (P ) =
not extend well to variable surfaces or overlapping 
 (di ) −u
local mapping, since it requires a complete data



 i=1
set to define the surface. A more general method

z
i if di = 0
is interpolation by inverse distance and, specifically, (1)
Shepard interpolation [1] which improves on it. where di is the distance from P to D numbered i in
the N known points set and zi is the known value at
Di . The exponent u is used to control the smoothness
of the interpolation. As P approaches a data point
Di , di tends to zero and the ith terms in both
the numerator and denominator exceeds all bounds
while other terms remain bounded. Therefore, the
limP →Di f1 (P ) = zi is as desired and the function
f1 (P ) is continuously differentiable even at the
6. SHEPARD INTERPOLATION junctions of local functions.
Shepard Interpolation is an inverse distance weighted
6.1. Global Shepard Algorithm Shortcomings and
scattered data interpolation algorithm. It is widely
Solutions
used in practise and has also been shown to
work well with noisy data [15]. Shepard defined a Shepard’s interpolation suffers from several short-
continuous function where the weighted average of comings imposed by the fact that each sample point
data is inversely proportional to the distance from has a radially symmetric influence despite the nature
the interpolated location. The algorithm explicitly of the underlying data [21]. Among the well known
implies that the further away a point is from an artifacts are cusps, corners, and flat spots at the data
interpolated location, P , the less effect it will have points, as well as the excessive influence of points
on the interpolated value. that are far away [22]. Further shortcomings include
Known points, Di , are weighted during interpo- that the global function necessitates all weights to
lation relative to their distance from P . Weighting be recomputed if any points are added, removed,
is assigned to data through the use of a weighting or modified. In WSNs this is impractical due to
power, which controls how the weighting factors drop the network dynamics such as: node failures, node
off as the distance from P increases. The greater the mobility, or deployment of new nodes. Shepard has
weighting power, the less effect far points have on the identified three main shortcomings of his method and
interpolation result. As the power increases, Shep- proposed modifications to deal with them as follows:
ard interpolation approaches the nearest neighbour
interpolation method [16] where the interpolated Building the Support Set
value simply takes on the value of the closest sample We define the support set as the set containing
point [16, 15]. all points used to calculate P . The global method
Many modifications to the original Shepard has a linear running-time O(N), which makes
algorithm have been proposed in the literature [17, it impractical and inefficient especially when the
18, 19, 20], however, most of these methods are number of nodes is large. To overcome this, a
designed for computer graphics and image processing local Shepard algorithm was defined. This algorithm
fields. These algorithms usually trade accuracy eliminates distant points from the calculation of any
with computation complexity. Nevertheless, when interpolated value since only nearby data points have
applying interpolation to WSN applications it is significant influence. To select nearby nodes, Shepard
desirable to keep the interpolation simple to reduce defined two criteria:
the amount of processing as well as communicated
information across the network. Therefore, we shall 1. Arbitrary distance criterion: All data points
use the original method with the modifications within radius r of the point P are included
proposed by Shepard that further reduce the amount in computation. This is computationally easy
of processing and data communication to achieve but allows the possibility that there are no
more energy savings using the limited available data points or a sufficiently large number of
bandwidth. These modifications are described in data points within the radius r. A collection of
subsection 6.1. points, Cp , within a search radius r is defined
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5. M. Hammoudeh et al. A map generation service for WSNs
as Cp = {Di |di ≤ r} and n(Cp ) is the total interpolation function f2 (P ) is given by:
number of data points in Cp . 
2. Arbitrary number criterion: Only the closest n 
 (si )2 zi

data points are considered in the computation  D ∈C
 i
 if di = 0 ∀Di
of any interpolated value. This approach f2 (P ) = (si )2 (4)
ignores the relative location and spacing of 
 D ∈C
 i

the points and requires deep searching and 

zi if di = 0
complex ranking procedure for data points. In
addition, it assumes that a single number, n, The function f2 (P ) requires less physical resources,
of interpolating points was optimal. If N is in terms of both computation and memory, than
the total number of data points, then, a the previous functions. The running-time is reduced
n
new collection of data points, CP , is defined to O(CP ). Consequently, the amount of memory used
n
as CP = {Di1 , Di2 ..., Din } where (n ≤ N ) and by the algorithm on data set of size N is reduced
the subscripts ij are defined such that 0 ≤ to inputs of CP . This reduction in computation and
di1 ≤ di2 ≤ ... ≤ diN . memory cost is invaluable in large scale WSNs which
Shepard has chosen a mix of the two criteria, which must be capable of in-network processing at all levels,
combined their advantages. An initial radius r is including the application level.
defined depending on the overall density of data Since topologies in WSNs changes frequently,
points such that seven data points are included on the MGS should be topology-independent and
average in a circle of radius r. r is written as follows: decentralised. That is, each node or subsystem, e.g.
cluster, uses only local information when making
7A mapping decisions. MGS can be implemented on
πr2 = (2)
N each node with varying the size of the support set
were A is the area of the largest polygon enclosed by and the way it is used. For instance, each node can
the data points. use the Cp to build a set of neighbours to collaborate
A function si = s(di ) is defined to guarantee the with in building a local map for their vicinity. This
local behaviour of the interpolating algorithm by local map can be used to respond to user quires,
calculating a surface model for any d ≤ r, and which to update the global map maintained at a cluster
weights the points at r ≤ r more heavily: head (partial) or a sink (global), used as a local
3
accuracy model, etc. The local map can be used to
calculate whether a new reading should be forwarded

1/d if 0 < d ≤ r
3
to upper nodes in the hierarchy by calculating the

s(d) = 4r2 ( r − 1)2
27 d
if r < d ≤ r (3)
 3 impact of the new reading on the local map. At
0 if r < d

cluster heads and the sink, the support set contains
all nodes within the cluster and all nodes in the
where r is a radius of influence about P chosen network respectively. With hierarchical approaches,
large enough to include n points and defined as the global map can be built and updated by cluster
n
/ n
r (Cp ) = min{dij |Dij ∈ CP } = din+1 . In order that heads. MGS deals efficiently with the addition or
the interpolation algorithm works realistically, if deletion of nodes due to the local mapping, i.e. any
the data points were girded, a minimum of four topological changes will be dealt with locally without
data points was chosen. A maximum of ten was recalculating the global map.
established to limit the complexity and amount of
computation required [1]. Thus Cp and rp are defined Including Direction
as follow: The current method ignores the direction factor

4 in computing the weightings. To make the
Cp if 0 ≤ n(Cp ) ≤ 4
 method intuitively reasonable, Shepard included the
Cp = Cp if 4 < n(Cp ) ≤ 10 direction in computing interpolated values. A new
 10
Cp if 10 < n(Cp )

directional weighting for each data point Di close
to P is defined by:
and 
4
r (Cp )
 if n(Cp ) ≤ 4 ti = sj [1 − cos(Di P Dj )]/ sj (5)
rp = r if 4 < n(Cp ) ≤ 10 Dj ∈C Dj ∈C
 10
r (Cp ) if 10 < n(Cp )

were the cos(Di P Dj ) is defined as: [(x − xi )(x −
The resulting function f2 (P ), has similar behaviour xj ) + (y − yi )(y − yj )]/di dj . The appropriateness of
to the original function but it is capable of the cosine function and computation ease makes
handling much larger data sets and it is much it a good measure of direction. The function sj
more suitable for parallel implementations. The is included in the new function to preserve the
Wirel. Commun. Mob. Comput. 2010; 00:1–17 © 2010 John Wiley & Sons, Ltd. 5
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6. A map generation service for WSNs M. Hammoudeh et al.
A new parameter v is defined with the distance
dimension to bound the maximum effect the slope
terms may have on the final interpolated value. For
a contour mapping application, v can be defined as:
0.1[max{zi } − min{zi }]
v= (9)
max{(A2 + Bi )}
i
2
Ai (x−xi )+Bi (y−yi )
An increment ∆zi = v is computed
v+di
for each Di ∈ CP as a function of P to include the
Figure 1. Fire spreading and wind direction.
effect of the slope in interpolating values at P . Thus
the latest version of the interpolation function is:
original weighting assumption. Within the direction

 wi (zi + ∆zi )
considered, a new weighting function wi = (si )2 ×


 D ∈C
 i
(1 + ti ) is defined and the final interpolation function
 if di = 0 ∀Di ∈ C
f4 (P ) = wi
is defined as: 


 Di ∈C

zi if di = 0
 
 w i zi / wi if di = 0 ∀Di
f3 (P ) = Di ∈C Di ∈C (10)

zi if di = 0 In function f3 (P ), the interpolated surface has a
(6) zero gradient at every Di . This modification is
This modification is useful for mapping modalities valuable to WSNs applications since it reduces noise
where the direction is vital, for instance wind and redundant small details in the image that are
direction in forest fire monitoring applications. The perceptually unnoticeable to human eyes.
two configurations in Figure 1, for example, would
yield identical interpolated values at location 2.
However, if the algorithm to be intuitively practical, 7. MAPPING IN HIGHER
the value at location 2 in the top configuration DIMINESIONAL-SPACE
should be closer to the value at sensor 1 than in the
lower configuration, because wind direction should This section defines a new metric for distance,
be expected to screen the effect of more distant point. suitable for higher dimensions (multi-modal sensing),
in which the concept of closeness is described in
Determining Slope terms of relationships between sets rather than in
The arbitrary and undesirable zero gradient at every terms of the Euclidean distance between points.
point Di still exists on the f3 (P ), generated surface. Using this distance metric, a new generalised
If di is very small, si will equal d−1 and wi will vary
i mapping function f , that is suitable for an arbitrary
as d−2 . To correct this, weighted averages of divided
i number of sensed modalities, is defined.
differences of zi about Di , Ai and Bi , were added In higher diminsional-space mapping every set Si
to sufficiently nearby data points to achieve partial corresponds to an input variable i.e. a sense modality,
derivatives at Di . Constants Ai and Bi represent the called i, and referred to as a dimension. The power
slope in the x and y directions at each data point Di , of such a generalisation can be seen when we include
Ai and Bi are defined as: the time variable as one dimension. The spatial map
generation problem can be stated as follows:
(zj − zi )(xj − xi ) Given a set of randomly distributed data points
wj
(d[Dj , Di ])2
Dj ∈Ci
Ai = (7) xi ∈ Ω, i ∈ [1, N ] , Ω ⊂ Rn (11)
wj
Dj ∈Ci
with function values yi ∈ R, and i ∈ [1, N ] we require
a continuous function f : Ω −→ R to interpolate
and unknown intermediate points such that
(zj − zi )(yj − yi ) f (xi ) = yi where i ∈ [1, N ] (12)
wj
(d[Dj , Di ])2
Dj ∈Ci
Bi = (8) We refer to xi as the observation points. The
wj integer n is the number of dimensions and Ω
Dj ∈Ci is a suitable domain containing the observation
points. When rewriting this definition in terms of
where Ci = CDi − {Di }. relationships between sets we get the following:
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7. M. Hammoudeh et al. A map generation service for WSNs
Lemma 7.1 Given N ordered pairs of separated sets by using Geometric Algebra (GA) in a special way
Si ⊂ Ω with continuous functions while using a point to set distance metric. Burley
et. al [25] discuss the usefulness of GA for adapting
fi : Si −→ R, i ∈ [1, n] (13)
uni-variate numerical methods to multivariate data
we require a multivariate continuous function f : using no additional mathematical derivation. Their
Ω −→ R, defined in the domain Ω = S1 ∪ S2 ∪ ... ∪ work was motivated by the fact that it is possible to
Sn−1 ∪ Sn of the n-dimensional Euclidean space define GAs over an arbitrary number of geometric
where dimensions and that it is therefore theoretically
possible to work with any number of dimensions.
f (xi ) = fi (xi ) ∀xi ∈ Si where i ∈ [1, n] (14) This is done simply by replacing the algebra of the
Proof of Lemma 7.1 The existence of the global real numbers by that of the GA. We apply the ideas
continuous function f can be verified as follows. in [25] to find a multivariate analogue of uni-variate
First, the data set is defined as interpolation functions. To show how this approach
 (0) works, an example of Shepard interpolation of this
(1) (n )

 v1 , v1 , · · · , v1
 
 form is given below:
 (0)
 v , v (1) , · · · , v (n )  Given a set of n distinct points X =
 

2 2 2
S= . . .  (15) {x0 , x1 , ..., xn } ⊂ Rs , the classical Shepard’s
 .
 . .
. . 
. 

  interpolation function is defined by
 (0) (1) (n ) 
vn , vn , · · · , vn n
o
where n ≤ N and vi = (xi , ri ) , i ∈ [1, N ] and ri is Sn,µ f (x) = wk (x) f (xk ) (18)
a reading value of some distinctive modality (e.g. k=0
temperature). Let Φ be a topological space on S and and
there exists open subsets Si , i ∈ [1, n] |x − xk |−µ
wk (x) = n (19)
(0) (1) (n )
S1 = v1 , v1 , · · · , v1 |x − xk |−µ
(0) (1) (n ) k=0
S2 = v2 , v2 , · · · , v2
. (16) where |.| denotes the Euclidean norm in Rs .
.
. 0
In the uni-variate case (s = 1) and Sn,2 f . The
(0) (1) (n ) 0
Sn = vn , vn , · · · , vn basic properties of Sn,µ f are:
0
which are topological subspaces of Φ such that 1. Sn,µ f (xi ) = f (xi ) , i = 0, ..., n;
0
2. doe Sn,µ f = 0, where doe is an abbreviation
ΦSi = {Si ∩ U |U ∈ Φ} (17) of degree of exactness.
Also define Ψ as a topological space on the co-
domain R of function f . Then there exists a
function, f , that has the following properties: 8. APPLICATION-BASED LOCAL MAP
GENERATION
1. Let f : S1 ∪ S2 ∪ ... ∪ Sn−1 ∪ Sn be a mapping
defined on the union of subsets Si , i ∈ [1, N ]
Because the accuracy level of generated maps
such that the restriction mappings f|Si are
may vary significantly depending on the specific
continuous. If subsets Si are open subspaces
application, e.g. existence of barriers, in this section
of S or weakly separated, then there exist a
we modify the distance metric to include the
function f that is continuous over S (proved
knowledge known about the application domain.
by [23]).
The proposed metric attempts to balance the size
2. If f : Ω → Ψ is continuous, then the restriction
of the support set with the interpolation algorithm
to Si , i ∈ [1, N ] is continuous (property,
computation complexity as well as interpolation
see [24]). The restriction of a continuous global
accuracy.
mapping function to a smaller local set, Si ,
We define the term scale for determining the
is still continuous. The local set follows since
weight of every given dimension with respect to P
open sets in the subspace topology are formed
based on a combined Euclidean distance criteria
from open sets in the topology of the whole
as well as information already known about the
space.
application domain a priori to network deployment.
Using the point to set distance generalisation, the While the term weight is reserved for the relevance
function f can be determined as a natural generali- of a data site by calculating the Euclidean distance
sation of methods developed for approximating uni- between P and Di .
variate functions. Well-known uni-variate interpola- We define a new scale-based weighting metric, mP ,
tion formulas are extended to the multivariate case which includes application domain information. The
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8. A map generation service for WSNs M. Hammoudeh et al.
support set is Ci , where Ci ⊆ Si , for each dimension Table I. Linux-class sensor node hardware platforms.
contains the nearest points for P using mP .
Symbolically, Ci is calculated as MicroServer Gumsense
CPU au1550 MarvellPXA270
Ci = L (d (P, Ej ) , δ (Si )) ∀Ej ∈ Si (20)
Clock speed 400MHz 100 to 600MHz
where i ∈ [1, n], L is a local model that selects the CPU power
consumption 0.5W 72µW
support set for calculating P , d is an Euclidean
distance function, Ej is an observation point in the Memory 128MB 128MB
dimension Si , and δ(Si ) a set of parameters for Flash ROM 128MB 32MB
dimension Si . These parameters are usually a set of
relationships between different dimensions or other
discontinuous as P crosses a barrier which result
application domain characteristics such as obstacles.
discontinuous interpolated surface at an obstacle.
In uni-dimensional distance weighting methods, the
The inclusion of barriers in the interpolation will
weight, ω can be calculated as follows
result in the selection of a different set of nearby
ω = d (P, Ej ) , Ej ∈ Si (21) data points, weightings, and slopes.
This function can be extended to multi-dimensional
distance weighting systems as follows 9. SHEPARD INTERPOLATION
ANALYSIS
ω = K (P, Si ) , i ∈ [0, n] (22)
In this section the effectiveness of the Shepard inter-
where K (P, Si ) is the distance from P to data set Si
polation algorithm is verified and its characteristics
and n is the number of dimensions in the system.
are studied quantitatively and qualitatively. The
Equation 22 can now be extended to include the
Shepard interpolation performance was compared
domain model parameters of arbitrary dimensional
with that of Triangulation with Linear Interpolation
system. Then the dimension-based scaling metric can
algorithm (TLI) [16].
be defined as
TLI was chosen for comparison because it is an
exact interpolator which uses the optimal Delaunay
mP = L (K(P, Si ), δ (Si )) i ∈ [0, n] & Si = CP triangulation. Delaunay triangulation is used exten-
i sively in the field of WSN. Uses include: adaptable
(23) network deployment [26], network coverage [27],
where CP is the dimension containing P . locating and bypassing routing holes [28], distributed
area computation [29], position-aware routing [30],
8.1. Example: Mapping Surfaces with Barriers and spatial clustering [31]. Furthermore, TLI is
widely referred to in the literature including in the
Shepard interpolation is based on the intuitive
image processing field [32] and reported to be one
assumption that there is a logical relationship
of the simplest and most efficient algorithms with a
between adjacent points. This assumption is,
good running time [16, 33].
however, violated if some barrier, such as a river,
ruptures the continuity of the surface. The effect 9.1. Hardware Requirements
of physical barriers can be simulated easily due to
the distance-dependent interpolation by including The following experiments target WSNs built from
virtual barriers. The user may specify discontinuities Linux-class devices that have higher storage and
in the metric space in which di is calculated using processing capabilities. The choice of less constrained
a different selection set of nearby data points and hardware platform was for two reasons:
different weightings and slopes are being calculated. 1. Distributed mapping is desirable but intro-
Given a detour of length b[P, Di] perpendicular to duces a considerable storage and computation
the line between P and Di , Shepard interpolation complexity on sensing devices when consider-
defines the effective distance to travel between the ing current sensor node capabilities.
two points as: 2. In-network visualisation has requirements typ-
1 ical of any non-trivial processing. For example,
di = {(d[P, Di ])2 + (b[P, Di ])2 } 2 (24) the MICA/MICA2 mote [34] microcontroller
has no support for floating point arithmetic or
where b[P, Di] is the strength of the barrier. When
integer multiplications.
a barrier exists, di replace di in all calculations.
Whereas, if there is no barrier between Di and P , The Gumsense [35] and EmStar MicroServers [36,
di = di and b[P, Di] = 0. The effective distance is 37], amongst other Linux boxes are example
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9. M. Hammoudeh et al. A map generation service for WSNs
hardware platforms that are available in the market method. The 2D maps depict the height where the
and are capable of running the MGS. Table I shows pixel intensities depict depth values.
the specifications of these hardware platforms.
In an extreme situation, assume that there is 9.4. Experimental Setup
a sensor node that store 500 observation points.
Assuming mapping data are represented as triplets
of 32-bit floats, the data alone requires 3.9KB
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 making a
simple estimate of the execution time. Combined
with the 600MHz clock speed, execution time to
calculate a partial map is estimated at 1.4583s.
What is defined as an acceptable execution time is
dependent on the application requirements.
Figure 2. Grand Canyon height map
9.2. TLI Algorithm Details
These experiments made use of the Grand Canyon
This method connects data points to form triangles height map [39]. The studied region is 15360m2 ,
that do not intersect with each other. The result with heights ranging from 165m to 284m above
of this process is a patchwork of triangular faces sea level. This map was sampled to 65536
over the extent of the grid. The slope and elevation points. The sensor nodes were randomly distributed
of the triangle is determined by the original data over the sensing field, i.e. the height map, at
points defining the triangle and all nodes within position (xi , yj ), where the pixel intensities depict
the triangular plane are defined by the triangular altitude values. The height map was chosen because
surface. Since the triangles are determined by the using numerous wide-distributed height points has
original data, the data must be sampled at a high been an important topic in the field of spatial
rate. TLI is fast with all data sets but it is not information [16]. Furthermore, the height is a static
effective with few points [16]. One advantage of measure which makes it suitable for the evaluation
triangulation is that, with enough data, triangulation of various interpolation algorithms. The primary
can preserve break lines defined in a data file. For purpose of these experiments is to take spatial
example, if a fault is delimited by enough data points interpolation to calculate the unknown heights by
on both sides of the fault line, the surface generated using the information of neighbouring points and
by triangulation will show the discontinuity [16]. to report results. Shepard Algorithm with all three
modifications is implemented and used in all of the
9.3. Comparison Metrics following experiements. Using the same data set, the
difference in quality and accuracy of generated maps
To determine the accuracy of the interpolation is determined by the interpolation method used.
quantitatively, the skewness and kurtosis of a
high resolution source data and the result of the
9.5. Experiment 1: The Effect of Network Density
interpolation using a subset of that data has been
chosen as a measure of the surface deviation. Aim: The effect of network density on the recon-
The kurtosis is a measure of the peakedness of a struction quality of both interpolation algorithms is
real-valued random variable where a high kurtosis studied.
distribution has a sharper peak and fatter tails and Procedure: Interpolation methods are run with
low kurtosis distribution has a more rounded peak different network densities and results are recorded.
with wider shoulders [38]. Skewness is a measure Results and discussion: Figures 3 and 4 show
of the asymmetry of the probability distribution of how the network density and choice of interpolation
a real-valued random variable [38]. A distribution algorithm affect the reconstruction results. It is
could have two kinds of skewness; positive skew or observed that higher network densities increase the
negative skew, where the mass of the distribution is smoothness in the re-constructed maps. For instance,
concentrated on the left of the figure or the right of contour maps made from high network density
the figure respectively. Further qualitative accuracy are visibly smoother due to shorter line segments
assessments are done using empirical peak profiling between data points.
to obtain peak information for studying the two Compared to the actual height map, Figure 2,
algorithms local behaviour. it is visually evident that both interpolation
Visually, we use 2D and 3D height maps to algorithms produced acceptable quality 2D and
determine the global accuracy of the interpolation 3D maps. However, the reconstruction quality of
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10. A map generation service for WSNs M. Hammoudeh et al.
TLI with the absence of sufficient data density (N ) Shepard TLI
(e.g. 50, and 100 node) is largely fictitious and
unconstrained especially on the map boundaries.
TLI requires the number of boundaries of the
observed area and 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
50
the reconstruction of the right hand side portion of
the map due to the smoothness of its surface. This
result is due to the assumption that TLI makes, that
the height is changing at constant rate, which was
the case in that portion of the map. Nevertheless,
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 100
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 200
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 300
Aim: In this experiment the local performance of
Shepard and TLI algorithms is to be evaluated
through application of image processing approaches.
Procedure: Peak profiling and statistical measures
are used to quantitatively characterise and compare
local features extraction capabilities of both
algorithms at various network densities. The highest 500
peak in the Grand Canyon height data, labelled in
Figure 2, was selected as the local feature that is
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
1000
the figures that Shepard interpolation appears to
be more visually plausible and has always rendered
Figure 3. (1): 2D maps produced by Shepard and TLI at various
a smoother surface than TLI. This is because that network densities (N ).
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 gets to sample points within the neighbourhood reduced
farther from the prediction location. While in TLI the effect of distant points and produced a final
curves, all locations within the relevant triangle surface that is much closer to the original for
get the same weight regardless of how far they some features. At low network densities (e.g. 50)
are from the prediction location. In local Shepard Shepard algorithm yields a surface which is much
interpolation, the enforced restriction of support set more representative of the original surface than that
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11. M. Hammoudeh et al. A map generation service for WSNs
(N ) Shepard TLI
50
100
Figure 5. Peak profiling with 100 nodes network density
200
300
Figure 6. Peak profiling with 200 nodes network density
500
1000
Figure 4. (2): 3D maps produced by Shepard and TLI at various
network densities (N ).
yielded by TLI. This is because TLI requires a Figure 7. Peak profiling with 300 nodes network density
medium-to-large number of data points to generate
acceptable results. With a highly variable surface Table II. Peak profiling statistical measures, where N is the
such as this, 50 data points are insufficient for TLI number of nodes.
to re-create the source data, despite the relatively
small size of the region in question. At all network N Skewness Kurtusis
densities, TLI suffer from edge effects because data Shepard TLI Shepard TLI
sets that contain sparse areas result in distinct 10 -0.340 -0.129 2.072 2.085
triangular facets on a surface plot or contour map. 50 -0.314 -0.105 2.088 1.782
At slightly higher network densities (100 and 200), 100 -1.698 -0.610 5.372 2.086
TLI was less representative of the original data range 200 -1.065 -1.069 2.862 3.380
than Shepard because it tends to capture broad 300 -1.635 -1.613 4.669 4.486
regional trends in the surface. TLI does not provide 500 -1.117 -1.186 4.211 3.465
the ‘flatness’ on the edges we would hope for. As 1000 -1.249 -0.964 4.211 3.092
the network density increases, both interpolation
algorithms give almost equal results with better
performance from the Shepard algorithm on the basis Table II presents a summary of statistics for
of adherence to the original surface. peak profiling results at various network densities
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12. A map generation service for WSNs M. Hammoudeh et al.
Original 10 50 100
200 300 500 1000
Figure 10. Contour maps drawn on maps produced by Shepard
Figure 8. Peak profiling with 500 nodes network density interpolation
9.7. Experiment 3: Acceptable Level of Data
Presentation
Aim: In this experiment the question of what is an
acceptable level of data presentation needed for a
particular application was investigated.
Procedure: The required accuracy level of inter-
polated maps may vary significantly depending on
the specific application. Contour map was chosen
as an application to determine the network density
required to reflect some terrain characteristics with
Figure 9. Peak profiling with 1000 nodes network density particular levels of accuracy and details. In this
experiment the effect of the network density on the
quality of the contour maps is to be studied.
We restrict the data representation quality experi-
ments to Shepard’s algorithm because Experiment 1
and Experiment 2 proved that it is more suitable
for spatial data interpolation at lower times and
processing complexities than TLI.
using Shepard and TLI interpolated maps. The Results and discussion: Figure 10, shows a
skewness and kurtosis values measured from the number of contour maps overlaying the height map
original map are −1.419 and 4.423 respectively. The generated using the Shepard interpolation algorithm
values recorded in table II shows that as the network at various network densities. By comparing contour
density increases, the quality of the produced maps maps constructed using low (10, 50, and 100),
increase. The skewness measurements in table II medium (200 and 300), and high (500 and 100)
confirm the results found in the previous experiment network densities, it is noticed that the reconstructed
that Shepard interpolation produced a more accurate maps are very similar to the original one. The
presentation of the interpolated surface at smaller lowest network density at which the selected peak
data sets. However, with bigger data sets (200 was successfully captured is 100, however it did
and 300) the peak deviation difference of Shepard not precisely identify the size of the peak. At 200
and TLI interpolated surfaces from the original nodes network density both the size and the height
surface is minimised. Looking at the kurtusis of the peak were represented correctly on the
measures, Shepard gives more accurate results of how contour map. With higher network densities, contour
peaked a distribution is. This success of Shepard was maps exactness increased rapidly and the difference
due to the use of a subset of the observation points between the contour maps generated using 200, 300,
which is more related to the interpolation location 500 and 1000 is insignificant. Thus a network density
and ignores the effect of distant points. of 200 is enough to give an acceptable presentation
Conclusion: The results of these experiments of that desired feature.
showed that Shepard interpolation was more capable Conclusion: From the contour map and the
of extracting local features of the interpolated terrain peak profiling results, it can be seen that most
than TLI. This result makes the Shepard method of the topographic variations of the terrain were
more suitable for implementing the localised MGS represented with accuracy levels enough to supply
in large WSNs. information on the topography of the land surface
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13. M. Hammoudeh et al. A map generation service for WSNs
at a 200 nodes 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 Figure 11. Heat diffusion map taken by IR camera.
Aim: This experiment aims to study the effect of
integrating the knowledge given by the application
domain into the multi-modal MGS.
Procedure: While 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 Figure 12. Heat map generated by the Shepard-mapping
mapping service. Therefore, to illustrate the benefits method.
of exploiting the domain model in map generation
we consider heat diffusion in metals model.
A FLIR ThermaCAM P65 Infrared (IR) camera [40],
is used to take sharp thermal images. A heat source
was placed on 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 Figure 13. Heat map generated by the modified Shepard-
mapping method.
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
seconds, a thermal image was taken for the sheet. reconstructed and has caused hard edges around the
This map has been randomly down-sampled to 1000 location of heat source. This is due to attenuation
points, that is 1.5% of the total 455 × 147 to be between adjacent points and the fact that some areas
used by the MGS to re-generate the total heat map. contain many sensor readings with almost the same
The mapping service integrates all the knowledge elevation.
given by the application domain. Particularly, the Figure 13 shows the map generated by modified
presence of the obstacle, its position, length, and Shepard-mapping method with knowledge about the
strength. It is assumed here that the obstacle (hole application model. A better approximation to the
excavation) is continuous and the existence of this real surface near the obstacle is observed. The
obstacle between two directly communicating nodes new details included in the domain model removed
will break the wireless links between them. This artifacts from both ends of the obstacle. This is due
means that the nearest neighbour triangulation RF to the inclusion of the obstacle width in weighting
connectivity map is used as a dimension by the MGS. sensor readings when calculating P which further
Results and discussion: The nearest neighbour reduces the effect of geographically nearby sensors
triangulation In this experiment, the RF connectiv- that are disconnected from P by the obstacle.
ity map is used as one dimension to predict the Conclusion: This experiment shows that the
heat map. Figure 11 shows the heat diffusion map incorporation of the application domain information
captured by the FLIR ThermaCAM P65 IR camera. in the MGS significantly improves the map
Given that the heat is applied at the middle of the production quality.
top edge of the brass sheet and the location of the
obstacle, by comparing the left side and right side
areas around the heat source, this figure shows that 10. EXAMPLE APPLICATION OF THE
the existence of the obstacle has strongly reduced the MGS
temperature rise in the area on its right side.
Figure 12 shows the map generated by the In this subsection, flood management application
Shepard-mapping. Compared with Figure 11, the is considered to demonstrates how MGS generated
obtained map conserves perfectly the global maps can be used in various applications. In this ap-
appearance and many of the details of the original plication an elevation data, Figure 14−(a), acquired
map with 98.5% less data. However, the area by the Shuttle Radar Topography (SRTM) [41] is
containing the obstacle has not been correctly used. The map is 42.2 × 40.4 kilometres where the
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14. A map generation service for WSNs M. Hammoudeh et al.
height is depicted as brightness. This map clarifies
the continuity of the drainage network, which can
be used for floodplain zoning (a procedure used to
identify areas of varying flood hazard). Consider a
WSN that is deployed for flood monitoring in which
nodes equipped with sensors to measure the river
and weather conditions. Measured information can
be integrated into maps that can be overlaid over
each other to provide more meaning of the mapped
data. The MGS can be used to generate high-risk
floodplain map (Figure 14−(b)) that can be used to
forecast, notify, plan, and manage floods. Such maps
can be used to answer questions related to the above
tasks; for instance, what is the deepest river channel
(a)Gotel Mountains, height as brightness. with the fasted water flow? The response generated
by the MGS based on simulated data (Figure 14−(c))
can be used to identify locations of where to install
portable inflatable tubes, e.g. at sharp corners, and
to inform emergency services.
To measure the cost of generating the flood
hazard map we used the same experimental setup
as above with MuMHR [42] as a commmunication
protocol. We instantiate unit transmission cost
on a communication link between two nodes
using the first order radio model values presented
in [43]. The typical energy consumption per bit
on the transmitter and receiver circuit is set
to 40nJ/bit. We simulate 16 different network
(b) Areas of flood hazard. topologies with various node densities because the
network topologies and nodes density will affect the
behaviour of different map generation algorithms.
We also performed the same experiment with
TLI, the generated maps are similar and omitted
here. Figure 14−(d) shows that the MGS with
Shepard interpolation expends less energy in map
generation than with TLI. This can be attributed to
the localised behaviour of the underlying Shepard
method, which reduces energy consumed by the
MGS for propagating mapping information to the
central location.
The MGS can be used to inform other network
services such as routing or calibration services. For
(c) The deepest river channel identified by the example, a node can estimate its reading from
MGS. its neighbours readings using the MGS. If the
1.1 difference between the estimated value and the
1
actual reading exceeds a certain threshold then the
0.9
0.8
node initiates the calibration service and indicates
that the sensor readings are erroneous. This example
Cost (mJ)
0.7
0.6 can be generalised to detect anomalies in the
0.5 network. Another example is when the MGS is used
0.4 TLI
MGS by the routing service to decide whether to forward
0.3
0.2
a reading or not by examining the impact of the new
25 100 175 250 325 400 reading on the local map.
Number of nodes
(d) Cost of generating map in mJ
Figure 14. MGS application.
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15. M. Hammoudeh et al. A map generation service for WSNs
11. CONCLUSION
This paper presented a new WSN service, the
MGS. Mapping was defined as a problem of
interpolation from sparse and irregular points.
Shepard interpolation method was identified and
emppirically proved to work well with the constriants
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 [44, 45] and it is easy to generalise to more than
two independent sensed 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 [44]. 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 a 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 the delivery
of field information visualisations.
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