Estimating Parameters of Multiple Heterogeneous Target Objects Using Composite Sensor Nodes

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Estimating Parameters of Multiple
Heterogeneous Target Objects Using Composite
Sensor Nodes
Hiroshi Saito, Fellow, IEEE, Shinsuke Shimogawa, Sadaharu Tanaka, and Shigeo Shioda, Member, IEEE
Abstract—We propose a method for estimating parameters of multiple target objects by using networked binary sensors whose
locations are unknown. These target objects may have different parameters, such as size and perimeter length. Each sensors, which is
incapable of monitoring the target object’s parameters, sends only binary data describing whether or not it detects target objects coming
into, moving around, or leaving the sensing area at every moment. We previously developed a parameter estimation method for a single
target object. However, a straight-forward extension of this method is not applicable for estimating multiple heterogeneous target objects.
This is because a networked binary sensor at an unknown location cannot provide information that distinguishes individual target
objects, but it can provide information on the total perimeter length and size of multiple target objects. Therefore, we propose composite
sensor nodes with multiple sensors in a predetermined layout for obtaining additional information for estimating the parameter of each
target object. As an example of a composite sensor node, we consider a two-sensor composite sensor node, which consists of two
sensors, one at each of the two end points of a line segment of known length. For the two-sensor composite sensor node, measures
are derived such as the two sensors detecting target objects. These derived measures are the basis for identifying the shape of each
target object among a given set of categories (for example, disks and rectangles) and estimating parameters such as the radius and
lengths of two sides of each target object. Numerical examples demonstrate that networked composite sensor nodes consisting of two
binary sensors enable us to estimate the parameters of target objects.
Index Terms—Sensor network, estimation, target object, composite sensor node, ubiquitous network, integral geometry, geometric
probability.
✦
1 I NTRODUCTION extract significant information. This sensing paradigm is
appropriate for the described situation.
Small and low-cost sensors can be used to build wireless
Unfortunately, we have not determined a theory that
sensor networks [1], [2], [3]. They have communication
enables us to implement this paradigm. As a typical
capabilities and built-in batteries to communicate via a
and challenging application, we investigated a problem
wireless link for transmitting sensed data of detected
of estimating parameters related to the shape and size
events of interest. A newly proposed development, such
of target objects moving in a monitored area by using
as a wide area ubiquitous network [4], supports sensors
many simple randomly distributed sensors. We address
with low performance and functionalities and a long-
this problem and present a theory for estimating the
range, low-speed wireless link with very low power con-
parameters of multiple target objects by using networked
sumption. These sensors can be implemented as single
binary sensors whose locations are unknown. An in-
chips, lowering their cost. As a result, we can deploy
dividual sensor is simple. It monitors its environment
such sensors like the scattering of dust [5]. Because there
and reports whether or not it detects a target object. It
are many sensors, we cannot carefully design the loca-
does not have a positioning function, such as a GPS, or
tion of each one. GPS cannot be used because the sensors
functions for monitoring the target object size and shape,
should have limited capability and power consumption.
such as a camera, and it can be placed without careful
This situation leads us to a new sensing paradigm.
design. However, by collecting reports from individual
That is, instead of having a few sensors with advanced
sensors, we can statistically estimate the parameters of
functions and high performance, many sensors with
target objects.
simple functions and low performance are distributed
In addition to theoretical contributions, this work is
randomly. They are networked and send reports, each of
useful to some application scenarios that require low
which includes only an insignificant amount of informa-
cost, a wide coverage, and low power consumption
tion, over a wireless link. However, as a whole, we can
and accept rough estimation of the size and shape of
target objects. For example, we may need to identify the
¯ H. Saito and S. Shimogawa are with NTT Service Integration Laboratories,
3-9-11, Midori-cho, Musashino-shi, Tokyo, 180-8585, Japan.
number and kinds of animals and vehicles in a large
E-mail: saito.hiroshi@lab.ntt.co.jp area to observe the wildlife, prevent poaching, or limit
¯ S. Tanaka and S. Shioda are with Chiba University. the number of tourist vehicles. We can easily obtain bi-
nary sensors of sound volume, acceleration, infrared, or
Digital Object Indentifier 10.1109/TMC.2011.65 1536-1233/11/$26.00 © 2011 IEEE

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magnetic sensors for such scenarios. These sensors with for such a purpose. In particular, [8] and [14] directly
normal detection ranges are not expensive. Therefore, we apply the results of integral geometry discussed in
can provide a large number of these sensors to cover a Chapter 5 and Section 6.7 in [15] to the analysis of
large area at a reasonable cost. By distributing these bi- detecting an object moving in a straight line and to the
nary sensors without expensive power-consuming GPSs evaluation of the probability of -coverage. In addition,
or time-consuming carefully-designed placement in a [16] applies integral geometry to the analysis of straight
large area, our proposed method can work. line routing, which is an approximation of the shortest
This research is an extension of our prior study [6], path routing, and [17] uses it to operate sensors in an
which was based on the coverage process theory [7] energy-conserving way.
and its application to sensor networks [8], [9], [10]. This The rest of this paper is organized as follows.
study emphasized the robustness of the derived formula Section 2 describes a basic model and previous results
for a number of sensors detecting a target object. In for the model. In Section 3, we introduce composite
[6], sensors that measure the size of a detected part sensor nodes and describe an extended model (static
of a target object are used for estimating the overall submodel). We discuss the analysis for deriving mea-
size. However, in [11], we developed a shape and size sures in Section 4. In Section 5, we discuss the dynamic
estimation method using binary sensors for sending submodel of the extended submodel. We then propose
reports on whether or not they detect the target object. an estimation method that uses composite sensor nodes
That study assumed that both the target object and the for target objects with different parameters in Section 6.
sensing area are convex or that the sensing area is disk- We show numerical examples in Section 7, and conclude
shaped. This developed method [11] was evaluated in this paper in Section 8.
an experiment where the target object was a box and
the sensor was an infrared distance measurement sensor
[12]. We also extended the estimation method in [11] to 2 P REVIOUS WORK
apply to non-convex target objects [13].
2.1 Basic model
These methods we developed in [11], [13] did not
take into account when multiple target objects are in the To describe the results of the previous work, which is
monitored area. This work addresses such a case, i.e., the basis of the current work, we discuss the basic model
multiple target objects that may have different shapes that the previous work used.
and sizes coming into, moving around, and leaving the A sensor network operator deploys sensors in a con-
area monitored by sensors. Each sensor, whose location vex area ¨ in 2-dimensional space ʾ , but the operator
is unknown, sends a report on whether or not at least does not know the sensor locations. Each sensor has
one target object is detected within its sensing area. The a sensing area, monitors the environment, and detects
sensor network operator collects these reports to identify events within that area. (This model is called the Boolean
the shape and to estimate parameters, such as sizes sensing model [9], [18], [19], [20] because it is clear
and perimeter lengths, of the target objects. However, whether a point is within a sensing area or not.) A sensor
identifying the shape and estimating the parameters of sends a detection report if a target object is in its sensing
multiple heterogeneous (i.e., having different parame- area, and it sends a no-detection report if there is no
ters) target objects is difficult, and a straight-forward target object in that area. (It is possible to assume that a
extension of the estimation method we previously de- sensor does not send a report if there is no target object
veloped [11] is not applicable. Thus, we introduce the in that area and that the sensor network operator judges
concept of composite sensor nodes where multiple sim- no-detection if there is no report within a timeout. This
ple binary sensors are arranged in a predetermined scenario can reduce the power consumption of sensors,
layout, such as a line. Sensors in a composite sensor but the sensor network operator cannot distinguish no-
node provide local and relative information even if the detection from the loss of reports. For simplicity, this
location of that node is unknown. This is because the paper assumes that no-report is sent if there is no target
relative locations of these sensors are predetermined. In object in that area.)
this sense, the composite sensor node is an intermediate Assume that the -th sensor is located at ´Ü Ý µ and
concept between a simple and an advanced sensor node the sensing area is rotated by from the referenced
equipped with GPS. With these sensors, we can estimate position. Let ´Ü Ý µ ʾ be the sensing area
the parameters of multiple heterogeneous target objects. where ½ ¾ . The -th sensor has communication
The estimation method using composite sensor nodes capability and can send a report Á . Here, Á is 1 if
is derived from integral geometry and geometric prob- it detects the target object and 0 if otherwise. That is,
ability, which are useful tools for analyzing a geometric Á ½´ ´Ü Ý µ Ì µ, where ½´ µ is an indicator
or spatial structure. Ubiquitous access networks, such function that becomes 1 if a statement is true and 0 if
as wireless sensor networks, require analysis of the otherwise. At Ø Ø , the -th sensor sends a report, Á ´Ø µ,
geometric or spatial structure of an object, and there is describing whether or not it detects the target object. The
literature (including one of our papers [11]) describing sensor network operator receives the report from each
the use of integral geometry and geometric probability sensor through the network.

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Sensors are classified into multiple types according to 3 C OMPOSITE SENSOR NODE
their sensing areas. That is, different types of sensors Estimating the unknown parameters of multiple target
have different sensing area sizes or shapes. (A different objects is difficult. Even when the number of target
sensing area can often be implemented by changing the objects is known, a straight-forward extension of the
sensitivity parameter of a sensor.) Type- sensors, of estimation method for a single target object to one for
which sensing area is denoted by , are deployed with multiple target objects is not applicable if the parameters
mean density . Assume that the first to Ò½ -th sensors of each target object are different. To estimate the pa-
are type-1 sensors, the ´Ò½ · ½µ-th to Ò¾ -th sensors are rameters of each target object, we introduce a composite
type-2 sensors, ... In general, Ò ½ · ½ ¡ ¡ ¡ Ò -th sensors
are type- sensors. Á
ØÑ
Ø Ø½
Ò È È
Ò ½ ·½ Á ´Øµ Ñ de-
sensor node, which consists of multiple sensors arranged
in a predetermined layout, such as a line. Instead of
notes the time average of the number of type- sensors deploying individual sensors, the sensor network op-
detecting the target object at each measurement epoch. erator deploys composite sensor nodes. Although the
We assume that the sensor network operator knows the locations of the composite sensor nodes are unknown,
sensor type of each sensor sending a report. local and relative information among multiple sensors
Consider a target object Ì in 2-dimensional space ʾ . in a predetermined layout of a composite sensor node
Its size is Ì , and its perimeter length is Ì . (In the becomes available because the relative locations of these
remainder of this paper, Ü denotes the size of Ü and sensors are known.
Ü denotes the perimeter length of Ü.)
Define a detectable area such that sensors will 3.1 Straight-forward extension to multiple target ob-
detect the target object if and only if they are located jects
in a detectable area with a certain rotation angle.
That is, ½´ ´Ü Ý µ Ì µ ½ if and only if
Before introducing composite sensor nodes, we explain
´Ü Ý µ ¾
an extension of Eq. (1) to an equation applicable to
. Equivalently, ´Ü Ý µ ´Ü Ý µ
Ì ÊÊÊ
. The detectable area size is defined as
multiple target objects.
Proposition 1: Assume that there are Ò Ì convex and
ÜÝ ¾
Ü Ý ¾ . We should note that
and Ì . When
bounded target objects and that the sensing area is
depends on both is disk-shaped,
also bounded and convex. Let be the detectable areas
of the -th target object and £
does not depend on . When does not depend on ,
´Ü Ý µ ´Ü Ý µ Ì
. If
we define for simplicity.
for any ,
Ì
Ò Ò Ì
2.2 Previous results for basic model
£ ½
´ Ì µ¡ · Ì · ÒÌ (5)
¾
½ ½
In this section, we summarize our previous results,
which are used in the current work. For the convex target
This is because
È
where Ì denotes the -th target object ´½
and because
ÒÌ µ. £
object, we have obtained the following equation, and a
shape and size estimation method was developed based satisfies Eq. (1).
Therefore, we can estimate
ÒÌ
½ Ì and
È ÒÌ
½ Ì
È
on this equation [11]: Assume that both the target object
Ì and the sensing area are bounded and convex. Then, instead of Ì and Ì in Eqs. (3) and (4), respectively.
Consequently, if we can assume that all the target objects
½
¾
Ì ¡ · Ì · (1)
have the same shape and size, we can estimate the size
and shape (size and perimeter length) of each target
object. However, if not, we cannot estimate the size and
Based on the fact that Á , we developed shape (size and perimeter length) of each target object
the following estimation method. (1) At Ø , receive the re-
È È
no matter how many types of sensors we introduce.
port Á ´Ø µ from each sensor whose location is unknown.
Ñ Ò
(2) Calculate Á ½ Ò ½ ·½ Á ´Ø µ Ñ for each
sensor type ( ½ ¾). (3) For two unknown parameters
3.2 Extended model (static submodel)
Ì and Ì , use Eq. (1) with the estimator Á of For analyzing an estimation method for multiple hetero-
and solve the following equations for ½ ¾. geneous target objects, we introduce a new model.
A sensor network operator deploys composite sensor
´
½
¾
Ì ¡ · Ì · µ Á (2) nodes in a convex area ¨ in 2-dimensional space ʾ , but
the operator does not know their locations. A composite
where Ì Ì are estimators of Ì Ì . That is,
sensor node consists of multiple sensors arranged in a
predetermined layout. We now consider a two-sensor
Ì ¾ ´Á½ ½ Á¾ ¾ ½ · ¾ µ
composite sensor node, i.e., a composite sensor node
½ ¾
(3) with two sensors. These two sensors are the same as
those described in the basic model. That is, they are
Ì ¾ ´ ½ Á½ ½µ · ½ ´ ¾ Á¾ ¾µ simple binary sensors. For simplicity, in the remainder
¾ ½
(4)
of this paper, we assume that the sensing area of each

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individual sensor in a composite sensor node is disk- 4 A NALYSIS FOR EXTENDED MODEL ( STATIC
shaped. SUBMODEL )
4.1 General results
There are multiple types of composite sensor nodes.
Type- composite sensor nodes are randomly deployed In this subsection, we provide general measure proposi-
with mean density ¼. Each of the two sensors in a tions that only one (both) of the sensors in a composite
type- composite sensor node has a disk-shaped sensing sensor node detects a target object. Let ѽ ´Ì µ be the
area with radius Ö and is located at each of the two end measure of a set of composite sensor nodes in which one
points of a line segment of length Ð . Assume that the first of the two sensors in each composite sensor node detects
to Ò½ -th composite sensor nodes are type-1 composite a target object Ì and the other does not. (According to
sensor nodes and the ´Ò½ · ½µ-th to Ò¾ -th composite
ѽ ´Ì µ
Ê
Appendix A, ѽ ´Ì µ can be written by an integral form:
Ø.) Roughly, ѽ ´Ì µ is a
sensor nodes are type-2 composite sensor nodes, ... In ´Ü½ ܾ µ¾ × Ô
general, Ò ½ · ½ ¡ ¡ ¡ Ò -th composite sensor nodes are non-normalized probability that one of the two sensors
type- composite sensor nodes for ½ ¡ ¡ ¡ Â , where in a composite sensor node detects a target object Ì and
 is the number of composite sensor node types. (For the other does not, where the normalizing constant is Ñ
½ ¾, Ð ½ Ð ¾ or Ö ½ Ö ¾ .) (the measure of a set of composite sensor nodes placed
Let Á be the report of the -th sensor of the -th A. Similarly, let Ѿ ´Ì µ
Ê
in (included in or intersects with) ¨), given in Appendix
´Ü½ ܾ µ¾ Ô Ø be the
composite sensor node for ½ ¾, and Á is 1 if it measure of a set of composite sensor nodes in which
È
detects the target object and 0 if otherwise. Let Á¾
Ò
Ò ½ ·½ ½´Á½ Á¾ ½µ be the number of type-
both sensors of each composite sensor node detect a
target object Ì . Ѿ ´Ì µ is a non-normalized probability
composite sensor nodes in each of which two sensors that both of the two sensors in a composite sensor node
Ò È
detect at least one target object at a single measurement detect a target object Ì . (When we need to indicate
epoch. Similarly, define Á½ Ò ½ ·½ ½´ Á½ ´½ the parameters Ð Ö of the composite sensor node to
Á¾ µ · ´½ Á½ µÁ¾ ½µ, which denotes the number evaluate the measures, we use the notations ѽ ´Ì Ð Öµ
of type- composite sensor nodes in each of which one and Ѿ ´Ì Ð Öµ. When we need to indicate the parameters
of two sensors detects at least one target object at a single of the -th target object in Ѿ for the type- composite
measurement epoch. sensor node, we use the notation Ѿ ´¢ Ð Ö µ instead
of Ѿ ´Ì Ð Ö µ, where ¢ is defined later.)
There are ÒÌ target objects in 2-dimensional space Proposition 2: Let ѽ ´ µ be the measure of a set of type-
ʾ , where ÒÌ is known. Let Ì be the -th target object composite sensor nodes in which only one of the two
(½ ÒÌ ). (We propose the stochastic geometric filter, sensors in each composite sensor node detects any target
which can estimate the number of target objects [21].) In object, and let Ѿ ´ µ be the measure of a set of type-
this static submodel, target objects do not move. composite sensor nodes in which both sensors of each
composite sensor node detect any target object. If there
For type- composite sensor nodes, define a is no overlap between composite-detectable areas ½
composite-detectable area such that the sensors of for any ½ ¾ (½ ½ ¾ ÒÌ ),
¾
the composite sensor nodes will detect the -th target ÒÌ
object if and only if they are located in a composite-
ѽ ´ µ ѽ ´Ì Ð Ö µ (6)
detectable area Ê . That is, when the locations ½
of the two sensors of a composite sensor node at the ÒÌ
two end points of a line segment of length Ð are Ü and Ѿ ´ µ Ѿ ´Ì Ð Ö µ (7)
their disk-shaped sensing areas are ´Ü µ ( ½ ¾), ½
½´´ ´Ü½ µ Ì µ ´ ´Ü¾ µ Ì µ µ ½ if and only if £
´Ü½ ܾ µ ¾ , where ܽ ܾ ¾ о . Similarly, we This is because, if there is no overlap of detectable
×
define a single detectable area (a double detectable areas, then the event that one of the two sensors (both
area ) such that only one (both) of the two sensors of the two sensors) in a composite sensor node detects
of type- composite sensor nodes will detect the -th Ì or Ì is the sum of the two events. One is the event
target object if and only if they are located in a single that one of the two sensors (both of the two sensors) in
detectable area × Ê (in a double detectable the composite sensor node detects Ì , and the other is
area Ê ). That is, when the locations of the the event that one of the two sensors (both of the two
two sensors of a composite sensor node at the two sensors) in the composite sensor node detects Ì .
end points of a line segment of length Ð are Ü and We can derive other measures on this basis. For ex-
their disk-shaped sensing areas are ´Ü µ ( ½ ¾), ample, the measure for at least one sensor in a type-
½´ ´Ü½ µ Ì µ ¡ ½´ ´Ü¾ µ Ì µ ½ if and only composite sensor node detecting a target object is
if ´Ü½ ܾ µ ¾ , where ܽ ܾ ¾ о . In addition, ѽ ´ µ · Ѿ ´ µ.
×
. (We may remove in , × , Remark 1: If there are overlaps of detectable areas of
to simplify the notation, when are not specified.) individual target objects (i.e., if ½ ¾
), the

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estimation error may increase (see Section 7. Numerical 4.2.1 Definitions and notations
examples). To avoid overlap, small sensing areas are Here, we provide a list of definitions and notations used
preferable. In a large sensing area, it may not be able in this subsection.
to detect a small gap between two target objects. This is
¯ ¢ : a vector of parameters describing the -th target
similar to the phenomenon of a large-sensing-area sensor
object.
not detecting a small hole or a small concave part of a
¯ ÒÔ ´ µ: the number of parameters in ¢ . (For simplic-
target object, causing a large estimation error [11], [13].
ity, ÒÔ ´ µ is constant in the following if not explicitly
For no overlaps between composite-detectable areas,
indicated.)
two conditions are required. The first is that a single
sensor in a composite sensor node should not simulta- ¯ ¢´ µ ´¢ ¡ ¡ ¡ ¢ µ.
neously detect multiple target objects. The second is that ¯ ¢ ¢´½ ÒÌ µ.
the two sensors in a composite sensor node should not ¯ Û ´Ð Ö µ.
simultaneously detect multiple target objects. The first ¯ Ï Û ÈÛ ´ ½ ¡ ¡ ¡ Â µ.
condition is identical to that in which the detectable
areas of individual target objects for a simple sensor
Ú Û
¯ ´¢ µ ÒÌ
½ Ѿ ´¢ µ. Û
do not overlap, and is also required in Proposition 1, Ú Ï
¯ ´¢ µ Ú Û Ú Û
´ ´¢ ½ µ ¡ ¡ ¡ ´¢ Â µµ.
which does not use a composite sensor node. However,
the second condition is not needed when we do not use
4.3 Definition and proposition of observability
the composite sensor node. This can be a new cause of
errors, although composite sensor nodes can obtain new Definition 1: A value vector ¢ of parameter vector ¢ is
information. To reduce this error, a shorter Ð is better. observable if there exists a set of composite sensor node
Both conditions can be easily satisfied when the target parameter values Ï ´Ð½ Ö½ µ ¡ ¡ ¡ ´Ð Ö µ satisfying
object’s density is low. £ Ú Ï
that ´¢ µ Ú
´¢¼ Ï
µ for any ¢¼ ¢ ¢¼ ¾ ËÔ for a
The following proposition means that the expected given feasible parameter space Ë Ô . Here, Ï
is called an
number of type- composite sensor nodes in which only observing parameter set. £
one (both) of the two sensors detects a target object is Under an ideal situation (that is, there are no
proportional to ѽ ´ µ (Ѿ ´ µ). This is natural because approximation or measurement errors), ´¢ µ Ú Ï
of the definition of ѽ ´ µ (Ѿ ´ µ). See Appendix A for ´Á¾ ½ ¡ ¡ ¡ Á¾  µ. Therefore, roughly speaking, Definition
mathematical details. In addition, the following propo- 1 implies that if obtained sensor reports can uniquely
sition is valid for any shaped target object. determine ¢ under an ideal situation when we use a
Proposition 3: Let Æ ´½ µ be the number of type- certain set of composite sensor node parameter values
composite sensor nodes in which one of the two sensors Ï , ¢ is said to be “observable.” The statement that
detects a target object, and let Æ ´¾ µ be the number obtained sensor reports can uniquely determine ¢ means
of type- composite sensor nodes in each of which two that any other value vector ¢¼ of parameter vector ¢ is
sensors detect a target object. If the composite sensor not consistent with the obtained sensor reports.
nodes are distributed in a sufficiently large area, The definition of observability requires the uniqueness
of the parameter value vector that is consistent to sensor
Æ ´½ µ ѽ ´ µ (8) reports. However, it does not require the uniqueness of
Æ ´¾ µ Ѿ ´ µ (9) Ú Ï
´¢ µ over the entire domain of ¢. In addition, an
observing parameter set can depend on ¢.
£ Thus, the following proposition is directly derived
Precisely, Eqs. (8) and (9) are affected by the shape from the definition of observability.
of ¨, the sensor-deployed area (Appendix A). However, Ú Ï
Proposition 4: If ´¢ µ is given and if ¢ is observable
if the border effect (the number of composite sensor with an observing parameter set Ï
, we can uniquely
nodes intersecting the border of ¨) is small, they are and exactly estimate ¢. £
independent of the shape of ¨. Practically, this is the Because ´¢ µ Ú Ï ´Á¾ ½ ¡ ¡ ¡ Á¾  µ if there is no
case. approximation or measurement errors, we can uniquely
Note that the sample of the random variable Æ ´½ µ and exactly estimate observable parameter values by
is Á½ and that of Æ ´¾ µ is Á¾ , and that Æ ´½ µ using an observing parameter set Ï
and sensor reports.
Á½ and Æ ´¾ µ Á¾ . Equivalently, if ¢ is observable, there exists an observing
parameter set, and we can uniquely and exactly estimate
4.2 Observability ¢ by using it and sensor reports if there is no approxi-
mation or measurement errors. On the other hand, even
In the remainder of this section, target objects are as- for observable parameter values, if the parameters of
sumed to be convex. Without loss of generality, we can composite sensor nodes are not appropriately chosen,
assume that ½ ¡ ¡ ¡ ÒÌ and н ¾Ö½ ¡ ¡ ¡ Р¾Ö . we may not be able to appropriately estimate them.
Here,
Ô
is the diameter of the -th target object, that is Unfortunately, we cannot judge whether a given is Ï
Ñ Ü´Ü½ ݽ µ ´Ü¾ ݾ µ¾Ì ´Ü½ ܾ µ¾ · ´Ý½ ݾ µ¾ . an observing parameter set without knowing values of

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Ï
¢ or cannot provide , which is an observing parameter 4.4 Derivation of measures
set for any values of ¢. In the following subsections, we derive the measures
If ½ ¡¡¡ ÒÌ , we can get a simplified sufficient ѽ and Ѿ for a certain class of target objects (disk-
condition for observability. This condition can be deter- shaped and rectangular target objects) as examples.
mined by an individual target object. (Consequently, through Eqs. (6), (7), (8), and (9), Ñ ½ and
Ѿ , Æ ´½ µ and Æ ´¾ µ can be obtained.) We first
Ï
Lemma 1: If there exists a set of composite sensor
node parameter values ´Ð½ Ö½ µ ¡ ¡ ¡ ´ÐÒÔ ÒÌ ÖÒÔ ÒÌ µ derive the measures ѽ and Ѿ for a disk-shaped target
satisfying that ½ д ½µÒÔ ·½ ¾Ö´ ½µÒÔ ·½ ¡¡¡
object. Second, we derive measures for rectangular target
д ½µÒÔ ·ÒÔ ¾Ö´ ½µÒÔ ·ÒÔ for ½
Û Û
ÒÌ
and that ´Ñ¾ ´¢¼ ´ ½µÒÔ ·½ µ ¡ ¡ ¡ Ѿ ´¢¼ ´ ½µÒÔ ·ÒÔ µµ
objects. Finally, they are derived when there are disk-
Û Û
´Ñ¾ ´¢ ´ ½µÒÔ ·½ µ ¡ ¡ ¡ Ѿ ´¢ ´ ½µÒÔ ·ÒÔ µµ when ¢¼
shaped and rectangular target objects.
¢ ¢¼ ¾ ËÔ ´ µ for a given feasible parameter space ËÔ ´ µ 4.4.1 Disk-shaped target objects
for ½ ÒÌ , ¢ is observable with an observing
parameter set Ï .£
Proof: Assume that ¢¼ £ ¢ £ and ¢¼ ¢ for £
When a target object is disk-shaped, we can obtain
explicit formulas. We derive ѽ and Ѿ under the as-
sumption that there is a single target object whose radius
ÒÌ . is Ê, the radius of each sensing area in a composite
Suppose that ½ д ½µÒÔ ·½ ¾Ö´ ½µÒÔ ·½ ¡¡¡
sensor node is Ö, and the distance between the two
д ½µÒÔ ·ÒÔ ¾Ö´ ½µÒÔ ·ÒÔ for ½ ÒÌ . sensors in the node is Ð.
Consider the type- composite sensor nodes where
´ £ ½µÒÔ · ½ ´ £ ½µÒÔ · ÒÔ . Note that, if From Appendix B,
Û È
£ , Ѿ ´¢ µ ¼Ò because Ð ¾Ö . There- ѽ ´Ê Ð Öµ
È Ú Û
fore, ´¢ Ú Û
µ Ì Ñ ´¢ µ and ´¢¼
£ ¾
¼ µ for ´ £ ½µÒÔ ·½
µ ¾ ¾´Ê · Öµ¾ × Ò ½ ´ ¾´Êзֵ µ
Ô
ÒÌ
£ Ѿ ´¢ ´ £ ½µÒÔ ·ÒÔ . ·Ð ´Ê · Öµ¾ о ¾´Ê · Öµ
Ú Û Ú Û
Hence, ´¢ µ ´¢¼ µ ´Ñ¾ ´¢ £ µ Ѿ ´¢¼ £ µµ. ¾ ¾ ´Ê · Öµ¾
for
for ¾´Ê · Öµ
Ð
Ð.
Ú Û Ú Û Ú Û Ú Û
According to the assumption of this lemma, for (10)
¢¼ £ ¢ £ , ´ ´¢ ´ £ ½µÒÔ ·½ µ ¡ ¡ ¡ ´¢ ´ £ ½µÒÔ ·ÒÔ µµ Ѿ ´Ê Ð Öµ
´ ´¢¼ ´ £ ½µÒÔ ·½ µ ¡ ¡ ¡ ´¢¼ ´ £ ½µÒÔ ·ÒÔ µµ. ´Ê Ð Ö µ for ¾´Ê · Öµ
Ú Ï Ú Ï
Ð,
Consequently, ´¢ µ ´¢¼ µ if ¢¼ ¢. £ ¼ otherwise,
(11)
In practice, we are likely to face the following situ-
ation: The target object shape can be categorized into where ´Ê Ð Öµ
Ô ´Ê · Öµ¾ ´ ¾ × Ò ½ ´ ¾´Êзֵ µµ
several categories, such as disks and rectangles, and Ð ´ Ê · Ö µ¾ Ð ¾ .
we may not know how many target objects belong to Remark 2: When there are Ò Ì disk-shaped target objects
each category. Note that ÒÔ ´ µ is likely to depend on the
Ï
and the radius Ê of the -th target object satisfies
category to which the -th target object belongs. Let be ʽ ¡¡¡ ÊÒÌ , that satisfies н ¾Ö½ ¾Ê½
the number of target objects in the -th category and Ò Ð¾ ¾Ö¾ ¾Ê¾ ¡ ¡ ¡ ÐÒÌ ¾ÖÒÌ ¾ÊÒÌ is an observing
the number of categories. ½ ¡ ¡ ¡ Ò are also unknown parameter set, due to Lemma 1. This is because ´Ê Ð Öµ
parameters. Similar to Proposition 4, the following corol- is an increasing function of Ê. £
lary shows that we can estimate ½ ¡ ¡ ¡ Ò as well as In the remainder of this paper, if we need to explicitly
other observable parameters ¢.
Ú Ï
indicate “disk-shaped target object” for these measures
Corollary 1: If ´¢ µ is given and if ¢ and values of ѽ and Ѿ , we use the notations ѽ and Ѿ .
½ ¡ ¡ ¡ Ò are observable, we can uniquely and exactly
estimate them. £ 4.4.2 Rectangular target objects
Proposition 4, Lemma 1, and the corollary mentioned
above mean that if we can provide more than
È
ÒÔ ´ µ
This subsection analyzes rectangular target objects. Con-
sider a single rectangular target object with two sides
types of composite sensor nodes with appropriate Ð Ö and a single type of composite sensor node whose sen-
and a sufficiently large number of samples of sensed re- sors’ sensing-area radius is Ö and where the distance be-
sults, we can estimate observable values of parameters of tween the sensors is Ð. The necessary and sufficient con-
any convex target object by using two-sensor composite dition of the first (second) sensor in a composite sensor
sensor nodes. To concretely obtain estimates, a calcu- node detecting the target object is that the location of the
lation method for Ѿ ´¢ Ð Öµ is required. As examples, first (second) sensor is in . Here, is the detectable area
we provide formulas to calculate Ѿ ´¢ Ð Öµ for a certain of this rectangular target object when a basic (i.e., non-
class of target objects. Theoretically, a simulation is ap- composite) disk-shaped sensing area with a radius Ö is
plicable by doing a simulation for various values of ¢ used. That is, ´Ü ݵ Ñ Ò ¾ ܼ ¾ ¾ ݼ ¾ ´Ü
for each pair of ´Ð Ö µ to obtain Ѿ ´¢ Ð Ö µ. However, ܼ µ¾ · ´Ý Ý ¼ µ¾ Ö¾ . To simplify the calculation, we
practically, the applicability of the simulation is limited introduce ´Ü ݵ ¾ Ö Ü ¾· Ö ¾ Ö
to special cases, for example, those in which the shapes Ý ¾ · Ö instead of (Fig. 1). That is, .
of the target objects and the ranges of parameter values Then, the necessary and sufficient condition of the first
are roughly known in advance. (second) sensor in a composite sensor node detecting the

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of ´Ð Ö µ, which satisfies Ð ¾Ö Ñ Ò´ µ is not
D
Ï
D included in an observing parameter set.
On the other hand, if ¾ · ¾
Ô ½ ½ ¡¡¡
¾ · ¾ ,
ÒÌ ÒÌ
that satisfies Ñ Ü´ ´ ½ · ¾Ö¾ µ ¾·´
½ · ¾Ö¾ µ¾ ·
Ô · ·¾
l
¾Ö¾ · ¾Ö¾ µ о ½ о ¾ ¾ Ö¾ Ö¾
t
p
H
r Ö¾ ½ о о ½ · Æ is an observing parameter set where
Æ is a sufficiently small positive scalar. See Appendix D
b for details. £
a In the remainder of this paper, if we need to explicitly
indicate “rectangle” for these measures Ñ ½ and Ѿ , we
use the notations ѽ Ö and Ѿ Ö .
r G
4.4.3 Combinations of disk-shaped and rectangular tar-
get objects
Let Ò be the number of disk-shaped target objects and
Fig. 1. Analysis of two-sensor composite sensor node for ÒÖ ÒÌ Ò be the number of rectangular target objects.
rectangular target object Ò and ÒÖ are unknown parameters. As the measures are
additive if ½ ¾
for any ½ ¾ (½ ½ ¾
ÒÌ ½ ¾ ), we can easily obtain
target object is approximately equivalent to the location Ò
of the first (second) sensor being in . We use this ѽ ´µ ѽ ´Ê Ð Ö µ
approximation and derive measures. Because brute-force ½
ÒÖ
but lengthy computations are needed, we show only the
results here. The computation details are in Appendix C. · ѽ Ö ´ Ð Ö µ (14)
Define · ¾Ö , · ¾Ö, « Ñ Ò´ µ, and Ò
½
¬ Ñ Ü´ µ. Ѿ ´ µ Ѿ ´Ê Ð Ö µ
½
ѽ ´ Ð Öµ ÒÖ
д · µ ¾Ð ¾ for Ð «, · Ѿ Ö ´ Ð Ö µ (15)
«¬ Ó× ½ ´« е · Ь ½
¬ о «¾ · «¾
Ô
for « Ð ¬ , where Ê is the radius of the -th disk-shaped target ob-
´Ô ½ ´ Ð µ · Ó× ½ ´ Ð µµ
Ó× (12) ject and are the side lengths of the -th rectangular
Ô¾
Ô
о ¾ о ¾ target object.
·¾´ · · µ ¾ ¾ о Ô ¾Ð
for ¬ · ¾,
¾ for · ¾ Ð, 5 E XTENDED MODEL ( DYNAMIC SUBMODEL )
5.1 Model description
Ѿ ´ Ð Ö µ
· ¾´ · µÐ¾ Ð for Ð «,
The difference between the static and dynamic sub-
¾«¬ ´ ¾ Ó× ½ ´« еµ ¾Ð¬
models is as follows: The target objects can move and
·¾¬ о «¾ «¾
Ô
every composite sensor node sends a report at each
for « Ð ¬,
¾ ´Ô ¾ Ó× ½ ´ Ð µ Ó× ½ ´ Ð µµ
measurement epoch. There are no other differences.
More precisely, the dynamic submodel is as follows.
·¾ о ¾ · ¾ о ¾
Ô
Each of the ÒÌ target objects may move along an
¾
¾ о for ¬ ԾРunknown route with unknown (maybe time-variant)
Ô ¾· ¾,
speed. Every composite sensor node sends a report at
¼ for · ¾
each measurement epoch. The -th sensor of the -th
Ð, composite sensor node sends the report Á ´Ø µ at time
(13)
Ø , where ½ ¾, ½ ¡ ¡ ¡  . Redefine Á¾ as the
Remark 3: When there are multiple rectangular target time average of the number of type- composite sensor
objects with side lengths (½ ÒÌ ) satisfy nodes in each of which two sensors detect at least
Ð ¾Ö Ñ Ò´ µ, Eqs. (7), (13), and (9) show È È
Ƚ È
Æ ´¾ µ
È
´ ´ · ¾Ö µ´ · ¾Ö µ · о
one target object at a single measurement epoch, that
ØÑ Ò
Ò ½ ·½ ´Á½ ´Øµ Á¾ ´Øµ ½µ Ñ.
´ È È´ · is Á¾
that
¾Ð ´ · · Ö µµ · ¾´ Ö Ð µ
Ø Ø½
Similarly, redefine Á½
ØÑ
Ø Ø½
Ò
Ò ½ ·½ ½
´ Á½ ´Øµ´½
µ · ´ Ö¾ · о Ð Ö µÒÌ µ . Thus, if we can use
Á¾ ´Øµµ · ´½ Á½ ´ØµµÁ¾ ´Øµ ½µ Ñ, which denotes the
Æ ´¾ È withÈ
µ various Ð Ö simultaneously, we can es- time average of the number of type- composite sensor
timate ´ · µ. However, we cannot estimate nodes in each of which one of two sensors detects at
each . Therefore, it is often the case that a pair least one target object at a single measurement epoch.

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5.2 Analyzed results for dynamic submodel where ÈÈ Ò
Ò
Ñ ½ ´Ê Û½ µ
ÈÈ
is an estimator of Ò ,
ÒÌ Ò
Ù¾
Û½ µµ
ÈÈ ÈÈ
´ ½´ · ѽ Ö ´
ÒÌ Ò
It should be noted that the analyzed results originally ½ ½
Ñ ½ ´Ê ÛÂ µ · ÛÂ µµ
Ò
¡¡¡ ´ ѽ Ö ´
ÒÌ Ò
derived for the static submodel are valid for the dy- Â ½ ½
½ Ѿ ´Ê Û½ µ Û½ µµ
Ò
namic submodel. The reasons are as follows. (1) At each ½´ · Ѿ Ö ´
ÒÌ Ò
½
Ѿ ´Ê Û µ · Û µµµ.
Ò
measurement epoch, the dynamic submodel is identical ¡¡¡ Â ´
½ ½ Ѿ Ö ´
to the static submodel. (2) Only the quantity affected
by multiple measurement epochs is Á , but it is no 7 N UMERICAL EXAMPLES
included in derived formulas. Æ´ µ Á is
This section provides numerical examples. The following
valid both for the static and dynamic submodels. The
conditions were used as a basic pattern for the simula-
fact Ú´¢ ϵ ´Á¾ ½ ¡ ¡ ¡ Á¾  µ under the assumption of
tion. We used a monitored rectangular area, ¾¼ ¼¼¼ ¢ ½¼¼
no approximation errors or measurement errors is also
square units, in which composite sensor nodes were
valid.
deployed. Three target objects moved at a speed equal to
10 units of length per unit time along a straight line that
6 E STIMATION METHOD was parallel to the bottom line of the monitored area.
Based on the analysis in the previous section, we propose Two of the objects were disk-shaped with radiuses of 3
an estimation method for multiple target objects that and 30, and the other one was rectangular with sides
may have different parameters. (3, 10). We used six composite sensor nodes of which
Note that Á Æ ´ µ for ½ ¾. Thus, Á¾ parameters ´Ð Ö µ were (3, 1), (4, 1), (9, 2), (12, 3), (20,
can be an estimator of Æ ´¾ µ . 2), and (22, 1) for ½ . We set ¼ per square
unit length for all , and composite senor nodes were
Á½ · ½ Æ ´½ µ (16) placed in a homogeneous Poisson process. (As a result,
Á¾ · ¾ Æ ´¾ µ (17) the mean density of the sensors was 1 per square unit
length.) The mean distance between the target objects
where Á Á is an error of Á from its was 1,000. One simulation yields 2000 measurement
expectation. By using Eqs. (6), (7), (8), and (9), epochs, and 10 simulation were run to obtain each result.
ÒÌ
Á½ · ½ ѽ ´Ì Ð Ö µ (18)
7.1 Approximation errors and sensitivities to vari-
½
ÒÌ ous conditions
Á¾ · ¾ Ѿ ´Ì Ð Ö µ (19) We first confirmed the agreement of the simulation re-
½ sults and the theoretically-derived results and evaluated
The right-hand sides of these two equations are given approximation errors under various conditions and sen-
by derived measures for each class of target object. For sitivities of Á½ (Á¾ ) to various conditions. (In 7.2.1 and
example, if the target objects are disk-shaped (rectan- 7.2.2, the impact of these conditions and errors on the
gles), Eqs. (10) and (11) (Eqs. (12) and (13)) can be used. estimation accuracy is shown.) We compared Á½ (Á¾ )
ÈÈ ÈÈ
When there may be both disk-shaped and rectangular with Æ ´½ µ ( Æ ´¾ µ ), that is, the right-hand side
objects, the right-hand sides of Eqs. (18) and (19) should of Eq. (8) (Eq. (9)). For the disk-shaped target objects,
Ò ÒÖ
be ´ ½ ѽ ´Ê Ð Ö µ · ½ ѽ Ö ´ Ð Ö µµ , Eqs. (10) and (11) were used, and for the rectangular
Ò ÒÖ
´
½ Ѿ ´Ê Ð Ö µ · ½ Ѿ Ö ´ Ð Ö µµ . target object, Eqs. (12) and (13) were used.
In general,
7.1.1 Basic pattern
Á· Ù (20)
For the basic pattern, Figure 2 shows the relative errors
Á ´Á½ ½ Á½ ¾ Á½  Á¾ ½ Á¾ ¾ Á¾  µ, of the theoretical values (that is, the relative error =
È È
where ¡¡¡ ¡¡¡
theoretical value/simulation result -1). Æ ´½ µ shows
´ ½½ ½¾ ¡¡¡ ½ Â ¾½ ¾¾ ¡¡¡ ¾ Â µ , and Ù
a positive bias because we approximated by for
´ ½
ÒÌ
½ ѽ ´¢ Û½ µ ¡¡¡ Â
ÒÌ
½ ѽ ´¢ Û µ Ú µ.
the rectangular target object (see Fig. 1). Æ ´¾ µ also
A set of that minimizes the square error Ì ´Á
¢
can have a positive bias, but it was within a range
Ù µ´Á Ù µÌ can be an estimator ¢ of ¢, where Ì is
of simulation error (see Figure 4 for the variance of
a transpose operator.
the simulation results). In total, the relative errors were
¢ Ö Ñ Ò¢ ´ Á Ù µ´ Á Ù µ
Ì
(21) small, and we concluded that the theoretical results are
valid.
When the target object shape can be categorized into sev-
eral categories, such as disks and rectangles, the number 7.1.2 Independence to speed, monitored area, and
of target objects in each category is also a parameter to moving directions
be estimated. For example, when there may be both disk-
shaped and rectangular objects, Fig. 3 provides Á¾ when one condition such as the target
object speed is modified among conditions used in the
´¢ Ò µ Ö Ñ Ò¢ Ò ´ Á Ù¾ µ´ Á Ù¾ µ
Ì
(22) basic pattern.