This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.IEEE TRANSACTIONS ON MOBILE COMPUTING 2 magnetic sensors for such scenarios. These sensors with for such a purpose. In particular,  and  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  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  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 , path routing, and  uses it to operate sensors in an which was based on the coverage process theory  energy-conserving way. and its application to sensor networks , , . 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 , 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 , 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  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 . We also extended the estimation method in  to 2 P REVIOUS WORK apply to non-convex target objects . 2.1 Basic model These methods we developed in ,  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 Ê¾ , 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 , , ,  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  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 Ê¾ 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 ) describing sensor network operator receives the report from each the use of integral geometry and geometric probability sensor through the network.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.IEEE TRANSACTIONS ON MOBILE COMPUTING 3 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 Ê¾ . 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 : 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 Ê¾ , 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
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.IEEE TRANSACTIONS ON MOBILE COMPUTING 4 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 Ñ½ ´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 Ñ½ ´Ì µ Ê Appendix A, Ñ½ ´Ì µ can be written by an integral form: Ø.) Roughly, Ñ½ ´Ì µ is a sensor nodes are type-2 composite sensor nodes, ... In ´Ü½ Ü¾ 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 Â 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 Ñ¾ ´Ì µ Ê in (included in or intersects with) ¨), given in Appendix ´Ü½ Ü¾ 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 . Ñ¾ ´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 Ñ½ ´Ì Ð Öµ of type- composite sensor nodes in each of which one and Ñ¾ ´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 Ñ¾ for the type- composite measurement epoch. sensor node, we use the notation Ñ¾ ´¢ Ð Ö µ instead of Ñ¾ ´Ì Ð Ö µ, where ¢ is deﬁned later.) There are ÒÌ target objects in 2-dimensional space Proposition 2: Let Ñ½ ´ µ be the measure of a set of type- Ê¾ , 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 .) In object, and let Ñ¾ ´ 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- Ñ½ ´ µ Ñ½ ´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 Ñ¾ ´ µ Ñ¾ ´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 Ü¾ µ ¾ , where Ü½ Ü¾ ¾ Ð¾ . 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 Ü¾ µ ¾ , where Ü½ Ü¾ ¾ Ð¾ . In addition, Ñ½ ´ µ · Ñ¾ ´ 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
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.IEEE TRANSACTIONS ON MOBILE COMPUTING 5 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 , . 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 Ñ¾ ´¢ 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 ¡ ¡ ¡ ´ÐÂ ÖÂ 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 Ñ½ ´ µ (Ñ¾ ´ µ). This is natural because approximation or measurement errors), ´¢ µ Ú Ï of the deﬁnition of Ñ½ ´ µ (Ñ¾ ´ µ). See Appendix A for ´Á¾ ½ ¡ ¡ ¡ Á¾ Â 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 Ñ½ ´ µ (8) reports. However, it does not require the uniqueness of Æ ´¾ µ Ñ¾ ´ 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 Â 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 Ð½ ¾Ö½ ¡ ¡ ¡ ÐÂ ¾ÖÂ . 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 Ý½ µ ´Ü¾ Ý¾ µ¾Ì ´Ü½ Ü¾ µ¾ · ´Ý½ Ý¾ µ¾ . an observing parameter set without knowing values of
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.IEEE TRANSACTIONS ON MOBILE COMPUTING 6 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 Ñ½ and Ñ¾ 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 Ñ¾ , 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 Ñ½ and Ñ¾ for a disk-shaped target satisfying that ½ Ð´ ½µÒÔ ·½ ¾Ö´ ½µÒÔ ·½ ¡¡¡ object. Second, we derive measures for rectangular target Ð´ ½µÒÔ ·ÒÔ ¾Ö´ ½µÒÔ ·ÒÔ for ½ Û Û ÒÌ and that ´Ñ¾ ´¢¼ ´ ½µÒÔ ·½ µ ¡ ¡ ¡ Ñ¾ ´¢¼ ´ ½µÒÔ ·ÒÔ µµ objects. Finally, they are derived when there are disk- Û Û ´Ñ¾ ´¢ ´ ½µÒÔ ·½ µ ¡ ¡ ¡ Ñ¾ ´¢ ´ 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 Ñ½ and Ñ¾ 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 Ð´ ½µÒÔ ·½ ¾Ö´ ½µÒÔ ·½ ¡¡¡ sensor node is Ö, and the distance between the two Ð´ 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 £ , Ñ¾ ´¢ µ ¼Ò because Ð ¾Ö . There- Ñ½ ´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´ÊÐ·Öµ µ Ô ÒÌ £ Ñ¾ ´¢ ´ £ ½µÒÔ ·ÒÔ . ·Ð ´Ê · Öµ¾ Ð¾ ¾´Ê · Öµ Ú Û Ú Û Hence, ´¢ µ ´¢¼ µ ´Ñ¾ ´¢ £ µ Ñ¾ ´¢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 Ñ¾ ´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´ÊÐ·Öµ 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 Ê½ ¡¡¡ ÊÒÌ , that satisﬁes Ð½ 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 Ñ½ and Ñ¾ , we use the notations Ñ½ and Ñ¾ . 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 Ñ¾ ´¢ Ð Öµ is required. As examples, ﬁrst (second) sensor is in . Here, is the detectable area we provide formulas to calculate Ñ¾ ´¢ 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 Ýµ Ñ Ò ¾ Ü¼ ¾ ¾ Ý¼ ¾ ´Ü for each pair of ´Ð Ö µ to obtain Ñ¾ ´¢ Ð Ö µ. However, Ü¼ µ¾ · ´Ý Ý ¼ µ¾ Ö¾ . To simplify the calculation, we practically, the applicability of the simulation is limited introduce ´Ü Ýµ 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
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.IEEE TRANSACTIONS ON MOBILE COMPUTING 7 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 Ð¾ ½ Ð¾ ¾ ¾ Ö¾ Ö¾ t p H r Ö¾ ½ Ð¾ Ð¾ 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 Ñ¾ , we use the notations Ñ½ Ö and Ñ¾ 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 Ñ½ ´µ Ñ½ ´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. · Ñ½ Ö ´ Ð Ö µ (14) Deﬁne · ¾Ö , · ¾Ö, « Ñ Ò´ µ, and Ò ½ ¬ Ñ Ü´ µ. Ñ¾ ´ µ Ñ¾ ´Ê Ð Ö µ ½ Ñ½ ´ Ð Öµ ÒÖ Ð´ · µ ¾Ð ¾ for Ð «, · Ñ¾ Ö ´ Ð Ö µ (15) «¬ Ó× ½ ´« Ðµ · Ð¬ ½ ¬ Ð¾ «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 Ð¾ ¾ Ð¾ ¾ target object. ·¾´ · · µ ¾ ¾ Ð¾ Ô ¾Ð for ¬ · ¾, ¾ for · ¾ Ð, 5 E XTENDED MODEL ( DYNAMIC SUBMODEL ) 5.1 Model description Ñ¾ ´ Ð Ö µ · ¾´ · µÐ¾ Ð for Ð «, The difference between the static and dynamic sub- ¾«¬ ´ ¾ Ó× ½ ´« Ðµµ ¾Ð¬ models is as follows: The target objects can move and ·¾¬ Ð¾ «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 Ð¾ ¾ · ¾ Ð¾ ¾ Ô Each of the ÒÌ target objects may move along an ¾ ¾ Ð¾ for ¬ Ô¾Ð unknown 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 ¡ ¡ ¡ Â . 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 È½ È Æ ´¾ µ È ´ ´ · ¾Ö µ´ · ¾Ö µ · Ð¾ 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 · Ð¾ 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.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.IEEE TRANSACTIONS ON MOBILE COMPUTING 8 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´ · Ñ½ Ö ´ ÒÌ Ò It should be noted that the analyzed results originally ½ ½ Ñ ½ ´Ê ÛÂ µ · ÛÂ µµ Ò ¡¡¡ ´ Ñ½ Ö ´ ÒÌ Ò derived for the static submodel are valid for the dy- Â ½ ½ ½ Ñ¾ ´Ê Û½ µ Û½ µµ Ò namic submodel. The reasons are as follows. (1) At each ½´ · Ñ¾ Ö ´ ÒÌ Ò ½ Ñ¾ ´Ê ÛÂ µ · ÛÂ µµµ. Ò measurement epoch, the dynamic submodel is identical ¡¡¡ Â ´ ½ ½ Ñ¾ 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´¢ Ïµ ´Á¾ ½ ¡ ¡ ¡ Á¾ Â 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 Ñ½ ´Ì Ð Ö µ (18) 7.1 Approximation errors and sensitivities to vari- ½ ÒÌ ous conditions Á¾ · ¾ Ñ¾ ´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 Ñ½ ´Ê Ð Ö µ · ½ Ñ½ Ö ´ Ð Ö µµ , Eqs. (10) and (11) were used, and for the rectangular Ò ÒÖ ´ ½ Ñ¾ ´Ê Ð Ö µ · ½ Ñ¾ 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 Â Á¾ ½ Á¾ ¾ Á¾ Â 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 Ñ½ ´¢ Û½ µ ¡¡¡ Â ÒÌ ½ Ñ½ ´¢ ÛÂ 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.