UCLA Papers. Title. Permalink. Authors. Publication Date. Localized Edge Detection in Sensor Fields.

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1 UCLA Papes Title Localized Edge Detection in Senso Fields Pemalink Authos K Chintalapudi Govindan Publication Date 3-- Pee eviewed escholashipog Poweed by the Califonia Digital Libay Univesity of Califonia

2 Localized Edge Detection in Senso Fields Kishna Kant Chintalapudi, amesh Govindan Univesity of Southen Califonia, Los Angeles, Califonia, USA, 97 Abstact A wieless senso netwok fo detecting lage-scale phenomena (such as a contaminant flow o a seismic distubance) may be called upon to povide a desciption of the bounday of the phenomenon (eithe a contou o some bounding box) In such cases, it may be necessay fo each node to locally detemine whethe it lies at (o nea) the edge of the phenomenon In this pape, we show that such localized edge detection techniques ae non-tivial to design in an abitaily deployed senso netwok We define the notion of an edge and develop pefomance metics fo evaluating localized edge detection algoithms We popose thee diffeent appoaches fo localized edge detection and pesent one example scheme fo each In all ou appoaches, each senso gathes infomation fom its local neighbohood and detemines whethe o not it is an edge senso We evaluate the pefomance of each of the example schemes and compae them with espect to the developed metics Intoduction Seveal physical phenomena (fo instance, contaminant flows [3] and seismic distubances) can span lage geogaphic extents Fine-gain sensing of these time-vaying phenomena can help scientists undestand what factos (eg, soil density vaiations) affect the spead of these phenomena One way to achitect an enegy-efficient senso netwok fo studying these phenomena is to stoe the detections of the phenomena within the netwok and povide a quey inteface which enables scientists to undestand the tempoal and spatial popeties of these phenomena We anticipate that one common quey will ask fo the spatial extent of the phenomenon at a given time: fo example, Which sensos saw the pimay wave befoe time T? Fo enegy-efficiency easons, it makes moe sense fo to design the spatial quey that etuns a bounday that captues all o most nodes that satisfy the quey pedicate A geometic epesentation of the bounday has the potential to be moe concise (and theefoe moe enegy-efficient) than an enumeation of all nodes Examples of such epesentations include contous, hulls, o bounding boxes An enegy-efficient bounday finding algoithm will need to caefully choose nodes in the senso netwok and compute the bounday in-netwok A key component of such an algoithm is a localized edge detection scheme: a technique by which each node locally detemines (pehaps by gatheing infomation fom othe nodes within its neighbohood) whethe it lies on o nea the bounday specified by the quey If a eliable technique existed fo localized edge detection, then, conceptually at least, bounday finding is simply a matte of sequentially tavesing all nodes that detemine themselves to be on the edge Localized edge detection will be an essential component of bounday tacking as well; as the phenomenon evolves with time, nodes at the edge may alet neighboing nodes (in a manne simila to taget tacking [4]) We think of localized edge detection as a poweful pimitive upon which a vaiety of applications might be built Edge detection has been widely studied in the context of digital image pocessing Filteing [] is one of the most common appoaches to detecting edges in images To detemine whethe an image pixel is at an edge o not, this appoach applies a filte to values of a set of neighboing pixels As such, these techniques can be diectly applied to localized edge detection Howeve, one fundamental diffeence between images and a senso field is the spatial egulaity of infomation A digital image is a egula gid of pixels, and infomation is sampled at egula intevals Almost all standad digital image pocessing techniques (Fouie tansfoms, high-pass filteing) ely on having infomation about the image at egula intevals Deployment and maintenance of thousands of sensos in a gid like egula fashion ove lage geogaphical extents is clealy infeasible It is expected that sensos will be abitaily placed in the senso field and will be pone to failues o even node displacement With this elaxation of egulaity, it is less clea that digital image filteing can be applied to localized edge detection In fact iegula node placement makes it had to even define pecisely whethe a node is at an edge o not In Section, we show how to cicumvent this difficulty and define some met-

3 ics fo localized edge detection Because the efficacy of edge detection based on digital image filteing is unclea, we conside two othe classes of edge detection schemes in Section 3: a statistical scheme and a classifie-based scheme (that is suggested by the patten ecognition liteatue []) We find that, ove a faily boad ange of opeating conditions, the classifie scheme out-pefoms the othe schemes We pesent ou evaluation esults in Section 4, and conclude in Section 5 To ou knowledge, no pio wok has consideed localized edge detection in senso netwoks Concuently, Nowak and Mita [5] descibe a scheme fo estimating the bounday of a lage-scale phenomenon by aggegating eadings along a pedefined hieachical stuctue within the netwok Thei appoach is somewhat complementay to ous, in that ou localized edge detection is a pimitive that might be used in a vaiety of bounday estimation applications (not just in thei algoithm, but also in applications that estimate moe concise, but appoximate, bounday desciptions such as ellipses and hulls) Edge Detection In this section we descibe ou model of the senso field and discuss definitions fo an edge We then develop metics to evaluate localized edge detection algoithms and discuss the tade-offs involved in the design of localized edge detection Assumptions, Models and Teminology In what follows, we make faily geneal assumptions about the capabilities of senso nodes and the stuctue of senso netwoks Senso nodes can be abitaily deployed, but each such node knows its location, pehaps using a localization system [6] Fo simplicity of exposition, we assume that the deployment of sensos is in the plane, and location can be specified by (x s,y s )) We use the tem senso field to both mean the geogaphical egion coveed by the deployment, and the set of sensos within the egion Sensos can make measuement eos, and ou localized edge detection schemes will need to be obust to these (Edge detection algoithms may also exhibit significant eo due to eos in localization [7]; we do not model such eos, since ou inteest is in undestanding how to compensate fo senso eo in edge detection) We model an edge as follows Conside a phenomenon that spans some abitaily shaped sub-egion of the senso field Each senso can, based on locally collected measuements, detemine whethe it belongs to the sub-egion coveed by the phenomenon o not We call the function that makes this decision the event pedicate, and denote the event pedicate at senso s by η s Taking ou example in Section, if the phenomenon of inteest is the geogaphical extent coveed by the pimay seismic distubance at time T, the event pedicate fo each senso is, infomally: Did I see a pimay seismic distubance at o befoe T? Given an event pedicate, we can then define the inteio of a phenomenon (I) to be the spatial egion such that, if a pefectly calibated eo fee senso wee placed in this egion its pedicate function would have evaluated to The exteio O of the phenomenon can be similaly defined Based on these, thee exists an idealized definition of the edge of a phenomenon E: the edge is the set of all points (x,y), such that evey non-empty neighbohood of (x, y) intesects with both I and O We call E the ideal edge, and E epesents the gound tuth that defines the bounday of the phenomenon This idealized definition is desciptive, but does not give us much insight fo designing o evaluating localized edge detection schemes The ideal edge has no thickness and theefoe constitutes a vey estictive definition of an edge Intuitively, we would like a senso to conside itself to be an edge senso if it is close to the ideal edge than any othe senso Fo this eason, we intoduce the notion of toleance of an edge detection scheme We define a senso to be an edge senso if it a) is in the inteio of the phenomenon, and b) lies within a pespecified distance of the ideal edge We call the toleance adius, and the aea aound a senso node coveed by a cicle of adius the toleance neighbohood The toleance adius oughly measues the thickness of the edge that the designe of a localized edge detection scheme is willing to toleate Fo a given toleance adius, we can define metics that enable us to compae the efficacy of diffeent edge detection schemes (Section ) A stonge definition an edge might be to equie continuity among the set of edge sensos that is, that thee exists a path between evey pai of edge sensos that only taveses edge sensos We have not adopted this definition because it seemed to be beyond the ealm of localized techniques to ensue this popety Of couse, if we could ensue this popety, it would be easy to define enegy-efficient bounday finding algoithms that simply tavesed edge nodes As such, whateve bounday finding algoithms that we build on top of localized edge detection will need to tavese some non-edge sensos to detemine a continuous bounday fo a phenomenon Such algoithms ae beyond the scope of this pape Finally, detecting whethe a node lies at the edge of a phenomenon is slightly diffeent fom detecting whethe a node lies on (o nea) a contou (o iso-lines; ie continuous cuves acoss the senso field defined by sensos having the same value of, say, tempeatue) In the geneal case, thee isn t a well-defined notion of the inteio and exteio of a contou, as much as thee is a distinction between whethe a senso detects a phenomenon o not

4 Metics Inteio Inteio senso node adius of toleance Edge Senso adius of toleance senso node Not an Edge Senso Figue : If the edge passes though the adius of toleance it is deemed an edge senso Thee ae, boadly speaking, two classes of desiable chaacteistics of localized edge detection: obustness, and pefomance These chaacteistics infom ou choice of metics fo localized edge-detection Thee ae seveal desiable obustness popeties of an edge detection algoithm Fist, as we shall show late, localized edge detection algoithms can intinsically exhibit eo, failing to detect an edge when thee is one, o detecting an edge when thee is none A good algoithm has low intinsic eo Second, localized edge detection algoithms must be elatively obust to easonable levels of senso calibation eo Finally, many localized edge detection schemes employ thesholds to decide on the existence of an edge We would pefe schemes which ae elatively insensitive to the theshold settings ove a boad ange of opeating conditions In the pefomance categoy, an obvious consideation is enegy expended in communication Thee exists a tade-off between enegy and accuacy in localized edge detection; intuitively, a node can get infomation fom a bigge neighbohood to incease the likelihood of a positive detection The second pefomance citeion is the quality of the esult, defined by the actual thickness of the edge Although ou definition of an edge above includes a toleance adius that nominally defines an edge thickness, an actual edge detection scheme might have a thickness that is lage o smalle than this adius Based on the above discussion, in this section we use the following metics to evaluate the pefomance of localized edge detection algoithms Exteio Edge Sensos Thickness Non Edge Sensos Let S be the set of all sensos Let E be the cuve epesenting the edge (as defined in Section ) Suppose set S tue be the set of sensos in the senso field which ae within a distance of fom E Let S det be the set of sensos maked as edge sensos by the algoithm and let N be the total numbe of senso nodes Pecentage Missed Detection Eos e m : This epesents the faction of sensos which lie within the adius of toleance (S tue ) but wee not maked as edge sensos (S det ) S tue S e m = det () S tue Figue : Having a toleance adius gives a cetain thickness to an edge False Detection Eos e f : This epesents the faction of nodes that declaed themselves to be edge sensos but should not have (S det S tue ) among the est of the (S S tue ) sensos Fo this eason, the denominato fo e f is diffeent fom that in ()

5 Sdet S tue e f = () N S tue Exteio edge Mean thickness atio e t : Let t(s,e) be the mean distance of all the sensos in set S to the edge E We define, Inteio e t = t ( S det,e ) t (S tue ) (3) t (S tue,e) To avoid the effect of andom outlies, we conside only the closest 95% edge sensos in the mean We ae now eady to discuss some localized edge detection schemes that illustate the tade-offs involved in localized edge detection 3 Thee appoaches to localized edge detection In this section we popose thee qualitatively diffeent appoaches to localized edge detection in a senso field: a statistical appoach, an appoach dawn fom image pocessing and an appoach dawn fom the patten ecognition liteatue Each appoach can be used to geneate a family of algoithms fo edge detection In all these appoaches, each senso gathes infomation fom sensos in its neighbohood and independently ties to detemine if an edge passes within its toleance adius Specifically, the senso gathes the location and the values of the event pedicate (that detemines whethe the senso is in the inteio o the exteio of a phenomenon) fom each node within the neighbohood One paamete that detemines the pefomance of all algoithms, to vaying extents, is the size of this neighbohood Abitay placement of the sensos coupled with senso eos can esult in detection eos In geneal, the pefomance of a scheme impoves as we collect infomation fom moe sensos (lage neighbohood) This is because the node gets moe samples fom the inteio and the exteio of the phenomenon, and can make moe confident estimates even in the pesence of senso eos Howeve, collecting moe infomation incus moe communication ovehead and hence inceases the enegy usage of the scheme We have aleady mentioned this enegy accuacy tade-off We epesent this paamete by a cicle of adius centeed aound the senso and call it the pobing adius Typically, the pobing adius is geate than the toleance adius ie > (see Figue 3) Geneally the geate the atio, the bette the pefomance of the algoithms in tems of Figue 3: Pefomance can be impoved by gatheing infomation beyond the toleance adius Hee, the senso gathes infomation in a cicle of adius, the pobing adius and is able to detect an edge which passes though the aea of toleance moe eliably eos and thickness atio, howeve the communication ovehead inceases oughly as In the est of the pape we shall efe to this neighbohood as the pobing neighbohood 3 The statistical appoach A geneal statistical scheme would gathe data fom the sensos in the pobing neighbohood and pefom statistical analysis to decide whethe o not the senso is an edge senso The advantage in this appoach is that statistical methods can be explicitly tailoed to be obust to eos, if eo chaacteistics ae known The geneal algoithm fo a statistical scheme then needs thee components to be specified The infomation to be collected fom the neighbos A set of statistics Γ,Γ,,Γ n based on the infomation collected fom the neighbos 3 A boolean decision function Ψ ( ) Γ,Γ,,Γ n to decide if the senso is an edge senso The decision function usually would involve compaing a value evaluated using {Γ i } i=n i= against a theshold which maybe statically o dynamically assigned 3 An example scheme In this pape we evaluate a specific statistical scheme that we designed fo edge detection The key idea behind the scheme is the obsevation that if one collects the event pedicate values

6 fom sensos in the neighbohood, and these values fom a bimodal distibution (spikes at and ) then an edge is pesent Let n + be the numbe of valued event pedicates and n be the numbe of zeo valued event pedicates in the neighbohood We calculate the following statistic: n+ n Γ = (4) n + + n Ψ(Γ) = i f S γ, i f S < γ Ou statistical scheme is intuitively simple, and theefoe foms a baseline fo compaison against othe schemes One salient featue lacking in the statistical scheme is that it does not take the geogaphical locations of sensos into account when making its decisions Fo abitaily placed sensos, as we shall see late, this can make a diffeence Designing the statistical scheme to be obust to senso eos is a bit ticky, as we now explain If the sensos wee pefectly calibated and eo fee, the pesence of an edge would be indicated by a non-zeo value of the statistic Γ and any γ > would suffice In a moe ealistic scenaio, with abitaily placed sensos having calibation and measuement eos, the statistic would yield non-zeo values in absence of edges because of senso eos Also if >, fo edges passing in the pobing neighbohood which do not lie in the aea of toleance, (4) would give a non-zeo value Then, the choice of an appopiate theshold γ would detemine the pefomance of the scheme In geneal the choice of γ (,) depends, ρ and the pefomance equiements of the application 3 Analysis of the scheme and choice of γ To gain some intuition about the choice of γ and how it elates to the toleance adius and the pobing neighbohood, we analyze the pefomance of the poposed statistical scheme Fo ou analysis we assume that nodes ae placed in the egion at locations dawn fom a unifom density function with a density ρ sensos pe unit aea Also we assume that the sensos make an eo in evaluating the value of the event pedicate with a pobability p We hope that this eo model encapsulates both calibation and measuement eos Futhe, we assume that the pobing neighbohood is so small in compaison to the aea coveed by the entie phenomenon that the edge can be appoximated by a staight line in this egion As discussed in Section, eos can be eithe false detections e f o missed detections e m False detections can aise in two ways One cause of false detections is when thee is no edge in the pobing adius but the algoithm detects an edge due to senso eos The second occus when thee is an edge in the pobing adius but not within the toleance adius We (5) call the fome kind of eos pue false detections (e p f ) and the latte unwanted detections (e ud ) e f = e p f + e ud (6) We make this distinction because a high e p f can esult in a lage numbe of sensos being deemed edge sensos even when they ae in the middle of a phenomenon because of senso eos and local vaiations in senso density On the othe hand, a high e ud simply inceases the thickness of the edge Fo cetain applications, the edge thickness may not be as hamful as identifying a senso as an edge senso when it is fa fom an edge Fo fixed values of and ρ, the eos e m, e p f and e ud depend on the choice of γ Now, the numbe of sensos pesent in an aea a can be modeled by a Poisson andom vaiable, aρ (aρ)n P(N = n) = e n! and the numbe of sensos making an eo M among N sensos can be modeled by a binomial andom vaiable, ( ) n P(M = m N = n) = (p) m ( p) n m (8) k Based on these assumptions, one can numeically calculate the pobability density function fo Γ defined in (4), fo given values of and ρ The pocedue fo calculating the density function is descibed in Appendix A Choosing γ : An Example Figue 4, computed fom ou analysis, shows the vaiation of the pecentage of tue, unwanted and false detections as γ vaies fom to, fo two diffeent values of ( and 5) The senso density is such that, the expected numbe of sensos in aea of toleance is 5 The senso eo pobability p = 5 Suppose an application equies a tue detection atio of at least 8% and also wants false detections to be below % Also suppose the application can do with slightly thicke edges and allows about % exta edge detections Fo =, coesponding to an eo of 8%, the S 5 Coesponding to this value of S, the false eo e p f 3% and e ud = The situation is consideably impoved when = 5 If we choose a theshold of 4, one can achieve 8% tue detections, but now e p f less than % and e ud 3% Hence, by inceasing the pobing adius we have impoved the pefomance of ou scheme Howeve, by inceasing the pobing aea we incued an incease in communication ovehead by 5% ( ) Clealy since pefomance depends on the choice of γ and, it becomes essential fo sensos to be able to figue out a suitable setting fo these paametes fo satisfactoy opeation It is conceivable that pe-calculated pefomance cuves (7)

7 pecentage = = 5 = 5 = = 5 tue unwanted false theshold Figue 4: Vaiation of pecentage tue, unwanted and false detections as theshold S vaies fom to fo values of and 5, an expected value of 5 sensos in the toleance egion and p = 5 simila to those in Figue 4 ae stoed in the senso nodes a pioi Nodes estimate the local senso density and based on the pefomance citeion desied, use the pefomance cuves to come up with an opeating theshold One staight-fowad way to map filteing techniques within the context of senso netwoks is to teat each senso as a pixel, and diectly apply Equation 9 Howeve, sensos may not exhibit pixel-like egulaity in placement To ovecome this, we obseve that Equation 9 is essentially a weighted aveage of all the neighboing values Ou appoach is to deive the weights fo the sensos based on the continuous vesion of the filte namely H(x,y) Let PA s be the set of all the sensos in the pobing aea of a senso s o Let V s be the value obtained fom a senso s, and (x s,y s ) its location Then the filteing output of senso s o is given by, V so = s PA s o W(x s,y s )H(x s,y s )V s () Hee, W(x s,y s ) ae weights to compensate fo the uneven weighing caused due to abitay positioning and vaiations in numbe of the sensos In geneal W(x s,y s ) is a function of senso locations and H Unlike the statistical filte, then, ou famewok fo using image pocessing techniques allows us to take the geogaphic locations into account In geneal, ou famewok allows fo diffeent kinds of H and W functions We do not exploe this space, choosing instead to evaluate one paticula localized edge detection scheme that fits into this famewok 3 The image pocessing appoach Numeous techniques fo edge detection have been developed and analyzed in the image pocessing liteatue [] It is theefoe tempting to attempt to apply such techniques to localized edge detection in senso netwoks In this pape we do not exhaustively examine all possible image pocessing techniques, but simply pick a famewok that can incopoate a class of high pass filteing techniques (a standad way of pefoming edge detection in images eg Pewitt, Sobel filtes) fo localized edge detection A high-pass filte etains only the high fequencies (abupt changes such as edges) pesent in the image and emoves all the unifomities Designing a filte with a desied fequency esponse is a matue at and seveal diffeent techniques exist In geneal, if a filte with a fequency esponse F( f x, f y ) is desied then a filte H(x, y) can be designed to appoximately match the desied fequency esponse Hee, f x and f y epesent the fequencies in the image in the x and y axes To detect edges, the image P(x, y) can be filteed by convolving with the filte H(x, y) Within the context of digital image pocessing, the x and y ae discetized into pixels The filte and the image ae epesented by matices H(i, j) and P(i, j) espectively The filteed image P is computed as a convolution of P and H P (i, j) = m=k m= n=k n= P(i + m k, j + n k )H(m,n) (9) 3 The Pewitt filte based scheme The Pewitt (diffeence) filte [] in digital image pocessing is a set of two matices, H x = () H y = () H x and H y ae based on the functions, H x (x,y) = i f x, i f x < H y (x,y) = i f y, i f y < (3) (4) σ x and σ y ae the gadients in the image, along the x and y diections espectively A high value of say σ = σx + σ y would indicate an edge We define V s as, V s = i f η s =, i f η s = (5)

8 Hee, η s is the event pedicate of senso node s Suppose we wish to filte at node s o based on (), we need to decide H x (x s,y s ), H y (x s,y s ), W x (x s,y s ) and W y (x s,y s ) to calculate σ x and σ y Calculation of H x and H y Fom (), fo calculating σ x, H x (x s,y s ) is - if x s < x so, if x s > x so and othewise Fo calculating σ y, H y (x s,y s ) is - if y s < y so, if y s > y so and othewise Exteio Patitioning Line Edge Senso Inteio Exteio Patitioning Line Not an Edge Senso Inteio Selection of W x and W y We calculate the weights to make the scheme moe toleant to the vaying numbe of sensos in the egion Conside, the filte H x fo calculating σ x Due to abitay placement, suppose the numbe of sensos to the left (x s < x so ) of s o ae n le ft and those on the ight ae n ight Suppose n le ft > n ight Then filteing at s o will be biased towad the left side This bias can be avoided if we choose W x such that V s fom sensos on the left by n and those on the ight le ft by n A simila stategy can by used fo calculation of σ y ight Let ( ) i=4 n i+,n i be the numbe of sensos with and values of event pedicates in the i th quadant of the pobing aea i= aound the senso in question Quadants ae numbeed in the anti-clockwise diection The weights then become, W x (x,y) = W y (x,y) = n + +n +n 4+ +n 4 i f x < x so, n + +n +n 3+ +n 3 i f x > x so n + +n +n + +n i f y > y so, n 3+ +n 3 +n 4+ +n 4 i f y < y so Based on (5),(6) and (7) we obtain, σ x = σ y = The algoithm can now be stated as: (6) (7) n + +n 4+ n n 4 n + +n +n 4+ +n 4 n + +n 3+ n n 3 n + +n +n 3+ +n 3 (8) n + +n + n n n + +n +n + +n n 3+ +n 4+ n 3 n 4 n 3+ +n 3 +n 4+ +n 4 (9) Collect the values ( ) i=4 n i+,n i in the pobing aea i= Calculate σ using (8),(9) and compae it against a theshold σ to decide whethe o not the senso is an edge senso Figue 5: Classifie-based schemes attempt to detemine a line which patitions all the event pedicates in the pobing aea into s and s If this line passes though the aea of toleance the senso is deemed an edge senso Analysis fo the image pocessing based scheme can be done simila to the statistical scheme (as in Section 3) The analysis is povided in Appendix B 33 The classifie-based appoach Ou last appoach comes fom the patten ecognition liteatue This classifie-based appoach elies on the infomation povided by sensos in the inteio I being significantly diffeent fom that by sensos in the exteio O Such a bi-patite data set will allow classification [] (patitioning) the data into two subsets, such that simila data lie in the same subset and dissimila data lie in diffeent subsets In a classifie, a senso would attempt to patition data gatheed fom its neighbohood into two classes The success of the patition may be assessed by a patition validity measue [] A successful patition implies the pesence of an edge The simplest classifie is a linea classifie This classifie attempts to find a line L(a,b,c) ax + by + c = such that all the sensos (in the pobing neighbohood) with η s = ae on one side of the line and those with η s = lie on the othe side A localized edge detection scheme based on a linea classifie is then quite simple If this line passes within a distance of fom the senso, the patition is accepted as valid and the edge is deemed as an edge senso Figue 5 depicts the scenaio Two impotant diffeences exist between classifie-based edge detection and ou two pevious appoaches Fist, the linea classifie explicitly encodes a notion of geogaphy Second, this classifie does not equie any thesholds fo opeation Classifie Definition In the event of senso eos, an exact patition may not exist In this case, we ty to find a line which maximizes the numbe of sensos with like values of

9 event pedicate on each side of the line Let PA s be the set of all sensos in the pobing aea of senso s Let s o be the senso pefoming edge detection Let L(a,b,c) be the line specifying the classifie Let V s be as defined in Equation 5 We define classifie scoe J s as, J so (a,b,c) = V s SN(ax s + by s + c) () SN(x) = s PA s o i f x < i f x = i f x > () In geneal thee can be seveal methods to find the optimal line based on J so In ou implementation, we sample (θ,c) in the egion c [,] and θ [,π] Hee, each sample specifies a line L(tan(θ),,c), which has a slope tan(θ) and intesects the x-axis at c We evaluate the value of J so at all these sample lines; the line L opt with the highest value of J so is chosen as the patitioning line We then deem the patition as valid if the optimal line L opt (a,b,c) satisfies ax so +by so +c ; that is, the line is within the adius of toleance (a +b ) The classifie based algoithm can now be summaized as: Collect all the coodinates and event pedicate values within the pobing adius Find a line L opt (a,b,c) which gives the maximum value fo J so (a,b,c) 3 If L opt passes within the adius of toleance, the senso is deemed an edge senso 4 esults In this section we compae the pefomance of the thee poposed schemes though extensive simulations We descibe the datasets used, followed by the details of the simulations We end this section by compaing the thee schemes descibed in the pevious section 4 The simulation famewok In all simulations, ou sensos ae located in a m by m aea, thei locations dawn fom a unifom distibution ove the aea The adio ange of all the sensos is m and assumed omni-diectional In all simulations, we abitaily chose the toleance adius equal to the adio ange of the sensos In this context, an =, oughly implies a -hop neighbohood The Data Sets Ou simulations wee conducted fo two diffeent data sets The fist, linea bounday data sets D l, compise of andomly chosen lines y+mx+c = c is dawn fom a unifom distibution ove the entie x-axis within the senso field m = tan θ is the slope of this line, geneated by dawing θ unifomly in (,π) Sensos with mx s + y s + c belong to the inteio egion (η s = ) and est belong to the exteio egion (η s = ) The edge (gound tuth) is defined by the line y+mx+c = The linea bounday foms a baseline fo evaluating ou scheme; an acceptable edge detection scheme should pefom well fo this data set The second, elliptical bounday data sets D e, consist of ellipses E(a,b,x,y,θ) = andomly chosen within the senso field a and b ae lengths of the majo and mino axes of the ellipse, unifomly dawn ove the length of the senso field (x,y ) is the cente of the ellipse dawn unifomly ove the entie senso field θ, which is the angle between the majo axis of the ellipse and the x-axis is dawn unifomly in (, π) Let (x s,y s) be the senso coodinates in a coodinate system natual to the ellipse (the majo and mino axes of the ellipse fom the x and y axes) (x s,y s ) can be obtained by fist tanslating the oigin to (x,y ) and then otating the axes by θ in the anti-clockwise diection If (x s) (y s), the senso is deemed to belong to the inteio egion (η s = ) and to a b the exteio (η s = ) othewise The edge (gound tuth) is defined by the ellipse E(a,b,x,y,θ) = Ellipses of diffeent eccenticities epesent continuously cuved edges, and can be seve to distinguish localized edge detection schemes Factos To examine the impact of density, we chose thee values of ρ: ρ = 6 sensos/sqmt (low density - about 5 senos within adio ange), ρ = 36 sensos/sqmt (modeate density - about 5 senos within adio ange), and ρ 3 = 7 sensos/sqmt (high density - about 3 senos within adio ange) To captue the impact of senso eos, we used a simple bit flipping technique In this model, a senso toggles its event pedicate value fom its tue value with a pobability p We used thee diffeent choices fo p p = % (low), p = 5% (modeate) and p 3 = % (high) Thus, fo the linea bounday data set, a single simulation un epesents one line chosen andomly, fo one value of density and senso eo Fo a given density and senso eo pobability, we aveage the pefomance metics fo a localized edge detection scheme ove diffeent uns coesponding to diffeent andomly chosen lines The same is tue fo the ellipse Paametes We chose five diffeent values of, namely, 5,, 5 and 3 We an simulations fo all the data sets, fo each of the thee schemes, fo the five values of The statistical and the image pocessing schemes equie choosing γ (,) and σ (,) espectively This choice

10 Detection Pobability Detection Pobability Data sets with linea edges Mean Thickness Eo Mean Thickness eo 5 5 Data sets with ellipical edges Stat Img Class False detection pobability Linea Elliptical Figue 7: False detections fo the classifie based scheme decease with incease in Figue 6: Enegy accuacy tade-off : As moe and moe neighbohood is examined ( the detection pobability inceases fo all the thee detection schemes) (as discussed in Section 3) can impact pefomance To be fai to all schemes, fo a given simulation un, we chose the best theshold value (using the analysis in Appendix A and B) defined thus: Choose the theshold which satisfies e p f % and minimizes e m Thus, fo schemes that equie thesholds, ou simulations epesent the fewest possible missed detections We evaluate ou schemes with espect to the metics descibed in Section 4 Simulation esults In this section we discuss the esults of ou simulations The space of paametes and factos we have exploed is lage athe than exhaustively pesent all of ou esults, we selectively descibe the simulation esults in an effot to give the eade an undestanding of the main diffeences between the schemes We stat by consideing (Figue 6) which shows the vaiation of e t (mean thickness eo) and e m (detection pobability) fo modeate eo (p = 5%) and modeate density (ρ = 36 ) with It depicts the basic natue of the enegy accuacy tade-off As seen in Figue 6, the edge detection pobability inceases with incease in Howeve, inceasing inceases the communication ovehead as o( ) and hence the enegy consumption Fo linea data sets all the thee schemes give simila detection atios at, while the classifie gives a thinne edge Fo elliptical data sets, at, the classifie pefoms slightly infeio to the othe two schemes, howeve it gives a much thinne edge The pefomance of statistical and image pocessing based schemes is simila The statistical and image pocessing based schemes, allow one to estict the false detection pobability by selecting an appopiate choice of theshold (γ and σ ) In all ou simulations, the choices esticted false detection to below % In the classifie based scheme, thee is no such diect way to estict false eo pobability Figue 7 shows the vaiation of false detections made by the classifie scheme with incease in fo the modeate density, modeate senso eos data sets The false detection pobability deceases with incease in The classifie scheme behaves qualitatively diffeently fom the statistical and image pocessing schemes Fo the latte, as the atio inceases, the edge thickness inceases while fo the classifie based schemes the edge thickness deceases Edge thickness eo esults fom pue false detections (e p f )and unwanted detections (e ud ) Since we esticted e p f < % fo ths statistical and image pocessing based schemes, edge thickness eo mostly aises out of e ud In the classifie based scheme e ud is small and does not change significantly with This is depicted in Figue 8 The incease in e ud causes an incease in thickness eo fo the statistical and image pocessing based schemes, while a decease in false detections leads to a decease in thickness eo fo the classifie based scheme Fo this egime, then, the classifie based scheme epesents a low-enegy technique fo achieving thin edges with high likelihood of tue detections What happens when we change density but keep senso eo constant? Pedictably the detection pobability inceases with incease in senso density fo both kinds of data sets This is shown in Figue 9 Thee was no σ which gave an e p f < %

11 Unwanted detections Unwanted detections Unwanted detections Image Pocessing Scheme Classifie Based Scheme Statistical Scheme Line Ellipse Detection pobability Detection pobability Detection pobabilty % 5% % Statistical Scheme Image Pocessing Scheme Classifie Based Scheme Figue 8: Vaiation of e ud with incease in fo the thee schemes fo modeately dense senso fields with modeate eos Figue : Vaiation of e m (detection pobability), with incease in senso eo p fo the thee schemes fo linea edge data sets Detection Pobability Detection Pobability Detection Pobability Statistical Scheme Image Pocessing Scheme Classifie Based Scheme Figue 9: Vaiation of e m (detection pobability), with incease in density fo the thee schemes fo linea edge data sets at ρ = 6, hence this point is missing We also found that while the thickness eo inceases with incease in density fo the statistical and image pocessing based schemes, it deceases fo the classifie based scheme The eason is that, unwanted eos, which dictate thickness eo fo the statistical and image pocessing schemes incease with incease in senso density Howeve, the false detections which dictate the edge thickness eo fo the classifie based scheme decease with incease in density How sensitive ae the schemes with espect to senso eos? Keeping density fixed, we notice an expected qualitative tend The detection pobability deceases with incease in senso eos fo all the thee schemes This is shown in Figue We also found that the thickness eo inceases with incease in senso eo fo all the thee schemes and both kinds of data sets Finally, how citical is the choice of appopiate thesholds? The classifie-based scheme does not equie selection of a theshold The othe schemes do, and in the esults we have pesented, we have chosen the best theshold possible fo each paticula scenaio The theshold based schemes might have been acceptable if thee existed one o a small ange of thesholds that wee acceptable ove the density and eo anges we conside Howeve, we found that the thesholds in the statistical scheme vay vey widely with changes in and p, especially at low densities Fo instance the optimal value of γ is 79 fo =, p = 5 and ρ = 6 The optimal value of γ is 7 fo = 3,p = 5% and ρ = 7 Fo the image pocessing based scheme, it tuns out that the vaiation in the choice of σ is vey small (within %) with espect to and p at low densities Howeve the scheme exhibits vaiations simila to the statistical scheme at highe senso densities fo changes in This discussion leads to the following conclusions Ove a ange of senso eo ates and densities, all the thee scheme can achieve tue detection ates of 9% o bette by using a two-hop pobing adius Among the thee schemes the classifie povides the thinnest edges and pefoms bette with inceasing pobing adius The classifie based scheme does not equie choosing appopiate thesholds Thus, fom a pactical pespective, the classifie-based scheme epesents a low enegy appoach to accuate localized edge detection Even though the classifie based scheme does not povide a diect contol ove false detection eos (as thesholds in othe two schemes do), one can incease the pobing adius (eg, 3-hop pobing) and achieve lowe false detections at the cost of moe communication cost ecall that localized edge detection will usually be a component of a lage system that, fo example,

12 Figue : Classifie based edge detection on a low density low eo data set Each unit on x and y axis epesent m o epesent edge sensos and + the inteio computes boundaies of a phenomenon We obseve that false detections can be disambiguated at the level of these bounday finding algoithm If the algoithm that constucts the bounday fom the obseved edges also uses infomation about each edge senso s patition line, it should be able to detect inconsistent patition line oientations and locations among neighboing edge sensos caused by false detections We believe that a bette scheme which elies on edge continuity infomation will esult in fewe false detections Fo this eason, we suggest that, of the thee schemes we conside, the classifie is the most pomising fo localized edge detection Figues,,3 show thee examples of the thee edge detection algoithms at wok Figue : Image pocessing based edge detection on a modeate density modeate eo data set Each unit on x and y axis epesent m o epesent edge sensos and + the inteio Conclusion In this pape we intoduced the poblem of localized edge detection in a senso field We discussed an edge and poposed metics to assess edge detection algoithms We poposed thee qualitatively diffeent appoaches to edge detection namely statistical, image pocessing based and classifie based appoaches We poposed an example scheme fo each of these appoaches Though numeous simulations we compaed the thee schemes with espect to the enegy accuacy tade-off, sensitivity to choice of paametes and pefomance Ou esults indicate that the classifie scheme pefoms much bette than the othe schemes Unde highe senso eo conditions, it is susceptible to moe false detections than othe schemes These false detections can be educed at the expense of highe communication cost o can pobably be disambiguated by a highe-level bounday finding algoithms The Figue 3: Statistical scheme based edge detection on a high density high eo data set Each unit on x and y axis epesent m o epesent edge sensos and + the inteio

13 statistical and image pocessing based scheme can exhibit simila pefomance but only if detection thesholds ae coectly set The coect detection thesholds vay widely with density and senso eo and we believe that dynamically setting thesholds by empiically obseving densities will be had to do As such, then, of the schemes we conside, the classifie based scheme seems to be the most pomising fo localized edge detection Exteio B a l A cos (l/ ) C Edge a efeences [] Bend Jähne, Digital Image Pocessing,, Spinge, 4th edition, 997 Inteio Figue 4: D [] O Duda, P E Hat, D G Stok, Patten Classification, John Wiley and Sons, nd edition, [3] B K De La Bae, T C Hamon, C V Chysikopoulos, Measuing and Modeling the Dissolution of a Nonideally Shaped Dense Non-aqueous Shaped Liquid Pool in a Satuated Poous Medium, Wate esouces eseach, [4] J Liu, J eich, F Zhao, Collaboative In-Netwok Pocessing fo Taget Tacking, Jounal on Applied Signal Pocessing, [5] Nowak, U Mita, Bounday Estimation in Senso Netwoks: Theoy and Methods, Poceedings of the Fist Intenational Wokshop on Infomation Pocessing in Senso Netwoks, Apil 3 [6] A Savvides, A, C-C Han, M Sivastava, Dynamic fine-gained localization in ad-hoc netwoks of sensos In Poceedings of the ACM/IEEE Intenational Confeence on Mobile Computing and Netwoking (ome, Italy, July ) [7] A Savvides, S Adlakha, Moses, M Sivastava, On the Eo Chaacteistics of Localization Algoithms, Poceedings of the Fist Intenational Wokshop on Infomation Pocessing in Senso Netwoks, Apil 3 APPENDIX A In this appendix we calculate e m, e uw and e p f to analyze the scheme descibed in Section 3 We assume that the phenomenon is lage enough and the edge can be appoximated by a line segment L within the pobing neighbohood Let L be l units distant fom the senso We assume that line segments at all values of l ae equally likely Suppose an edge passes at a distance l fom the senso ( as depicted in Figue 4) The senso collects infomation about the exteio fom aea a (ABC) and about the inteio fom a (ADC) a = cos ( l ) l l, (A-) a = π a, (A-) The numbe of sensos N and N in these egions can be modeled as Poisson andom vaiables P(N i = n) = e a i ρ (a i ρ)n n! (A-3) The numbe of senso eos K (in ABC) and K (in ADC), can be modeled by a binomial andom vaiable with p as senso eo pobability ( ) n P(K i = k N i = n) = (p) k ( p) n k (A-4) k The value of the statistic in (4) can now can be witten in tems of K i and N i as, N + K Γ = N K (A-5) N + N Using equations (A-)-(A-5), we can numeically calculate the pobability density function, P(Γ = γ l = q) Calculation of e m and e uw : A miss-detection occus when l and Γ < γ P(l = q l ) = q P(Γ = γ l ) = e m = γ P(Γ = γ l = q) q dq, P(Γ = γ l )dγ An unwanted detection occus when l > and Γ γ P(l = q l > ) = q (A-6) (A-7) (A-8) (A-9)

14 P(Γ = γ l > ) = P(Γ = γ l = q) e uw = γ P(Γ = γ l > )dγ q dq, (A-) (A-) Calculation of e p f : Now suppose thee is no edge with the pobing neighbohood (l > ) Γ can assume non-zeo values only because of senso eos The pdf of numbe of sensos in the neighbohood egion N can be expessed as a Poisson andom vaiable ( P(N = n l > ) = e π ρ π ρ ) n (A-) n! The numbe of senso eos K in the pobing neighbohood can be expessed as a binomial andom vaiable ( ) n P(K = k N = n) = (p) k ( p) n k (A-3) k The value of the statistic in (4) can now can be witten in tems of K and N as, N K Γ = (A-4) N We numeically calculate the pdf P(Γ = γ l > ) A pue false eo occus when, Γ > γ e p f = N ρ P(Γ = γ l > )dγ γ N is the total numbe of sensos in the field APPENDIX B (A-5) In this appendix we calculate the eos e f, e uw and e p f fo Pewitt filte example in Section 3 Let the line segment L(l,θ) appoximate the edge (as agued in Appendix A) intesect the pobing neighbohood Hee, l is its distance fom the senso and θ is the angle the y-axis makes with the nomal fom the senso (see Figue 5) We divide the neighbohood into 4 aeas, a (DCHF), a (ABCD), a 3 (ADEG) and a 4 (EDFI) as depicted in Figue 5 The fou aeas (a i ) i=4 i= fo θ (, π ) can be calculated by the equations, a = a = ( cos ( ( ) θ) tanθ + ) θ < cos ( ) (B-) θ cos ( ) (B-) ( cos ( ) + θ) ( tanθ + ) θ < cos ( ) (B-3) cos ( ) θ cos ( ) Exteio A G B a a 3 l E C D a H θ F a 4 Inteio Figue 5: Figue fo appendix B a 3 = π a (B-4) a 4 = π a (B-5) The numbe of sensos (N i ) i=4 i= in the fou egions and the numbe of senso eos (K i ) i=4 i= can be modeled as in A-3 and A-4 The value of σ x in (8) can now can be witten in tems of K i and N i as, σ x = N K + N 3 K 3 + N K + N 4 K 4 N + N 3 N + N (B-6) 4 The pdf P(σ x = µ l,θ) can be numeically calculated using, (B-)-(B-6), (A-3) and (A-4) It tuns out that P(σ y = µ l,θ) is same as P(σ y = µ l, π θ) The pdf P(σ = µ l,θ) can be calculated numeically fom σ = σx + σy Calculation of e m and e uw : A miss-detection occus when l but σ < σ P(σ = µ l ) = e m = 8 π σ π 4 I P(σ = µ l,θ)dl dθ(b-7) P(σ = µ l < )dµ An unwanted detection occus when l > but σ σ P(σ = µ > l ) = e uw = (B-8) π 8 4 π( P(σ = µ l,θ)dl dθ, ) (B-9) σ P(σ = µ > l )dµ (B-) Hee, we integate on θ only ove (, π 4 ) since, P(σ = µ l,θ) is identical in any section ( nπ 4, (n+)π 4 )

15 Calculation of e p f : Now suppose thee is no edge with the pobing neighbohood (l > ) and a non-zeo value of σ occus due to senso eos Let N i be the numbe of sensos in the i th quadant (quadants numbeed in anti-clockwise diection) Let K i be the numbe of senso eos in the i th quadant Then, ( ) n π P(N i = n) = e π ρ ρ (B-) ( ) n! n P(K i = k N i = n) = (p) k ( p) n k (B-) k The value of σ can be calculated in tems of (N i ) i=4 i= and (K i ) i=4 i= σ x = σ y = σ = σx + σy N +K 3 N 3 K N +N 3 + N +K 4 N 4 K N +N 4, N +K N K N +N + 4 N 3 +K 4 N 3 K 4 N 3 +N, 4 (B-3) (B-4) (B-5) Using (B-)-(B-5), we can numeically calculate the pdf P(σ = µ l > ) A pue false eo occus when, σ > σ e p f = N ρ P(σ = µ l > )dµ σ (B-6) N is the total numbe of sensos in the field