MULTI-CAMERA POSITIONING FOR AUTOMATED TRACKING SYSTEMS IN

November 16, 2010 15:12 WSPC-IJIA S0219878910002208.tex

International Journal of Information AcquisitionVol. 7, No. 3 (2010) 225–242c© World Scientific Publishing CompanyDOI: 10.1142/S0219878910002208

MULTI-CAMERA POSITIONING FOR AUTOMATEDTRACKING SYSTEMS IN DYNAMIC ENVIRONMENTS

YI YAOVisualization and Computer Vision Lab

GE Global [email protected]

CHUNG-HAO CHENDepartment of Mathematics and Computer Science

North Carolina Central [email protected]

BESMA ABIDI∗, DAVID PAGE†,ANDREAS KOSCHAN‡ and MONGI ABIDI§Imaging, Robotics, and Intelligent Systems Lab

University of Tennessee, Knoxville∗[email protected]†[email protected]

‡[email protected]§[email protected]

Received 6 July 2010Accepted 12 October 2010

Most existing camera placement algorithms focus on coverage and/or visibility analysis,which ensures that the object of interest is visible in the camera’s field of view (FOV).According to recent literature, handoff safety margin is introduced to sensor planning sothat sufficient overlapped FOVs among adjacent cameras are reserved for successful andsmooth target transition. In this paper, we investigate the sensor planning problem whenconsidering the dynamic interactions between moving targets and observing cameras. Theprobability of camera overload is explored to model the aforementioned interactions. Theintroduction of the probability of camera overload also considers the limitation that agiven camera can simultaneously monitor or track a fixed number of targets and incor-porates the target’s dynamics into sensor planning. The resulting camera placement notonly achieves the optimal balance between coverage and handoff success rate but alsomaintains the optimal balance in environments with various target densities. The pro-posed camera placement method is compared with a reference algorithm by Erdem andSclaroff. Consistently improved handoff success rate is illustrated via experiments usingtypical office floor plans with various target densities.

Keywords : Sensor planning; visual surveillance; multiple camera systems; camera handoff.

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226 Y. Yao et al.

1. Introduction

With the increased scale and complexityinvolved in most practical surveillance appli-cations, it is almost impossible for any singlecamera (either fisheye or PTZ) to fulfill auto-mated and persistent tracking with an accept-able degree of continuity and/or reasonableaccuracy. Systems with multiple cameras findextensive use in surveillance applications. Theneed for sensor planning emerges when thequestion of how to place multiple cameras to ful-fill given tasks with given performance require-ments arises.

In literature, most sensor planning algo-rithms are proposed for such applications as3D object inspection and reconstruction [Royet al., 2004; Roy et al., 2005; Chen et al.,2008]. Sensor planning for surveillance systemshas received increasing attention in recent years[Quereshi and Teropoulos, 2005; Cai and Aggar-wal, 1999; Isler et al., 2005; Pavlidis et al.,2001]. Cameras are placed such that a full orspecified coverage of the environment or objectis achieved. A probabilistic camera planningframework with visibility analysis of dynamicocclusions was proposed by Mittal and Davis[2004]. Erdem and Sclaroff [2006] defined dif-ferent types of coverage problems and devel-oped corresponding solutions. The conventionalrequirements in sensor planning, such as cover-age and visibility, cannot by themselves ensurean automated tracking in real-time surveillancesystems. A uniform and sufficient amount ofoverlap between the FOVs of adjacent cam-eras should be reserved so that consistent label-ing and camera handoff can be executed suc-cessfully. To achieve such a camera placement,sensor planning algorithms achieving the opti-mal balance between coverage and handoff suc-cess rate are discussed in [Yao et al., 2010;Yao et al., 2008].

In this paper, additional considerationsregarding the targets’ dynamics including theirdynamic interaction with the observing cam-eras are incorporated into sensor planning viathe probability of camera overload. The opti-mal camera placement is obtained accord-ing to the targets’ statistical distribution in

the environment. More overlapped FOVs aresecured for more clustered environments so thata target has more freedom to be transferred toanother camera when experiencing camera over-load or dynamic occlusion.

In addition, the introduction of the prob-ability of camera overload also simplifies themodeling of PTZ cameras. Most existing sen-sor planning algorithms find it difficult to prop-erly model PTZ cameras. Let the instant FOVdenote the FOV that a PTZ camera can see atany given time instance and the achievable FOVthe FOV that a PTZ camera can survey givena sufficient period of time. The camera’s lim-ited pan and tilt speeds lead to the discrepancybetween the instant and achievable FOVs, whichin consequence introduces difficulties in model-ing PTZ cameras for sensor planning. Some algo-rithms simply use the achievable FOV for thesensor planning of PTZ cameras.

In brief, the major contribution of this paperis the derivation of the probability of cameraoverload and the application of this probabil-ity to sensor planning. The resulting sensorplanning algorithm has the advantage of con-sidering both the dynamics of the observingcamera, especially for PTZ cameras, and thedynamic interactions between the objects andcameras. The algorithm presented in this paperis closely related to our previous work describedin [Yao et al., 2010; Yao et al., 2008; Yao et al.,2008]. The major differences are listed as fol-lows. (I) The algorithms in [Yao et al., 2010;Yao et al., 2008] construct an objective func-tion without considering the dynamic interac-tion between objects and observing cameras,which is, however, the major additional consid-eration introduced in this paper. (II) The algo-rithm in [Yao et al., 2008] concentrates on theplacement of PTZ cameras, which indeed is aspecial case of the scenarios addressed by ouralgorithm. In [Yao et al., 2008], the maximumnumber of targets that can be tracked simulta-neously by a camera, Nobj , is set to 1, whereas inthis paper we investigate the general case withNobj ≥ 1.

The remainder of this paper is organized asfollows. Table 1 lists the major notations used inthis paper. A brief introduction to related work


Multi-Camera Positioning for Automated Tracking Systems in Dynamic Environments 227

Table 1. Major notations.

λj Arrival rate in the FOV of the jth camera configurationµj Average residence time in the FOV of the jth camera configurationA, AC Coverage matrix with aij = 1/aC,ij = 1 indicating that the ith grid can be seen by the

jth camera configurationAH Handoff safety margin with aH,ij = 1 indicating that the ith grid is in the handoff safety

margin of the jth camera configurationAV Visible area with aV,ij = 1 indicating that the ith grid is in the visible area of the jth

camera configurationci The cost function of the ith gridKo,j Target density in the FOV of the jth camera configurationMD , MD,ij Distance component of the observation measure of the ith grid observed by the jth

camera configurationMR, MR,ij Resolution component of the observation measure of the ith grid observed by the jth

camera configurationNobj , Nobj,j The maximum number of objects that can be tracked simultaneously by the jth camera

configurationPdo,ij Probability of dynamic occlusion on the ith grid observed by the jth camera

configurationPdo,i Probability of dynamic occlusion on the ith gridPco,i Probability of camera overload on the ith gridQi,j Observation measure of the ith grid observed by the jth camera configurationQF Failure threshold that separates the handoff safety margin and invisible areasQT Trigger threshold that separates the handoff safety margin and visible areasx The solution vector with xj = 1 indicating that the jth camera configuration is selected

is given in Sec. 2. Section 3 defines the obser-vation measure and the objective functions usedfor the search of the optimal camera placement.Section 4 introduces the probability of cam-era overload and the modified objective func-tion considering targets’ dynamics. Section 5demonstrates our experimental results and com-parisons with the reference algorithm. Section 6concludes this paper.

2. Related Work

In literature, most placement algorithms usingvisual sensors are proposed for such applicationsas 3D object inspection and reconstruction. Royet al. [2004] reviewed existing sensor planningalgorithms for 3D object reconstruction andproposed an online scheme using a probabilis-tic reasoning framework for next-view planningand object recognition [2005]. A more recentand thorough discussion regarding sensor plan-ning algorithms for 3D object reconstructionand recognition can be found in [Chen et al.,2008]. The authors also pointed out promising

directions for future research, such as the combi-national optimization of the placement of bothcameras and illumination sources. Wong et al.[1999] defined a metric evaluating the unknowninformation in each group of potential view-points and used it in the search of the nextbest view for 3D modeling. Yous et al. [2006]designed an active scheme for the assignmentof multiple PTZ cameras so that each cam-era observes a specific part of a moving object,mainly pedestrians, and achieves the best vis-ibility of the whole object. The selection ofsets of omnidirectional views for the represen-tation of a 3D scene is discussed in [Tosic andFrossard, 2006]. In [Saadatseresht et al., 2005],fuzzy logic inference is employed for cameraplacement considering the uncertainty in theanalysis of visibility, accessibility, and camera-object distance. Chen and Li [2004] addressedthe placement of active sensors in the contextof robot vision. With the increased complexityof multiple camera systems, sensor planning isalso conducted in a larger scale and a higherlevel similar to sensor networks. Guo et al.[2008] modeled observability as a decreasing


228 Y. Yao et al.

exponential function of the observation distanceand used this model in camera placement forthe monitoring and tracking of mass objects.Dunn et al. [2006] employed the Parisian evo-lutionary algorithm to search for the opti-mal camera placement for 3D object recon-struction aiming at a reduced computationalcomplexity.

Sensor planning for surveillance systems hasreceived increasing attention in recent years[Quereshi and Teropoulos, 2005; Cai and Aggar-wal, 1999; Isler et al., 2005; Pavlidis et al., 2001].Erdem and Sclaroff [2006] defined different typesof coverage problems and developed correspond-ing solutions using perspective cameras. Theirmethods have been implemented in a simulatorwith a genetic algorithm as the optimizationengine [David et al., 2007]. Several placementalgorithms are developed based on Erdem andSclaroff’s method. Angella et al. [2007] pre-sented solutions for the more generalized M-coverage problem, where it is desired that theobject of interest can be observed by at leastM cameras. Horster and Lienhart [2006] alsoaddressed the M-coverage problem and trans-formed their nonlinear objective function to alinear one so that linear binary programmingcan be used in the search of the optimal cameraplacement.

Literature also mentions sensor placementalgorithms focusing on additional considerationssuch as path observability, dynamic occlusion,and frontal view availability. Bodor et al. [2007,2005] presented a camera placement algorithmto maximize the observability of a path. In thesimilar vein, Fiore et al. [2008] used the dis-tance and foreshortening constraints to describethe observability of a path and defined thecorresponding objective function to guide thesearch for optimal camera placement. Success-ful camera placement and online repositioningare demonstrated for tracking pedestrians mov-ing along a regular path using two fixed cam-eras mounted on remotely controllable mobileplatforms. A probabilistic camera planningframework with dynamic visibility analysis wasproposed by Mittal and Davis [2004]. Anothermetric describing the likelihood of dynamicocclusion is also discussed in [Chen and Davis].

Ram et al. [2006] introduced frontal view prob-ability to coverage analysis and demonstratedreal-time camera selection for a better observa-tion of a pedestrian.

Online camera selection, also referred to asthe focus of attention problem by Isler et al.[2005], is introduced as a result of the improvedmobility of cameras. Since the object of interestcan be observed by multiple cameras, an onlineresource management mechanism is necessary toguide the coordination among multiple camerasfor an optimal system performance. The opti-mal performance is two-fold: the optimal obser-vation of every object of interest and the optimalcomputational load for every camera deployed inthe environment. Gupta et al. [2007] discusseda unified approach, referred to as COST, whichselects a set of cameras to be used for the infer-ences for each person in a group of pedestriansconsidering occlusions and visual confusion. Isleret al. [2005] proposed a selection framework toassign cameras to track the object of interestfor a minimized expected error in the estima-tion of the object’s location. The visibility inter-val is explored as one criterion for online cameraselection in [Lim et al., 2005]. The online cam-era selection or scheduling is discussed for largescale camera networks, where up to 80 camerasare deployed [Lim et al., 2007].

Since Erdem and Sclaroff’s [2006] sensorplanning algorithm is selected as the base-line method for performance comparison, moredetails are presented as follows. In [Erdem andSclaroff, 2006], it is assumed that a polygonalfloor plan is represented as an occupancy grid,a binary vector b ∈ B

Ng can be obtained byletting bi = 1 if the corresponding grid can beseen by at least one camera and bi = 0 oth-erwise, where B

.= {0, 1} and Ng denotes thenumber of grid points. We construct a binarymatrix A ∈ B

Ng×Nc with aij = 1 if the ithgrid is covered by the jth camera configuration,where Nc denotes the number of camera configu-rations. Each camera configuration specifies onecombination of camera intrinsic and extrinsicparameters, including the camera’s focal lengthf , pan/tilt angle θP /θT , and position Tc. Thesolution vector x ∈ B

Nc has xj = 1 if the jthcamera configuration is selected and xj = 0,



otherwise. The following relation holds: bi = 1 ifb′i > 0; and bi = 0, otherwise, with b′ = Ax. Letthe cost associated with the jth camera config-uration be ωj. Given the maximum cost Cmax,the Max-Coverage sensor planning problem canbe described by

max∑

i

bi, subject to∑

j

ωjxj ≤ Cmax. (1)

3. Sensor Planning

As we can see from the previous section, abinary model, visibility matrix A, is sufficientto tackle the optimization of coverage. How-ever, this binary model is no longer adequate toincorporate the additional requirement of uni-form and sufficient overlapped FOVs betweenadjacent cameras. Therefore, in this section,we first define a continuous observation mea-sure to describe the suitability of the targetbeing tracked by the current camera. After-wards, this continuous observation measure isthresholded to generate three regions includ-ing, invisible, visible, and a gray region. Thegray region, to which we refer as the hand-off safety margin, defines the area in the cam-era’s FOV where a target needs a transfer.The basic idea of our sensor planning algo-rithm is to utilize this gray area so thatuniform and sufficient overlapped FOVs areachieved. Note that the definition of the obser-vation measure and the formulation of theobjective function for static environment aredirectly inherited from our previous work pre-sented in [Yao et al., 2010; Yao et al., 2008].To save space, a brief introduction is given inthis section for the completeness of the algo-rithm description. Interested readers please referto our previous publications for more detailedderivations.

3.1. Observation measure

To describe the observation of a tracked tar-get in addition to visibility, we consider twocomponents: the resolution MR and the dis-tance to the edges of camera’s FOV MD. TheMR component is designed to evaluate a validobservation for the viewer. For a persistent

object tracking and smooth camera handoff, thetracked target should be at a reasonable distancefrom the edges of the camera’s FOV. The MD

component, therefore, considers the safety mar-gin before the object falls out of the camera’sFOV.

To begin our study, the camera and worldcoordinates are defined and illustrated in Fig. 1.A point gi = [gx,i gy,i gz,i]T in the worldcoordinates is projected onto a point pij =[x′

i,j y′i,j z′i,j]T in the jth camera’s coordi-nates by

x′i,j

y′i,jz′i,j

=

cos θT,j 0 − sin θT,j

0 1 0

sin θT,j 0 cos θT,j

×

1 0 0

0 cos θP,j sin θP,j

0 − sin θP,j cos θP,j

×

gz,i − Tz,j

gx,i − Tx,j

gy,i − Ty,j

, (2)

with Tc,j = [Tx,j Ty,j Tz,j]T . Assuming zeroskew, unit aspect ratio, and zero image cen-ter, the projected point in the image plane is

X

Y

Z

y’

z’

x'

T C

World coordinates

Camera coordinates

Pθ

Tθ

Fig. 1. Illustration of the camera and world coordinates.


230 Y. Yao et al.

given by: {xi,j = fjx

′i,j/z

′i,j

yi,j = fjy′i,j/z

′i,j

.

Letting gz,i = 0 (in the ground plane),the target depth z′i,j, the distance between theobject’s centroid and the camera’s optical cen-ter, can be estimated by:

z′i,j =−Tz,j

xi,j/fj cos θT,j + sin θT,j(3)

and the resolution component MR,ij is thenexpressed as:

MR,ij =

αRfj

z′i,jz′i,j > − Tz,j

tan θT,j

αRfj

(z′i,j + Tz,j/tan θT,j)2 −Tz,j/tan θT,j

z′i,j ≤ − Tz,j

tan θT,j

,

(4)

where αR is a normalization coefficient.In practice, for better observation and to

reserve enough computation time for handoff,the target should remain at a distance fromthe edges of the camera’s FOV. Moreover, thismargin distance is affected by the target depth.When the target is at a closer distance, itsprojected image undergoes larger displacementin the image plane. Therefore, a larger marginshould be reserved. In our definition, differentpolynomial powers are used to achieve varyingdecreasing rates of the MD component as theobject of interest approaches the edges of thecamera’s FOV. The MD,ij is then given by:

MD,ij =

{αD

[(1 − |xi,j |

Nrow/2

)2

+(

1 − |yi,j|Ncol/2

)2]}β1z′i,j+β0

, (5)

where Nrow and Ncol denote the image’s widthand height, αD is a normalization coefficient,and β1 and β0 are used to adjust the polyno-mial degree.

The observation measure is then given by:

Qij =

{wRMR,ij + wDMD,ij [xi,j, yi,j]T ∈ Π

−∞ otherwise,

(6)

where wR and wD are importance weights andΠ denotes the image plane.

As for PTZ cameras, considering the addi-tional flexibility from the camera’s adjustablepan and tilt angles, the resolution componentMR,ij is given by MR,ij = αRfj/z

′i,j . We assume

that the target is always maintained at theimage center by panning and tilting the camera.Therefore, the MD,ij component can be elimi-nated from the computation of the observationmeasure.

3.2. Objective function

A failure threshold QF and a trigger thresholdQT are derived to define three disjoint regions:(I) invisible area with Qij < QF where Qij

represents the observation measure value of theith grid observed by the jth camera configura-tion, (II) visible area with Qij ≥ QT , and (III)handoff safety margin with QF ≤ Qij < QT .The failure threshold QF segments the invis-ible areas and is used for coverage analysis.The trigger threshold QT separates the visibleareas and handoff safety margins. It is intro-duced for handoff rate analysis, where necessaryoverlapped FOVs between adjacent cameras areoptimized. The trigger threshold QT is given byQT = QF + κuobj tH , where uobj represents theaverage moving speed of the object of interest,tH denotes the average duration for a successfulhandoff, and κ is a conversion scalar.

Let AC ∈ BNg×Nc represent the grid cover-

age with aC,ij = 1 if Qij ≥ QF and aC,ij = 0otherwise. Matrix AC resembles matrix A inthe conventional coverage analysis discussed inthe previous section. For the purpose of hand-off analysis, two additional coefficient matricesare constructed AH and AV . The matrix AH hasaH,ij = 1 if QF ≤ Qij < QT and aH,ij = 0 oth-erwise. The matrix AV has aV,ij = 1 if Qij ≥ QT

and aV,ij = 0, otherwise. Matrices AH and AV

represent the handoff safety margin and visiblearea, respectively. Recall that the solution vector



x specifies a set of chosen camera configurationswith the corresponding element xj = 1 if theconfiguration is chosen and xj = 0 otherwise.Let c′C = ACx, c′H = AHx, and c′V = AV x.The objective function of the ith grid is formu-lated as:

ci = wC(c′C,i > 0) + wH(c′H,i = 2)

−wV (c′V,i > 1), (7)

where wC , wH , and wV are predefined impor-tance weights. The operation (c′C,i > 0) means

c′C,i ={

1 c′C,i > 0

0 otherwise. The first term in the

objective function considers coverage, the sec-ond term produces sufficient overlapped hand-off safety margins, and the third term penalizesexcessive overlapped visible areas. Our objectivefunction achieves a balance between coverageand sufficient margins for camera handoff. Theoptimal sensor placement for the Max-Coverageproblem can then be obtained by:

max∑

i

ci, subject to∑

j

ωjxj ≤ Cmax. (8)

Note that the optimization of camera place-ment can be formulated in an alternative waywhere additional constraint is used instead ofconstructing an objective function as a weightedsum of three terms as in Eq. (7). To be morespecific, the optimization formulation can bechanged to:

max∑

i

ci, subject to∑

j

ωjxj ≤ Cmax

and AHx ≥ 2, (9)

(a) (b)

Fig. 2. Schematic illustration of the problem of dynamic occlusion, (a) target 2 is occluded by target 1 for bothcameras, (b) target 2 can be observed by camera 2 when it is occluded by target 1 in the FOV of camera 1.

where ci = wC(c′C,i > 0) − wV (c′V,i > 1) andthe term 2 represents a vector with all ele-ments as 2. From our experiments, the hardconstraint approach suffers from a slower con-vergence speed in comparison with the schemewith an additive objective function. Therefore,in the following discussion, we have chosen touse the additive objective function approach.

4. Dynamic Considerations

Environments with multiple moving objectsimpose additional difficulties on sensor planning.Multiple moving objects cause dynamic occlu-sions depending on their real-time relative posi-tions. Figure 2 compares two camera placementsin terms of the ability to handle dynamic occlu-sion. It is obvious that the camera placement inFig. 2(a) is unable to deal with dynamic occlu-sion since target 2 is blocked by target 1 in theFOVs of both cameras. On the contrary, in thecamera placement shown in Fig. 2(b), target 2can be seen from camera 2 when it is occludedby target 1 in camera 1. From the above illus-tration, we could see that the probability ofdynamic occlusion can be reduced by a propercamera placement. Due to the non-deterministicnature of dynamic occlusion, analysis regard-ing such occlusions is conducted in a proba-bilistic framework. The probability of dynamicocclusion is derived and incorporated into sensorplanning.

Another important issue in sensor planningfor environments with multiple dynamic targetsis the coordination among multiple cameras. Inpractice, a single camera can track a limitednumber of targets simultaneously because of the


232 Y. Yao et al.

1

2

34

New target

1

4

2

3

Camera’s FOVTracked objectUntracked object

(a) (b)

Fig. 3. Schematic illustration of the problem of camera overload. Assume that the camera is able to track four targetsat maximum simultaneously due to limited computational capacities, (a) the maximum number of targets is achieved,(b) camera overload occurs when a new target enters the camera’s FOV. Target 3 is dropped due to camera overload.

limited resolvable distance and computationalcapacities [Chen et al., 2008]. The camera maynot be able to detect and/or track new objectswhen its maximum computational capacity hasbeen reached. This scenario is referred to as theproblem of camera overload and is demonstratedin Fig. 3. Assume that the camera is able totrack four targets at maximum simultaneously.When a new target enters the camera’s FOV, adecision is to be made so that an appropriatetarget is dropped due to the limited computa-tional capacity. In Fig. 3(b), since target 3 is far-ther away from the camera, it is dropped so thatthe camera can track the new target. The goalof sensor planning is to automatically minimizethe number of dropped targets due to cameraoverload.

4.1. System model

For camera overload analysis, we model themulti-object tracking system as an M/M/N/Nqueuing system. Following the conventions inqueuing theory, an M/M/N/N system sug-gests that: (1) the arrival process follows a Pois-son distribution; (2) the residence time followsan exponential distribution; and (3) the numberof servers and buffer slots is N .

Poisson processes are employed to modelobjects’ arrival and departure. The Poisson pro-cess has been proved effective in modeling ran-dom events, such as a customer arrival and thearrival of a cellular phone call that emanatescontinuously and independently at a constantaverage rate. It has found wide applicationsin systems such as bank service and wireless

communication [Panneerselvam, 2004; Tivedi,2001]. The case of an object of interest enteringthe FOV of a surveillance system for the serviceof “tracking” and “monitoring” is similar to thecase of a customer entering a bank for the ser-vice of account transactions and the case of amobile call entering the base station for the ser-vice of wireless communication. Therefore, thePoisson process is chosen to formulate objects’arrival and departure in a surveillance system.The Poisson and exponential probability distri-bution describes two aspects of a Poisson pro-cess. The Poisson probability function presentsthe distribution of the number of events thatoccur in a time interval of fixed length whereasthe exponential probability function records thedistribution of the length of the time intervalbetween consecutive events. The different ratesof the objects’ arrival and departure determinethe target density in the environment. A higherarrival rate and a lower departure rates resultsin an environment with higher target density.

Assume that the average arrival rate in theFOV of the jth camera is λj and that the meancamera-residence time is 1/µj . Let Nobj,j be themaximum number of targets that can be trackedsimultaneously by the jth camera. From theM/M/N/N queuing theory, the system can bedescribed by a Markov chain as shown in Fig. 4.

4.2. Dynamic occlusion

For dynamic occlusion analysis, we borrow thedefinition of the probability of dynamic occlu-sion defined by Mittal and Davis [2004]. Objectsare modeled as a cylinder with a radius of robj



1 2 Nobj,jNobj,j-10

µj 2µ j Nobj,j µ j

n

nµj

jλ jλ jλ jλ

Fig. 4. Illustration of the state transition of an M/M/N/N queuing system, which is used to model a multi-objecttracking system as more objects come into a camera’s FOV.

and a height of hobj . Let the area of their pro-jection onto the ground plane be fixed as Aob =πr2

obj . Assume that the object of interest cen-tered at the ith grid is observed by the jth cam-era from a distance Dij . Its region of occlusionis Ao,ij = 2robjhobj

Dij

Tz,j. The occlusion proba-

bility depends on the target density, which canbe derived from our Markov chain based sys-tem model. Let Aj denote the FOV of the jthcamera. The target density in the jth camera isgiven by:

Ko,j =λj

Ajµj. (10)

The occlusion probability at the ith gridobserved by the jth camera Pdo,ij can then beexpressed as [Mittal and Davis, 2004]:

Pdo,ij = 1 − exp{−KoAo,ij(2 − KoAob)

2(1 − Ko,jAob)

},

(11)

where Ko denotes the object density.Given the probability of dynamic occlusion

at the ith grid observed from the jth camera, wecan compute the overall probability of dynamicocclusion at the ith grid Pdo,i by:

Pdo,i =∏

j,aC,ijxj=1

Pdo,ij . (12)

The objective function becomes:

ci = wC(c′C,i > 0) + wH(c′H,i = 2)

−wV (c′V,i > 1) + wdo(Pdo,i ≤ Pdo,th), (13)

where Pdo,th is a predefined threshold and wdo isthe importance weight.

4.3. Camera overload

Given the probability of the (n − 1)th statePn−1,j of the Markov chain, the probability ofthe nth state Pn,j is expressed as:

Pn,j =λj

nµjPn−1,j =

1n!

(λj

µj

)n

Po,j, (14)

for 1 ≤ n ≤ Nobj,j, where Po,j is a normalizationterm to make the sum of the probabilities of allpossible states as one:

Po,j =

Nobj,j∑

n=0

1n!

(λj

µj

)n−1

. (15)

The probability that the jth camera reachesits maximum computational capacity is theprobability that the Markov chain reaches the(Nobj,j)th state:

Pmax,j =1

Nobj,j !

(λj

µj

)Nobj,j

∑Nobj,j

n=01n!

(λj

µj

)n . (16)

Let the average arrival rate at the ith grid be λg,i

and the mean camera-residence time be 1/µg,i.The ith grid can be observed from multiplecameras. The probability of camera overload atthe ith grid Pco,i is the probability that all theobserving cameras have reached the maximumcomputational load when new objects appear atthe ith grid:

Pco,i = (1 − e−λg,i/µg,i)∏

j,aC,ijxj=1

Pmax,j . (17)

The objective function becomes:

ci = wC(c′C,i > 0) + wH(c′H,i = 2)

−wV (c′V,i > 1) + wco(Pco,j ≤ Pco,th), (18)

where Pco,th is a predefined threshold andwco is the importance weight. Note that


234 Y. Yao et al.

okt θ

1+okt

θ1+okt

tt okt

1kt

11 +kt 2kt

Target 1

Target 2

(a) (b) (c)

Fig. 5. (a) Illustration of a PTZ camera’s instant (lightly shaded lines) and achievable (black circle) FOVs, (b) illustra-tion of reachable regions of a PTZ camera as defined in Erdem and Sclaroff’s algorithm, (c) illustration of the proposeddynamic modeling of a PTZ camera with Nobj,j = 1. Red circles and green dots depict untracked and tracked objects’trajectories.

the computation of the probability of cam-era overload depends on the target’s arrivalrate and residence time, which describes thetarget’s dynamic distribution in the environ-ment. Therefore, the resulting optimal cameraplacement not only depends on the environ-ment’s geometry but also adjusts to the target’sdynamics.

4.4. Dynamic modeling of PTZcameras

The significance of introducing camera over-load analysis becomes obvious especially forPTZ cameras. As mentioned before, cameraplacement algorithms always find it difficult toproperly model a PTZ camera’s instant andachievable FOVs. Erdem and Sclaroff [2006]defined the reachable region to model PTZ cam-eras. It is assumed that a PTZ camera has twoend points for panning. The reachable regioncorresponds to the intersection areas that thePTZ camera can pan from these two end pointsduring a given period of time, as shown inFig. 5(b). The reachable region represents thecamera’s coverage in the worst case, which leadsto a camera placement with excessive over-lapped achievable FOVs.

In this paper, the discrepancy in model-ing a PTZ camera’s instant and achievableFOVs is solved elegantly by letting Nobj,j =1. That is at a given time instance, a singlePTZ camera is able to track a single objectin its 360◦ × 90◦ achievable FOV. Figure 5(c)

illustrates scenarios where a PTZ camera istracking an object in its instant FOV when asecond object enters its achievable FOV. Oncethe first object leaves the camera’s achievableFOV, the camera turns toward the second objectand begins tracking. We observe that a seem-ingly trivial assumption of Nobj,j = 1 sufficientlydescribes the dynamic interaction between aPTZ camera and the objects of interest. Theachievable FOV remains the same under theassumption that the number of objects thathave been tracked is zero, whereas the limitedinstant FOV can be described as the achiev-able FOV under the assumption that the max-imum number of objects that can be trackedsimultaneously has been reached. Therefore, theachievable FOV with Nobj,j = 1 is sufficient tomodel PTZ cameras’ both instant and achiev-able FOVs for sensor planning.

In addition to the power of resolving thedilemma of PTZ cameras’ achievable and instantFOVs, our modeling also incorporates the anal-ysis of PTZ cameras into a unified frameworkalong with the static perspective cameras. Theonly difference is the assumption regarding themaximum number of targets that can be trackedsimultaneously. The maximum numbers of tar-gets for a static camera and a PTZ camera areNobj,j ≥ 1 and Nobj,j = 1, respectively.

5. Experimental Results

In this section, our experimental results usingstatic perspective and PTZ cameras are



presented and compared with the reference algo-rithm proposed by Erdem and Sclaroff [2006].Both dynamic occlusion and camera overloadare included in the objective function for thesearch of the optimal camera placement:

ci = wC(c′C,i > 0) + wH(c′H,i = 2)

−wV (c′V,i > 1) + wdo(Pdo,i ≤ Pdo,th)

+ wco(Pco,j ≤ Pco,th). (19)

Two criteria are used to evaluate and comparethe performances of various algorithms: coverageand handoff success rate. Handoff success ratedenotes the ratio between the number of suc-cessful handoffs and the total number of hand-off requests. Compared to Erdem and Sclaroff’smethod, a consistently improved handoff successrate is expected from our proposed algorithm.

5.1. Parameter selection

There are two sets of parameters: normal-ization coefficients (αR and αD) and impor-tance weights (wR, wD, wC , wH , wV , wdo,and wco). The goal of choosing the appropriatenormalization coefficients is to provide a uniformcomparison basis for different types of camerasand cameras with various intrinsic and extrin-sic parameters. In so doing, sensor planning andcamera handoff can be conducted independently

11m

10m 11m

16m

10m

: Entrance

Height: 3m

: Obstacles

: Simulated trajectory

A

B

: Point of interest

(a) (b)

Fig. 6. Tested floor plans. Two office floor plans: (a) without path constraints and (b) with path constraints.

of the actual types of cameras selected. In gen-eral, we normalize the MR,ij and MD,ij com-ponents in the range of [0, 1], which leads toαR = 1

maxj{−Tz,j/tanθT,j} and αD = 0.5.

Different from the selection of the normaliza-tion coefficients, which depends on the charac-teristics of the cameras used, the selection of theimportance weights is application dependent.We purposefully reserve the freedom for users tochoose different importance weights according totheir special requirements to increase our algo-rithm’s flexibility. Meanwhile, default values canbe used if the corresponding variables are notspecified by users. The default values of wR/wD

and wdo/wco are simply 0.5 and 0.2, respectively.We could compute wC , wH , and wV such thatthe turning point is placed at the middle pointbetween the contours defined by QT and QF . Tosave space, please refer to [Yao et al., 2008] fordetails on the selection of parameters. In the fol-lowing experiments, the importance weights areset to default values.

5.2. Experimental methodology

The floor plans under test are shown in Fig. 6.The floor plan in Fig. 6(a) represents two typesof environments commonly encountered in prac-tical surveillance: space with obstacles (region


236 Y. Yao et al.

A illustrated in yellow) and open space wherepedestrians can move freely (region B illustratedin green). Region B is deliberately includedbecause it imposes more challenges on cam-era placement when considering handoff successrate. Camera handoff is relatively easier whenthere is a predefined path compared with thescenarios where subjects are assumed to movefreely, since camera handoff may be triggeredat any point in the camera’s FOV. Figure 6(b)illustrates an environment with a predefinedpath where workers proceed in a predefinedsequence. In the following experiments, we willrefer to these two floor plans as plan A and B.

To obtain a statistically valid estimation ofhandoff success rate, simulations are carried outto enable a large amount of tests under vari-ous conditions. A pedestrian behavior simula-tor [Antonini et al., 2006; Pettre et al., 2002]is implemented so that we could have a closeresemblance to the experiments in real environ-ments and in turn an accurate estimation ofthe handoff success rate. To save space, inter-ested readers can refer to the original papers fordetails. In our experiments, the arrival of thepedestrian follows a Poisson distribution. Theaverage walking speed is 0.5 (meters/second).Several points of interest are generated ran-domly to form a pedestrian trace. Figure 6depicts some randomly generated pedestriantraces.

Since one of the major advantages ofour algorithm is the consideration of targets’statistical distribution, we focus on testing andcomparing the algorithms’ performances in envi-ronments with a variety of target densities. Intheory, given the proper estimation of the tar-get’s arrival rate in the environment, the samevalue should be used in sensor planning. How-ever, in our experiment, we purposefully usetwo sets of arrival rates. One set of arrivalrates is used in sensor planning. Three cameraplacements are generated with λ = 0.01 per-son/second (pps), 0.025 pps, and 0.05 pps. Theother set is used in the simulation of targetbehavior. The tested arrival rates vary from0.01 pps to 0.05 pps, representing environmentswith low to high target density. The residencytime depends on the pedestrian walking speed

and the average trajectory length. From a statis-tical study regarding the geometry of the envi-ronment, the average residency time is 80 s.

According to queuing theory, it is requiredthat λj

µjNobj,jbe less than one so as to main-

tain the equilibrium state of the Markov chain.Therefore, given a fixed µj, the object’s arrivalrate is bounded by µjNobj,j. Such a requirementrelates the maximum number of objects Nmax inthe FOV of one camera to the maximum num-ber of objects that one camera can handle. Inour experiments, we have Nobj = 4 and Nobj = 1for static and PTZ cameras. Under such a con-straint, an arrival rate of 0.05 pps is the max-imum rate that our sensor planning algorithmcan handle. For more crowded scenarios, Nobj

should be increased. Although the arrival rateis one controlling parameter in our experiment,it does not directly describe how crowded theenvironment is. Therefore, we chose to use themaximum number of objects in the FOV ofone camera Nmax to describe the target den-sity instead of the arrival rate. In addition, toremove the dependence of the performance eval-uation on the camera’s maximum tracking capa-bility, Nmax is divided by the maximum numberof objects that a camera can handle. In brief, wehave used the ratio Nmax/Nobj to describe thetarget density. A simulation with Nmax/Nobj > 1suggests a scenario that exceeds the trackingcapability of a static camera.

It is desirable to investigate the ability ofthe proposed algorithm in handling crowdedscenes. It has been shown that 0.30 personsper square meters is a typical target density forfree flow traffic scenarios where pedestrians stillcan freely select their own walking speed, canbypass slower-moving people, and can readilyavoid conflicts when crossing in front of others[Fruin]. Beyond 0.30 persons per square meters,traffic congestion may occur. Therefore, we haveselected 0.30 as the average target density forcrowded scenes. Given an FOV of 50 m2, themaximum number of targets is 15 persons. Acamera with the ability to track 15 targets isneeded, Nobj = 15.

We employ different values of target arrivalrates to verify that the camera placement with a



certain arrival rate is able to maintain the hand-off success rate for environments with an arrivalrate up to the value used in sensor planning. Forinstance, the handoff success rate of a cameraplacement with λ = 0.025 pps should be main-tained for environments with an arrival rate lessthan 0.025 pps or a maximum number of targetsless than four.

5.3. Results

Figures 7 and 8 illustrate the optimal cameraplacements from the reference algorithm andour algorithm with various target densities forfloor plans A and B, respectively. Note that theoptimization of the camera’s parameters is notrestricted to 2D. The optimized position of acamera includes both Tx,j/Ty,j and Tz,j (height).For clear presentation, Figs. 7 and 8 illustratethe FOVs of the cameras at the optimal posi-tions that are projected onto the 2D groundplane.

We can see from Fig. 7 that as the maximumnumber of targets in the environment increasesfrom one to six, the handoff success rate of thereference method drops gradually from 74.1%to 44.2%. On the contrary, the handoff successrate of our method is maintained within 90.0%for camera placement with λ = 0.025 pps (λ =0.05 pps) till the maximum number of targetsreaches four (six). As expected, with differentdensity parameters used in Eq. 16, the result-ing camera placement yields different capacityin handling clustered environments.

To examine the ability of the proposed sen-sor planning algorithm in handling crowdedscenes, the simulated target density is increasedto 0.3 persons per square meters. The corre-sponding Nobj is set to 15. Using the same sen-sor placement, the handoff success rate is shownin Fig. 7(e), which presents similar performanceas the scenarios with lower target density andNobj = 4. This verifies the capability of the pro-posed sensor planning algorithm in dealing withcrowded scenes given that the camera is able tohandle crowd as well.

Figure 9 shows and compares sample framesfrom two cameras with and without sufficienthandoff margins. If only coverage is taken into

account as shown in Fig. 9(a), the object ofinterest is lost before the left camera is ableto identify the subject and cooperate with theright camera. With sufficient handoff margins asshown in Fig. 9(b), the object of interest can bedetected and labeled correctly before it becomesunidentifiable in the right camera.

Figure 10 illustrates our experimental resultsusing PTZ cameras. Figures 10(b) and 10(c)show the camera placement obtained from ourmethod with different target densities. A largerarrival rate/target density setting leads to acamera placement with more overlapped FOVsbetween adjacent cameras so that the trackedtarget has more freedom to be transferred toanother camera when experiencing dynamicocclusion and/or camera overload.

The advantage of our method over the ref-erence method becomes clear when we look intothe handoff success rate with respect to the max-imum number of targets in the environment, asshown in Fig. 10(d). When the maximum num-ber of targets is one, our method elevates thehandoff success rate from 48.7% to 100% andmaintains a similar coverage. As the maximumnumber of targets in the environment increasesfrom one to six, the handoff success rate of thereference method drops gradually from 48.7% to10.2%. On the contrary, the handoff success rateof our method is maintained within 90% for cam-era placement with λ = 0.025 pps (λ = 0.05 pps)till the maximum number of targets reaches four(six). Therefore, the proposed algorithm is ableto achieve and maintain a significantly higherhandoff success rate according to the targets’density in the environment.

In addition to simulation results, we alsoconducted real-time experiments based on thesensor placement derived from the proposedalgorithm for floor plan A. However, produc-ing results using real sequences requires speciallydesigned handoff algorithms, which is out of thescope of this paper. Interested readers pleaserefer to [Chen et al., 2009], where a camerahandoff algorithm designed based on the obser-vation measure and the probability of cameraoverload is proposed and real-time experimentalresults of multi-camera multi-object tracking aredemonstrated.


238 Y. Yao et al.

(a) (b)

0.4 0.6 0.8 1 1.240

50

60

70

80

90

100

Nmax

/ Nobj

Han

doff

succ

ess

rate

(%

)

Erdem & SclaroffOur method λ=0.01Our method λ=0.025Our method λ=0.05

(c) (d)

0 0.5 1 1.540

50

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100

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/Nobj

Han

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succ

ess

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(%

)


(e)

Fig. 7. Sensor planning results for floor plan A considering various target densities. The optimal camera positioningfrom: (a) the reference method (coverage: 98.5%), (b) our method with λ = 0.01 pps (coverage: 89.9%), and (c) ourmethod with λ = 0.05 pps (coverage: 89.6%). System performance comparison based on handoff success rate withvarious target densities (d) Nobj = 4 and (e) Nobj = 15. Target density is described by the maximum number of targetsto be tracked simultaneously in the environment.



(a) (b) (c)

0.4 0.6 0.8 1 1.240

50

60

70

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90

Nmax

/Nobj

Han

doff

succ

ess

rate

(%

)


(d)

Fig. 8. Sensor planning results for floor plan B considering various target densities. The optimal camera positioningfrom: (a) the reference method (coverage: 94.1%), (b) our method with λ = 0.01 pps (coverage: 90.5%), and (c) ourmethod with λ = 0.05 pps (coverage: 86.7%). (d) System performance comparison based on handoff success rate withvarious target densities.

The computational complexity of the pro-posed and reference algorithms consists of thecomputation of the objective function and theoptimization process. With the same optimiza-tion algorithm deployed, the computationalcomplexity of computing the objective functionbecomes dominant, which is O(NgNc) for bothalgorithms. Therefore, the proposed algorithmhas similar computational complexity as the ref-erence method for solving the Max-Coverageproblem.

6. Conclusions

A sensor planning algorithm designed for bothstatic perspective and PTZ cameras was pro-posed to achieve the optimal balance betweencoverage and handoff success rate. The prob-ability of camera overload was derived andemployed to solve the discrepancy between aPTZ camera’s instant and achievable FOVs. Theprobability of camera overload also introducedadditional considerations regarding the dynamic


240 Y. Yao et al.

Frames collected at tLeft camera Right camera

Areas covered by both cameras

Frames collected at t +∆ to

Frames collected at tLeft camera Right camera

Frames collected at t +∆ t

Areas covered by both cameras

(a) (b)

Fig. 9. Illustration of sufficient safety margin for continuous and automated handoff using perspective cameras. Sampleframes from two cameras (a) when only coverage is considered and (b) when both coverage and handoff are considered.The object of interest is visible in the right camera at to. In (a), the object of interest is lost in the right camera asit moves and becomes visible in the left camera at to + ∆t. There is no sufficient margin for a successful handoff. In(b), the object of interest remains visible in the right camera at to + ∆t, which ensures a successful handoff to the leftcamera.

interaction between targets and observing cam-eras to sensor planning. The proposed sensorplanning algorithm not only considered the PTZcamera’s dynamics from panning and tilting but

(a) (b) (c)

Fig. 10. Sensor planning results for PTZ cameras considering various target densities. The optimal camera positioningfrom: (a) the reference method (coverage: 100.0%), (b) our method with λ = 0.01 pps (coverage: 99.5%), and (c) ourmethod with λ = 0.05 pps (coverage: 100.0%). (d) System performance comparison based on handoff success rate withvarious target densities.

also incorporated the target’s dynamics. Exper-imental results demonstrated a significantlyand consistently improved handoff success ratein comparison with the reference algorithm



100

1 2 3 4 5 60

20

40

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/Nobj

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)

Reference methodOur method λ=0.01Our method λ=0.025Our method λ=0.05

(d)

Fig. 10. (Continued)

described by Erdem and Sclaroff in environ-ments with various target dynamics. The pro-posed algorithm presents superior and robustperformance regardless of the target densities.

Acknowledgment

This work was supported in part by the Univer-sity Research Program in Robotics under grantDOE-DE-FG52-2004NA25589.

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