Modeling Coverage in Camera Networks: A Survey

Abstract

Modeling the coverage of a sensor network is an important step in a number of design and optimization techniques. The nature of vision sensors presents unique challenges in deriving such models for camera networks. A comprehensive survey of geometric and topological coverage models for camera networks from the literature is presented. The models are analyzed and compared in the context of their intended applications, and from this treatment the properties of a hypothetical inclusively general model of each type are derived.

Full text

Int J Comput Vis
DOI 10.1007/s11263-012-0587-7
Modeling Coverage in Camera Networks: A Survey
Aaron Mavrinac ·Xiang Chen
Received: 16 June 2011 / Accepted: 9 October 2012
© Springer Science+Business Media New York 2012
Abstract Modeling the coverage of a sensor network is
an important step in a number of design and optimization
techniques. The nature of vision sensors presents unique
challenges in deriving such models for camera networks. A
comprehensive survey of geometric and topological coverage
models for camera networks from the literature is presented.
The models are analyzed and compared in the context of their
intended applications, and from this treatment the properties
of a hypothetical inclusively general model of each type are
derived.
Keywords Camera networks ·Coverage geometry ·
Coverage topology ·Sensor planning ·Calibration
1 Introduction
Visual coverage is an important quantifiable property of cam-
era networks, describing from a pragmatic standpoint what
the system can see—that is, what visual data it is physically
capable of collecting—and thus informing the most funda-
mental requirement of any computer vision task. Virtually
all camera network applications depend on or can benefit
from knowledge about the coverage of individual cameras,
the coverage of the network as a whole, or the relationships
between cameras in terms of their coverage.
It is therefore not surprising that there exists a sig-
nificant body of literature on modeling camera network
A. Mavrinac ·X. Chen (B
)
Department of Electrical and Computer Engineering,
University of Windsor, Windsor, ON, Canada
e-mail: xchen@uwindsor.ca
A. Mavrinac
e-mail: mavrin1@uwindsor.ca
coverage, spanning back to the earliest days of camera net-
work research. With such diverse applications as sensor
planning, optimal camera placement, camera reconfigura-
tion, camera selection, calibration, tracking correspondence,
and optimal load distribution, and often broad variation of
specific objectives and constraints within each, numerous
structures have emerged for capturing coverage informa-
tion. Given the mixed lineage of camera networks, these
models are influenced by earlier work in computer vision
and in sensor networks; they also exhibit their own inno-
vations to meet the unique challenges of the field. In this
correspondence, we examine the state of the art in mod-
eling camera network coverage, with a view to how the
properties of various models relate to their intended appli-
cations.
Coverage models can be classified into two different major
types. Geometric coverage models, the focus of Sect. 2,are
concerned with the physical area or volume of the scene cov-
ered by a camera network. Given some information about the
camera viewpoints, the physical structure of the scene, and
the task to be performed, such a model seeks to quantify
whether or not a particular stimulus (minimally described
as a point in Rn) is covered, and, in some cases, how well.
A set structure with geometric definitions lends itself natu-
rally to this purpose. Topological models are combinatorial
structures describing the relationships between cameras in
a network with respect to their coverage. This survey con-
siders two types: coverage overlap models in Sect. 3, which
describe pairs or groups of cameras with mutual coverage of
the scene, and transition models in Sect. 4, which describe
the more abstract relationships arising from the possibility (or
probability) of a moving agent transiting from one camera’s
region of coverage to that of another. A topological cover-
age model is typically formalized as a graph—or as some
more general graph-like structure (e.g., simplicial complex,
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Int J Comput Vis
Fig. 1 Hierarchy of information in coverage models
hypergraph)—in which vertices represent the individual
camera nodes, and edges indicate coverage relationships.
The ideal geometric coverage model is derived from four
primary sources of information: the viewpoint parameters
(including position, orientation, and intrinsic parameters) of
all cameras in the network, a model of static objects in the
scene, a probabilistic model of agent dynamics, and a set of
independent task-specific requirements. The coverage over-
lap topology may be thought of as a distillation of some of
this information into a combinatorial structure; ideally, it is
derived from the quantified geometric overlap between indi-
vidual cameras or coverage cells. Transition topology is ide-
ally derived from both the geometric coverage (at some level
of granularity) and from the agent dynamics model, captur-
ing information not present in either the geometric coverage
model or the overlap topology. Figure 1illustrates the ideal
hierarchy of information. All three types of model can be esti-
mated directly from various forms of captured visual infor-
mation in the absence of some or all of the primary sources.
We take a common approach in examining each type of
coverage model. First, we describe the abstract, fundamental
structure of the model class, a comprehensive generaliza-
tion of all reviewed models which provides the reader with
an overall understanding and consistent terminology for the
ensuing discussion. Next, we introduce the various surveyed
works presenting models of this type, in the context of their
target applications. The models being thus introduced, we
compare and contrast the realizations (or lack thereof) of
each individual aspect of the general model. Finally, with
respect to the ultimate goal of producing a general, accurate
model of the given class, we discuss the state of the art and
expose the open research questions by highlighting the most
promising contributions to date.
1.1 Applications
Geometric coverage models are used in a variety of sen-
sor planning applications. In single-camera vision systems,
the objective is to find a viewpoint which satisfies coverage
requirements, often on a well-defined target in a controlled
scene. Analytic solutions are common. By contrast, the opti-
mal camera placement problem in multi-camera systems is
usually far less structured, with an objective of maximizing
coverage under a cost constraint or minimize cost under a
coverage constraint; typical approaches include optimization
by linear programming or search heuristics such as genetic
algorithms. In this context, the coverage of multiple cameras
is modeled simultaneously within the scene, and the over-
all coverage performance of the network can be quantified.
A special case of this problem is camera reconfiguration,
wherein only certain viewpoint parameters of a fixed set of
cameras are variable, and must be optimized for maximum
coverage performance in an online context (which carries
its own unique constraints). Camera coverage models can
also be used for online camera selection; typical problem
instances involve determining one or more optimal views of
a given parametric target, subject to energy and other con-
straints.
Coverage overlap topology models have several impor-
tant applications in computer vision and camera networks.
The offline problem of multi-view registration has been
approached by applying combinatorial optimizations to an
equivalent structure, derived from view overlap, to control
the registration process. Similarly, a number of multi-camera
calibration algorithms, which estimate the relative poses of
the cameras in the network, proceed according to optimized
paths in a combinatorial representation of coverage overlap.
Knowledge of overlap topology may also greatly improve
the performance of direct tracking correspondence, in which
tracked agents are matched between cameras with simultane-
ous coverage. This is a subproblem of the more general pre-
dictive tracking problem and is the basis for camera handoff
across overlapping cameras. Coverage overlap models have
also been applied to scheduling problems in camera networks
with energy constraints, such as duty cycling, triggered wake-
up, and load distribution.
Transition topology models are tailor-made for online pre-
dictive tracking applications in camera networks. The objec-
tive here is to estimate the probability and duration of an
agent transition from one region of coverage (e.g., a camera)
to another. In a sense, this is a generalization of the direct
tracking correspondence problem to cameras which do not
necessarily have coverage overlap, and is likewise used for
camera handoff and other tracking tasks.
1.2 Scope of this Survey
Our interest in this survey is in geometrical and topological
models of (generally) multi-camera coverage, and, where
applicable, on methods of estimating their parameters. We
cover posterior applications only insofar as they elucidate
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Int J Comput Vis
the motivations behind decisions in the design of the mod-
els; the inclusion of a full exposition of every application for
its own sake would produce a prohibitively long work, tanta-
mount to a survey of a considerable cross-section of the entire
camera network field, and more to the point, would not be
particularly germane to an understanding of the theory and
state of the art of coverage models.
In order to remain focused on the stated purpose, the works
surveyed here are primarily drawn from literature specifically
on camera networks or other multi-camera systems. Where
appropriate, the origins of certain concepts, often from the
broader computer vision and sensor network fields, are men-
tioned to provide historical context.
The survey of geometric coverage models in Sect. 2
includes some discussion of single-camera sensor planning
and next best view work from the computer vision litera-
ture, as the level of detail in some of these models provides
a comparative baseline for later multi-camera work. It also
includes mention of some general sensor network models
where appropriate, although the focus is primarily on direc-
tional sensor networks (usually a label for camera networks).
Section 3considers some coverage overlap models devel-
oped for multi-view registration, which is not necessarily
a camera network application, as the views are typically
obtained from a single camera in a video sequence or multi-
image scan. However, since multiple views are theoretically
equivalent to multiple cameras, the model structures and
techniques are of interest here.
The transition models presented in Sect. 4address an issue
largely endemic to camera networks, and accordingly, all of
the works surveyed are specific to the field.
2 Coverage Geometry
2.1 Anatomy of a Geometric Coverage Model
Ageometric coverage model describes the coverage of a geo-
metric space by a sensor or sensor system. Losing some gen-
erality, an intuitive example is a region of a three-dimensional
Euclidean space which is imaged by a camera sensor satis-
factorily for a given image processing task. A precise defini-
tion of a generalized model derived from the works surveyed
follows.
2.1.1 Coverage Model
Asensor system is an entity which detects stimuli for the pur-
pose of executing a task. In general, this system may physi-
cally comprise a single sensor, multiple sensors, or part of one
or more sensors’ ranges, with one or more sensing modali-
ties. Stimuli are uniquely defined in a stimulus space S;in
most cases, stimuli are 2D or 3D points in the stimulus space
R2or R3, respectively, but exceptions exist when geometri-
cal characteristics of the stimulus other than spatial position
(e.g., direction) affect coverage as well.
A stimulus p∈Sis considered covered by a sensor sys-
tem if it yields a response sufficient to achieve the given
task. An ideal coverage function, therefore, is a mapping
C:S→{0,1}, where C(p)=1 indicates that a point is
covered. Equivalently, one can speak of the covered volume
C⊂S. A more general definition C:S→Rencom-
passes models which handle uncertainty and/or consider cov-
erage quality with a coverage grade; this allows for (at least)
relative assessment of coverage. Defining the function as
C:S→[0,1], or equivalently, over any closed range which
can be mapped linearly thereto, also allows absolute assess-
ment; as well, extension of the subset notion is possible if
one considers Ca fuzzy subset of S.
2.1.2 Coverage Criteria for Vision
All visual stimuli considered in the work reviewed herein can
be reduced to point features, which have a single point of
origin in Euclidean space and possibly other characteristics.
Based on well-studied geometric imaging models (Faugeras
1993;Ma et al. 2004), various researchers have identified
one or more of the criteria described here and incorporated
them into their coverage models. We assume that the reader is
familiar with the terminology and parameters of the standard
camera model, and with basic camera optics. The collective
set of intrinsic and extrinsic parameter values of a camera is
termed a viewpoint, and the associated parameter space is the
viewpoint space V(Figs. 2and 3).
We identify three basic criteria (Tarabanis et al. 1994)
which depend only on the viewpoint and a feature point in
R3; two dimensional coverage models can be thought of as
projecting these criteria onto the R2plane.
–Field of View: The infinite subspace of R3which can
theoretically be imaged by the camera, determined by the
horizontal and vertical apex angles (in turn, by the optics
and physical image sensor size) and the pose (extrinsics)
of the camera.
–Resolution: A constraint on the minimum 1required res-
olution; translates directly into an upper limit on depth.
–Focus: A constraint on the acceptable sharpness of the
image; given a maximum blur circle diameter, imposes
upper and lower depth limits around the focus distance
(this range is termed the depth-of-field).
1A maximum resolution constraint is conceivable, e.g., for privacy
purposes, but we have not encountered this in the literature.
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Fig. 2 Imaging criteria (Erdem et al. 2003;Tarabanis et al. 1994)—α
and βare the field of view angles, zris the depth limit for resolution,
and znand zfare the near and far depth of field limits for focus
Fig. 3 Visual stimulus space—the spatial and view angle coverage
criteria induce a stimulus space comprising three-dimensional position
and direction
The field of view in combination with the depth constraints
of resolution and/or focus are also sometimes termed the
viewing frustum.
Considering the view angle to the feature—the direction
of the surface normal at the feature point with respect to the
camera’s optical axis or the ray joining its principal point
with the feature point—adds a fourth coverage criterion, and
up to two additional (angular) dimensions to S.
A feature may also be occluded, thus not covered, if
the ray from the feature to the optical center of the cam-
era is interrupted by an opaque physical object. We consider
two criteria for occlusion, differing primarily in the type of
information about the scene used to evaluate them: static
occlusion, caused by static and/or deterministically dynamic
objects, such as walls, and dynamic occlusion, caused by
probabilistically dynamic objects, such as humans. Note that
self-occlusion is typically handled by the view angle crite-
rion, by imposing a maximum angle of 90◦.
2.1.3 Task Definition
Some of the imaging criteria in the previous section require
task-specific parameters in addition to the camera model
parameters. These typically include the minimum resolution
and maximum acceptable blur circle diameter for the reso-
lution and focus criteria, respectively, as well as a maximum
acceptable view angle. The scene information required by
the occlusion criteria—a deterministic scene model and/or
a probabilistic model of scene agent dynamics—is also part
of the task definition. It should be noted that the information
conveyed by the static scene model and the agent dynamics
model may not be mutually independent; for example, agent
motion may be constrained by the presence of walls, which
also factor in static occlusion.
Besides this coverage model information, it is worth men-
tioning two other aspects of a task definition often encoun-
tered, both of which are known under various names.
The first is the relevance function, which indicates the sub-
set of Swhich is of interest, and may be prioritized (graded).
In general, this takes a form similar to that of a coverage
function, R:S→R. Depending on the task, it may indi-
cate a large volume of the stimulus space, representing e.g.,
a set of rooms and hallways to observe, or a small, localized
region of interest, representing e.g., a part to be inspected or
a person to be tracked.
The second is the allowable viewpoint set, a subset of V
encompassing all allowable viewpoints. Some dimensions
may be constrained to single or discrete values dictated by
hardware properties or fixed settings of the cameras, inde-
pendent of the task. In general, however, such constraints are
task-specific.
2.2 Geometric Coverage Models by Application
The basic function of a geometric coverage model is to
evaluate the coverage of some region of the stimulus space
(viz. a relevance function) by a sensor system. While
not an end in itself, some form of this evaluation is a
clear prerequisite for a family of camera network coverage
problems.
One motivating application which predates camera net-
works is the offline sensor planning problem. The objective
is to find a viewpoint which adequately covers the relevance
function for a given task. This may be found via a generate-
and-test search, or else the entire set of viewpoints may be
solved analytically. Although the output of such methods is
a subset of viewpoints in Vwhich cover some R⊂S,itis
generally straightforward to invert the criteria to obtain the
coverage C⊂Sfor some specific viewpoint in V.Tarabanis
et al. (1995) give an excellent survey of the topic. Typically,
the target systems employ a single camera observing a rel-
atively structured scene, and thus require (and can afford)
highly accurate coverage models. Cowan and Kovesi (1988)
and Tarabanis et al. (1995) present good examples.
In the multi-camera context, one encounters a simi-
lar problem most commonly known as optimal camera
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placement. The exact approach used in single-camera sen-
sor planning does not scale well to multiple cameras, and
there are typically additional design variables such as the
number of cameras (with cost constraints), so nonlinear opti-
mization techniques and search heuristics are the typical
tools of choice, encouraging much simpler coverage models.
Typically, the objective is to search for either the solution
with maximum coverage given a fixed cost (or number of
cameras), or the solution with minimum cost yielding some
minimum coverage. The problem appears similar to the clas-
sic art gallery problem (O’Rourke 1987;González-Banos
and Latombe 2001) frame it so, with their model assuming
omnidirectional visibility and infinite range. Limiting visi-
bility and range yields a more accurate model of coverage,
but fundamentally changes the problem. Following the sen-
sor network approach, Ma and Liu (2005b,2007) propose a
so-called boolean sector coverage model (derived from the
common 2D disc model Wang 2010), enabling them to treat
optimal camera placement similarly to a set covering prob-
lem (Tao et al. 2006;Liu et al. 2008). Qian and Qi (2008),
Wang et al. (2009), and Jiang et al. (2010) further develop this
direction. Erdem et al. (2003) and Erdem and Sclaroff (2006)
approach the problem with a more realistic two-dimensional
model; subsequent results using different coverage models
and optimization techniques but similar basic method have
been reported by Hörster and Lienhart (2006,2009), Angella
et al. (2007), and Zhao et al. (2008,2009). Malik and Bajcsy
(2008) similarly address optimal placement of stereo camera
nodes. Yao et a l . (2008) adapt this type of approach to sur-
veillance networks with tracking and handoff tasks, adding
a “safety margin” to their coverage model to enforce the
necessary coverage overlap. The work of Mittal and Davis
(2004,2008) and Mittal (2006) extends the set of constraints
to include dynamic occlusion, important in a significant sub-
set of applications involving relatively high densities of mov-
ing agents.
The problem of online camera reconfiguration is funda-
mentally similar to optimal camera placement, but restricts
the allowable viewpoints to those which can be arrived at
by varying some set of parameters which can be controlled
online (e.g., mobile platforms, pan-tilt mounts, motorized
lenses), and allows for a dynamic relevance objective based
on feedback from camera data. The coverage models and
optimization techniques used may reflect the need for real-
time online performance. Bodor et al. (2005,2007) and Fiore
et al. (2008) seek to optimize the configuration of cam-
eras mounted on mobile robots for global scene coverage.
Piciarelli et al. (2009,2010) address reconfiguration of pan-
tilt-zoom (PTZ) cameras, common in surveillance applica-
tions. Ram et al. (2006) and Erdem and Sclaroff (2006)
both also touch on PTZ reconfiguration; the latter do so by
introducing a time constraint to the optimal camera place-
ment problem. Chen et al. (2010) focus on the view angle
criterion in optimizing the configuration of rotating (panning)
cameras.
Coverage evaluation is also useful in an online context for
camera selection, which chooses an optimal subset of view-
points for a localized relevance function, often subject to
constraints such as energy costs. In the single-camera realm,
this can be related to the next best view problem, approached
by Reed and Allen (2000) and Chen and Li (2004) using cov-
erage models similar to those used in sensor planning. Park
et al. (2006) use a fairly realistic three-dimensional coverage
model for camera selection, and acknowledge that a yet more
sophisticated model could be substituted. The approach of
Shen et al. (2007) is notable for assigning a scalar coverage
metric to the stimulus space and for allowing task-specific
weighting of the individual factors; they also touch on a ver-
sion of the optimal camera placement problem. Soro and
Heinzelman (2007) approach a slightly different problem:
given a desired viewpoint directly, rather than a relevance
function, their algorithm attempts to find the closest actual
viewpoint (subject to energy costs).
For completeness, it is worth mentioning the geometric
component of the topological coverage overlap model of
Kulkarni et al. (2007), which differs from other geometric
models surveyed here in that it is not analytically derived
from a camera model. Instead, it is purely empirical: through
a Monte Carlo process whereby a structured target is placed
at an arbitrary number of random points in the scene, each
camera with a view to the target at a given position estimates
its pose, and each Voronoi cell around a target position forms
a part of the geometric coverage of each camera that observed
that position. In combination with the topological model, it
is applied to scheduling problems. This model is discussed
further in Sect. 3.
2.3 Analysis and Comparison of Geometric Models
Table 1compares the nature and properties of a number
of camera network coverage models from the literature,
grouped by application. Since most of these models have
been developed with specific applications in mind (indicated
in the first column), it should be interpreted as a comment
on the generality – not necessarily the validity or quality—of
the models. The second column indicates the dimensional-
ity of the model; a dimensionality of 2.5 indicates that the
final representation is two-dimensional, but is derived from
three-dimensional characteristics of the cameras and scene.
The third column indicates whether the model is graded,
i.e., whether it assigns to a point a scalar measure of cov-
erage in some form (weighted, probabilistic, fuzzy, etc.);
non-graded models are bivalent. The following four columns
indicate which of the imaging coverage criteria (field of view,
resolution, focus, view angle) are included. The final two
columns indicate which occlusion criteria (static, dynamic)
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Tabl e 1 Comparison of selected geometric camera network coverage models
Model Appl. Properties Imaging criteria Occlusion
Dim. Graded FOV Resol. Focus Angle Static Dynamic
Cowan and Kovesi (1988)SP•••    
Tarabanis et al. (1995)SP•••  
González-Banos and Latombe (2001)OCP •• 
Wang et al. (2009)OCP•• 
Jiang et al. (2010)OCP•• 
Erdem and Sclaroff (2006)OCP••  
Hörster and Lienhart (2009)OCP••  
Angella et al. (2007)OCP•••    
Zhao et al. (2009)OCP••• 
Malik and Bajcsy (2008)OCP•••  
Mittal and Davis (2008)OCP••
◗
  
Bodor et al. (2007)CR••
◗
 
Piciarelli et al. (2010)CR••
◗

Park et al. (2006)CS•••  
Shen et al. (2007)CS••
◗
 
SP sensor planning, OCP optimal camera placement, CR camera reconfiguration, CS camera selection
are included. It should be noted that, in some cases, the
authors do not provide quantitative descriptions of some
criteria or means of obtaining the information required to
derive them.
2.3.1 Dimensionality
Although vision is an inherently three-dimensional phenom-
enon, many coverage models in various applications are two-
dimensional. In such cases, to simplify the problem at hand,
it is assumed (either implicitly or explicitly) that
– all cameras are positioned in a common plane,
– all targets are constrained to a common plane, and
– the scene consists of occluding vertical “high walls.”
In models derived from the art gallery problem formula-
tion, e.g., González-Banos and Latombe (2001), the choice
reflects the fact that three-dimensional AGP is NP-hard
(Marengoni et al. 2000). The vast majority of work on sen-
sor network coverage problems (Meguerdichian et al. 2001)
has employed two-dimensional disc models (Wang 2010)
(although the three-dimensional case has been studied Huang
et al. 2007), assuming a roughly planar environment. Some
camera network models, including those of Ma and Liu
(2005b,2007), Liu et al. (2008), Wang et al. (2009), and
Jiang et al. (2010), follow directly from this tradition, simply
restricting the disc to a sector (Wang 2010) for direction-
ality. Erdem and Sclaroff (2006) and Hörster and Lienhart
(2009) do not appear to share this lineage, and explicitly cite
the complexity of their respective optimization methods as
motivating their restriction to two dimensions. The model of
Yao et a l . (2008) appears to be heavily influenced by that of
Erdem and Sclaroff. In all of the preceding cases, the domain
of camera coverage is explicitly planar.
In contrast, some two-dimensional models are not devel-
oped from the ground up as such. Bodor et al. (2005,2007)
and Mittal and Davis (2008) begin with three-dimensional
analytic treatments of their respective constraints, but subse-
quent assumptions about the scene and viewpoint restrictions
effectively reduce their models to the plane without loss of
information. Shen et al. (2007) present a similar treatment
of view angle—in particular, including the inclination angle
between the sensor and a human subject’s head with respect to
the ground plane—in an otherwise two-dimensional model.
Piciarelli et al. (2010) account for a three-dimensional field
of view criterion by projecting the elliptical cross-section of
their conical visible region onto the plane.
Early coverage models used in sensor planning, such as
those of Cowan and Kovesi (1988) and Tarabanis et al.
(1995), are fully three-dimensional: the gains in generality
and accuracy clearly outweigh the added complexity in the
single-camera case. These advantages have induced a num-
ber of multi-camera coverage models across the application
spectrum to follow suit. Cerfontaine et al. (2006) describe
a multi-camera method employing a three-dimensional cov-
erage model presumably derived from the pinhole camera
model, but give no details on the criteria. Park et al. (2006)
fully describe their model with a three-dimensional view-
ing frustum; the multi-camera complexity is handled by
dividing the covered volume into discrete parts and gen-
erating look-up tables for coverage grade. Angella et al.
(2007) employ a three-dimensional model drawing heavily
on the sensor planning literature. The models of Malik and
Bajcsy (2008) and Zhao et al. (2009) are also fully three-
dimensional.
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(a) (b)
Fig. 4 Coverage valuation schemes—coverage may be graded either
using a bivalent indicator function, or a real-valued function (without
loss of generality, bounded to [0,1])
2.3.2 Valuation
Real-world sensor planning applications typically have well-
defined requirements, and the goal is simply to find any view-
point which meets these requirements. Accordingly, models
such as those of Cowan and Kovesi (1988) are bivalent: either
the viewpoint is acceptable or it is not, or equivalently, either
a relevance function is covered or it is not. Tarabanis et al.
(1995) discuss not only this admissibility of a viewpoint,
but also its optimality, proposing an overall coverage quality
metric based on the robustness in individual criteria (Fig. 4).
In solving the camera selection problem, one is interested
in finding the best view of a relevance function, to which a
real-valued coverage metric clearly lends itself. In Park et al.
(2006), the quality of coverage of a point pfrom a camera Ci
is considered to vary inversely with the distance from pto the
center of the viewing frustum of Ci. The authors point out
that developing an accurate coverage quality metric is not
their focus, and allow that a more sophisticated definition
could be substituted. Shen et al. (2007) explicitly set out to
define such a metric for the restricted problem case of human
surveillance; theirs takes the form of a real-valued function.
Soro and Heinzelman (2007) study several coverage-
based valuations of viewpoints for camera selection, but as
previously mentioned, their formulation is notably different
than others discussed here. Roughly speaking, each valua-
tion can be thought of as a distance metric in V. If one were
to assign an ideal viewpoint to every p∈S, these metrics
would effectively constitute a coverage grade of the form
C(p):S→R+.
By contrast, in solving the optimal camera placement and
reconfiguration problems, bivalent coverage valuations are
used almost exclusively, to enable the use of various opti-
mization techniques (e.g., binary integer programming) that
would otherwise not be applicable. Wang et al. (2009)pro-
vide one counterexample, applying a multi-agent genetic
algorithm over a graded coverage model simple enough to
make the optimization computationally feasible. Continuous
grading functions defined by Yao et a l . (2008) assign reduced
coverage values to the edges (i.e., regions near the limits of
field of view and resolution) of a camera’s model, in order to
encourage their optimization process to yield solutions with a
substantial margin of overlap between cameras for improved
tracking and handoff. Shen et al. (2007) notably use their
coverage grade as a constraint in solving a restricted case of
the optimal camera placement problem using a greedy algo-
rithm.
2.3.3 Field of View
The coverage model employed by González-Banos and
Latombe (2001) is unique among those surveyed in assuming
omnidirectional viewing capabilities, and thus not including
a field of view criterion. The directional nature of camera
coverage is a recurring key point in the literature, and field
of view is the most commonly modeled constraint.
The simple sector-based models of Ma and Liu (2005b),
Ma and Liu (2007), Liu et al. (2008), Qian and Qi (2008),
Wang et al. (2009), and Jiang et al. (2010) describe field
of view with a single angle parameter, which corresponds
roughly to the horizontal apex angle. The boundary rays are
symmetric about the optical axis, implying an assumption
of non-oblique projection. This turns out to be a satisfactory
definition in two dimensions; Erdem and Sclaroff (2006) and
Hörster and Lienhart (2009) arrive at the same by way of
the pinhole camera model, perhaps elucidating how its value
should be determined from a given camera system.
Erdem et al. (2003) also describe the three-dimensional
field of view using two apex angles. Malik and Bajcsy (2008)
and Mittal and Davis (2008) handle field of view similarly.
Cowan and Kovesi (1988) and Tarabanis et al. (1995) both
effectively limit field of view to the smaller of the two apex
angles and assume non-oblique projection. Piciarelli et al.
(2010) model the field of view as a cone, presumably with
aperture angle equal to the smaller apex angle. While this
representation facilitates their algorithm by projecting to a
circle of constant radius on a transformation of the scene
plane, it lacks accuracy and no justification is given in the
context of their application.
The apex angles are derived from a more elementary char-
acterization of the field of view. In general, a point p∈R3
is within the field of view of a camera if its projection lies
somewhere on the physical sensor surface. The field of view
induced by a rectangular sensor is a pyramid bounded by the
rays from each of its four corners through the optical center
of the camera. Zhao et al. (2009) use this constraint directly,
and can theoretically handle oblique projection. The visible
pyramid volume can also be thought of as divided into an
infinite set of visible “subplanes” orthogonal to the optical
axis; Park et al. (2006) simply assume that the dimensions of
the visible subplanes at the near and far depth of field lim-
its are known, and that these subplanes are centered at the
optical axis (implying non-oblique projection).
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2.3.4 Resolution
The sector-based models proposed by Ma and Liu (2005b,
2007), Liu et al. (2008), Wang et al. (2009), and Jiang et al.
(2010) have a radial sensing range limit; although there is
no explicit relationship to a resolution constraint, it seems
its most likely justification. Cowan and Kovesi (1988) model
their resolution constraint as an arc in two dimensions and
as a spherical cap in three dimensions.
In fact, this circular/spherical representation unnecessar-
ily complicates the matter: since the projected image is planar
and orthogonal to the optical axis, resolution is a function
of depth along the optical axis rather than distance along
the ray from the optical center (Tarabanis et al. 1994). The
triangle-shaped model of Hörster and Lienhart (2009)isa
more accurate two-dimensional representation of the resolu-
tion constraint, although it is not explicitly parameterized as
such. Erdem and Sclaroff (2006), Bodor et al. (2007), Malik
and Bajcsy (2008), Yao et a l . (2008), and Mittal and Davis
(2008) all use distance along the optical axis as the single
parameter for the resolution constraint. The last also sug-
gest that such a resolution criterion could be used as a “soft”
constraint informing a quality measure.
2.3.5 Focus
While focus is a staple constraint in sensor planning coverage
models (Tarabanis et al. 1995), it has not been included in
most coverage models developed for other purposes. Angella
et al. (2007) mention it, but as with their other imaging cri-
teria, they provide no details. Park et al. (2006) are the other
exception; their model is bounded in depth along the optical
axis by the near and far depth of field limits.
Park et al. also use focus as part of their coverage grade
computation (discussed in Sect. 2.3.2), to some extent: if the
center of the viewing frustum is taken as an approximation
of the focus distance, the distance of a point along the opti-
cal axis from the center varies approximately proportionally
to the blur circle diameter. A similar interpretation can be
applied to the valuation function of Wang et al. (2009).
2.3.6 View Angle
A constraint on view angle (Fig. 5) is present in some sensor
planning coverage models (Tarabanis et al. 1995), such as that
of Cowan and Kovesi (1988). In the multi-camera context,
it has been included where the target task depends on view
angle. For example, the task of the camera network in Zhao
et al. (2009) is the identification of planar tags, the perfor-
mance of which degrades with increasing view angle. Sim-
ilarly, Shen et al. (2007) are interested in surveillance tasks
such as face tracking, so view angle features prominently in
their model. Mittal and Davis (2008), drawing on the earlier
Fig. 5 View angle—the view angle to a point feature on a surface,
shown as pwith a corresponding surface normal, is measured as αin
some sources and as βin others
sensor planning models, include the criterion, anticipating
that some tasks will have such requirements.
Special cases of task view angle requirements give rise to
a few alternate—but equivalent—forms of the view angle cri-
terion. Bodor et al. (2007) are interested in observing paths,
where foreshortening effects due to the view angle to a path
degrade performance; their view angle criterion is based on
both the angle between the path normal and the camera posi-
tion, and the angle between the path center and the optical
axis. Some applications, such as those of Malik and Bajcsy
(2008) and Chow et al. (2007), require 360◦coverage of a tar-
get, and define a maximum view angle for mutual coverage
of a point by two cameras. If the view angle to a feature on an
opaque surface exceeds 90◦, the surface occludes the feature
from view; this phenomenon is known as self-occlusion and
is sometimes treated as a separate criterion, such as by Chen
and Li (2004) and Zhao et al. (2009).
An interesting question that arises in defining this criterion
is whether to measure the view angle between the feature
surface normal and the optical axis, or between the feature
surface normal and the line-of-sight ray from the camera’s
optical center. Both approaches have merit in terms of validity
with respect to task requirements. The former is taken by
Chen and Li (2004) and Bodor et al. (2007); the latter, by
Cowan and Kovesi (1988), Shen et al. (2007), Malik and
Bajcsy (2008), and Zhao et al. (2009).
Soro and Heinzelman (2007), in one of their models, grade
views primarily based on view angle.
2.3.7 Static Occlusion
Occlusion by static scene objects factors heavily in most
multi-camera coverage work (Fig. 6). Malik and Bajcsy
(2008), whose model does not include a static occlusion
criterion, assume a simple rectangular room with nonzero
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Fig. 6 Static occlusion—the
two white boxes represent the
static scene model. Coverage
without the constraint is
outlined, with actual coverage in
gray
relevance somewhere near its center, which suits their tar-
get task, but in most multi-camera applications the scene
is assumed to be more complex. The “high wall” occlusion
model common in two-dimensional approaches has its origin
in the art gallery problem, exemplified by González-Banos
and Latombe (2001). This constraint is enforced as follows:
given a scene model consisting of line segments in the plane,
a point p∈R2is occluded (not covered) if the line of sight
from the camera’s optical center to pintersects any such line
segment. Erdem and Sclaroff (2006) propose an algorithm to
construct a continuous “visibility polygon” set which con-
tains all non-occluded scene points. Hörster and Lienhart
(2009), Mittal and Davis (2008), and Shen et al. (2007)sim-
ply check for line-of-sight on each discrete relevance point.
Jiang et al. (2010) approximate static occlusion by simply
excluding obstacle regions from the field of view of a cam-
era; in confined spaces and using cameras with realistic field
of view, this would likely result in poor performance.
The three-dimensional analog to the line segment scene
model is composed of opaque surfaces. A continuous,
analytic solution has been employed in sensor planning
(Tarabanis et al. 1996) and next best view (Maver and Bajcsy
1993) applications. In the multi-camera context, discrete
line-of-sight checking is more common, as is done by Angella
et al. (2007) and Zhao et al. (2009).
Piciarelli et al. (2010) handle static occlusion directly in
the relevance function. Each camera node has its own copy of
the global relevance function, with all points occluded (via
two-dimensional line of sight) from that camera removed
from the model.
2.3.8 Dynamic Occlusion
Mittal and Davis (2008) have pioneered handling dynamic
occlusion in a geometric coverage model. They use a prob-
abilistic model of agent occupancy and some assumptions
about agent height and allowable camera viewpoints to for-
mulate a probabilistic visibility criterion, which is then inte-
grated with their other (static) constraints. Angella et al.
(2007) use this model. Chen and Davis (2008) independently
propose their own probabilistic metric for dynamic occlu-
sion, under similar assumptions about the agents and cam-
eras. Qian and Qi (2008) also propose a probabilistic model,
with targets modeled as 2D discs (analogous to Mittal and
Davis’ representation) and using a simple sector-type cover-
age model.
Zhao et al. (2009) include a “mutual occlusion” criterion in
their model, which approximates worst-case dynamic occlu-
sion by specifying a range of view angles within which a
point is assumed to be occluded by another agent.
2.3.9 Combining Criteria and Multi-Camera Coverage
Cowan and Kovesi (1988) treat coverage criteria as con-
straints on the viewpoint, so in order to find the solution
set which satisfies all constraints (i.e., the set of viewpoints
which adequately cover the relevance function), it suffices to
intersect the solution set for each individual criterion. Biva-
lent coverage models have taken much the same approach,
intersecting the sets of covered points generated by each cri-
terion, exemplified by the “feasible region” result of Erdem
and Sclaroff (2006). In the multi-camera context, the over-
all coverage of the scene is of interest; this is usually found
by taking the union of the coverage sets for each individual
camera, as Erdem and Sclaroff also show.
Mittal and Davis (2008) integrate their probabilistic
dynamic occlusion metric with their other “static” constraints
to obtain an overall (graded) quality metric for each point and
orientation.
Several models in the literature also provide mechanisms
to compute overall k-coverage of a scene. Erdem and Sclaroff
(2006) show a similar approach in their experimental fig-
ures, but none of their experimental problem statements
require multi-camera coverage. Liu et al. (2008)alsouse
an intersection-union approach in their work, which focuses
specifically on k-coverage.
Mittal and Davis (2008) discuss more complex “algorith-
mic constraints” involving the interplay of various constraints
between multiple cameras, for such tasks as stereo matching.
To some extent, particularly on the view angle criterion, this
is realized in the k-coverage model of Shen et al. (2007).
2.3.10 Task Parameters
A recurring motif in the literature is that the quantification
of visual coverage depends as much on the task as it does
on the parameters of the imaging system. Generally, given a
computer vision algorithm used in a task, it is at least the-
oretically possible to quantify soft or hard requirements on
imaging properties such as resolution, focus, and view angle.
The actual values of the imaging constraints in sensor plan-
ning models, such as those of Cowan and Kovesi (1988) and
Tarabanis et al. (1995), are assumed to be direct task
requirements. Erdem and Sclaroff (2006) emphasize the task-
specific nature of the constraints in the optimal camera place-
ment context.
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One form of the optimal camera placement problem con-
strains the minimum required proportion of the relevance
function covered by the solution (while maximizing or min-
imizing some other variable, such as cost), a task-specific
requirement. This is one of the four variations studied by
Hörster and Lienhart (2009). The weighted form of this pro-
portion, sometimes called the coverage rate (Jiang et al.
2010), may fill a similar role, as in the optimal placement
problem studied by Shen et al. (2007).
2.3.11 Relevance Function
A relevance function is most commonly used in optimal
camera placement and camera reconfiguration applications,
where it comprises the coverage objective. Often, the nonzero
relevance function is implicitly the working volume (as in the
art gallery problem); in order to support a general problem
definition, however, a model should separate the coverage
target from other considerations such as the scene model for
static occlusion and the allowable viewpoint set. Jiang et al.
(2010), Hörster and Lienhart (2009), Angella et al. (2007),
Zhao et al. (2009), and Malik and Bajcsy (2008) all allow
specification of a relevance function, in some form, that is
distinct from the scene model and/or allowable viewpoint
positions.
It is also useful to allow prioritization of coverage in the
relevance function. One of the experiments of Erdem and
Sclaroff (2006) specifies a higher resolution requirement on
certain parts of the floor plan. Hörster and Lienhart (2009)
use a continuous weighted relevance function in their prob-
lem instance definition; in the actual discrete domain of their
algorithm, this informs the sampling frequency of control
points. Jiang et al. (2010) retain a similar continuous defini-
tion, using distinct regions with integer weights to simplify
the weighted coverage computation. Piciarelli et al. (2010)
define a relevance function as a mapping of discrete points
to real values.
2.3.12 Allowable Viewpoint Set
Generally, explicit restrictions on viewpoints in sensor plan-
ning and optimal camera placement applications are on the
position component of the viewpoint only. There are usu-
ally no restrictions on orientation; a notable exception is the
work of Chen and Li (2004), where both position and orien-
tation are constrained by kinematic reachability by the robot
on which the camera is mounted. Restrictions or specifica-
tions on other aspects of the viewpoint, such as the intrinsic
parameters of the camera, are usually implicit in the problem
instance.
Sensor planning coverage models translate coverage cri-
teria into constraints on the solution set of viewpoints, so the
allowable viewpoint set is just a directly-defined constraint
to be intersected with the rest. The “prohibited regions” con-
straint employed by Cowan and Kovesi (1988) is an example.
In the art gallery problem, the volume of relevance to be
covered and the allowable Rnpositions of the guards (cam-
eras) are implicitly the same, as exemplified by González-
Banos and Latombe (2001). Erdem and Sclaroff (2006) and
Wang et al. (2009) also use the relevance volume as the
allowable viewpoint position set. Jiang et al. (2010) spec-
ify a relevance function, but place no restriction on camera
position. Malik and Bajcsy (2008) constrain camera positions
to a rectangular volume which is implied to be a superset of
the relevance volume. Shen et al. (2007) restrict viewpoints
to the outer boundary of a rectangular relevance volume.
Hörster and Lienhart (2009) specify the relevance function
and the allowable viewpoint set separately, as subsets of a
larger working volume.
Camera reconfiguration applications typically place tighter
restrictions on allowable viewpoints. Bodor et al. (2007)
allow full online control of position and orientation, as their
cameras are mounted on mobile robots. Piciarelli et al. (2010)
allow online control of orientation (pan and tilt) as well as
some intrinsic parameters (zoom), but constrain the cameras
to fixed positions. Chen et al. (2010) allow horizontal rotation
(pan) only.
2.4 State of the Art and Open Problems
To date, no geometric model has fully captured the phe-
nomenon of visual coverage in a representation suitable for
the general multi-camera context. While some of the single-
camera sensor planning models we have discussed are quite
accurate and general enough to apply to a wide set of tasks,
they are ill-suited to modeling typical systems and environ-
ments involving multiple cameras, and in their present form
would likely put prohibitive computational requirements on
optimizations involving even relatively small networks. Con-
versely, in expressly designing multi-camera models in forms
suitable for specific optimization techniques, the remainder
of the authors mentioned have restricted applicability to rel-
atively specific problem classes. Mittal and Davis (2008)
appear to have designed the most accurate and general model
to date which is still suitable for multi-camera optimization,
but it is still somewhat restricted by certain assumptions,
notably its two-dimensional final representation, and its lack
of a focus criterion.
The ideal geometric coverage model would not only
accurately model visual coverage in a form convenient for
multi-camera systems and their environments, with as few
assumptions as possible and allowing for generalized task
requirements, but also provide this information in a form
accomodating powerful optimization techniques. It is clear
from the preceding discussion that the factors involved
in a model achieving the former goal would be highly
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complex, complicating success in the latter goal. The prevail-
ing approach to this problem has been to design the model to
be as accurate and general as possible for one specific opti-
mization technique from the outset, but this has failed to pro-
duce the ideal model. We suggest that attempting to achieve
the first goal in isolation could, at the very least, produce a
tool for evaluation, but may also yield new insights into the
nature of multi-camera coverage that may lend the model,
or some derivative thereof, to an appropriate optimization
scheme.
Most sources surveyed have assumed that the coverage
model employed reflects a posteriori task performance, with
little or no validation of the model itself. In order to eval-
uate accuracy and generality, a generic scheme for relating
the coverage metric to a task performance metric should be
developed and adopted. A simple statistical measure, such as
the Pearson product-moment correlation coefficient, might
suffice; depending on the nature of the coverage and perfor-
mance metrics, other measures might be more illuminating.
3 Coverage Overlap Topology
3.1 Anatomy of a Coverage Overlap Model
Acoverage overlap model describes the topology of a cam-
era network in terms of coverage overlap (mutual coverage
of some part of the scene). Typically, the camera node is
the atomic entity, and of interest are the node-level coverage
overlap relationships. It is often desirable to capture not only
the fact but the degree of overlap.
In the most general form, such a model is a weighted
undirected hypergraph H=(C,E,w), where Cis a set of
coverage cells,E⊆P(C)(where Pdenotes the power set)
is a set of hyperedges, and w:E→R+is a weight function
over E. A coverage cell may represent an individual camera
node’s coverage model or some portion thereof. The exis-
tence of a hyperedge e∈Eindicates that the nodes in e
share mutual coverage of the scene, with a k-hyperedge cor-
responding to k-coverage. In a weighted model, w(e)quanti-
fies the degree of shared coverage. In an unweighted model,
implicitly, w(e)=1ife∈Eand w(e)=0 otherwise; the
existence of an edge indicates sufficient mutual coverage for
the given task, by some task-specific quantitative definition.
The most common form is the vision graph (Fig. 7), which
is an ordinary graph (a 2-uniform hypergraph) and thus con-
siders only pairwise coverage overlap.
3.2 Coverage Overlap Models by Application
The earliest examples of coverage overlap models are
found in multi-view registration applications, including video
sequence registration and 3D range image registration. Since
Fig. 7 Vision graph—from 2D coverage geometry (left) to overlap
topology (right). Note that the pairwise overlaps of A,B,and Eare
represented, but their 3-overlap is not
the objective is to align visual data from multiple views, it is
clearly useful to know which views overlap and thus might
have some corresponding features for registration. Although
this is not necessarily a camera network application, multi-
ple views are theoretically equivalent to multiple cameras.
Sawhney et al. (1998) propose a graph formalism of the cov-
erage overlap relationships between multiple views for video
sequence registration, with each frame (view) represented by
a vertex, addressing the fact that frames (views) which are not
temporally adjacent may still be adjacent in terms of overlap
topology. Kang et al. (2000) construct a similar graph repre-
sentation of the overlap topology of frames, in which edges
indicate either temporal or spatial (overlap) adjacency. Their
algorithm searches for an optimal path in this graph to min-
imize error in global registration. Huber (2001) constructs
a graph for registration of partial 3D views using an over-
lap criterion on the range images, analyzes the registration
problem through its connectivity properties, and performs
reconstruction over a spanning tree. Sharp et al. (2004)also
study 3D range image registration using a similar graph for-
malism, which they assume exists a priori. They approach the
global registration problem by first considering registration
over basis cycles within the graph, then merging the results
using an averaging technique.
Knowledge of camera network topology in terms of cov-
erage overlap is a useful precursor to full metric multi-
camera calibration.Antone et al. (2002) require, as input
to their calibration algorithm, a graph of node adjacency;
although the criterion for edge presence is based on posi-
tion (from GPS), since the algorithm targets omni-directional
cameras, this is supposed to approximate coverage over-
lap and is thus a vision graph. Brand et al. (2004)fur-
ther develop this work, using directionally-constrained graph
embeddings. Devarajan and Radke (2004) name and explic-
itly describe the vision graph, pointing out its distinctive-
ness from the communication graph (a departure from sensor
networks), and demonstrating its usefulness in informing a
full calibration algorithm as to which camera pairs should
attempt to find a homography. However, they offer no
means of obtaining the vision graph automatically, instead
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Tabl e 2 Comparison of selected topological coverage overlap models
Model Appl. Properties Construction data
Struct.2Weight k-View Part. Geom. Reg. Feat. Occup. Motion
Sawhney et al. (1998)RG •
Huber (2001)RG •
Sharp et al. (2004)RG
•
Cheng et al. (2007)CG •
Kurillo et al. (2008)CG
•
Bajramovic et al. (2009)CG •
Mavrinac et al. (2010)CG
•
Stauffer and Tieu (2003)DTG •
Mandel et al. (2007)DTG •
Van Den Hengel et al. (2006)DT G •
Lobaton et al. (2010)DTSC  •
Kulkarni et al. (2007)SHG
 •
Mavrinac and Chen (2011)SHG •
Rmulti-view registration, Ccalibration, DT direct tracking correspondence, Sscheduling, Ggraph, HG - hypergraph, SC simplicial complex
making the temporary assumption that it is available a pri-
ori. Cheng et al. (2007) address this issue by approximat-
ing the vision graph via pairwise feature matching, and
describe a full calibration algorithm also employing the fea-
ture data following the procedure of Devarajan and Radke.
Kurillo et al. (2008) construct a weighted vision graph based
on the number of shared calibration points, then optimize the
set of calibration pairs by finding a shortest path spanning
tree. Bajramovic et al. (2009) perform multi-camera cali-
bration over connected components of their vision graph,
which they construct independently using the normalized
joint entropy of point correspondence probability, one of sev-
eral methods described by Brückner et al. (2009). Mavrinac
et al. (2010) describe the vision graph as a theoretical upper
bound for the connectivity of their grouping and calibration
graphs.
Overlap topology can be used to help establish direct
tracking correspondence, a subproblem of tracking cor-
respondence involving agents simultaneously visible in
multiple cameras. This is useful for camera handoff among
overlapping cameras (Javed et al. 2000;Khan and Shah
2003). In this context, overlap topology is usually consid-
ered to be a subset of a more general transition topology,
models for which are covered in Sect. 4.Stauffer and Tieu
(2003) describe a “camera graph” which identifies with the
vision graph, estimating camera overlap from sets of likely
correspondences between tracks. This graph is then used as
feedback to improve tracking correspondence. Mandel et al.
(2007) use a probabilistic approach on motion correspon-
dence to establish overlap topology for tracking purposes.
In a series of papers on the topic, Van Den Hengel et al.
(2006,2007), Detmold et al. (2007,2008), Hill et al. (2008)
describe the exclusion approach, whereby the vision graph
begins complete and edges are removed based on contra-
dictory occupancy observations, with a target application of
tracking correspondence in surveillance networks. Lobaton
et al. (2009a,b,2010) propose a simplicial complex repre-
sentation of overlap topology dubbed the CN-complex,pri-
marily targeted at tracking applications. Overlap topology is
employed by Song et al. (2010) as part of their consensus
approach to tracking and activity recognition.
Camera networks are often composed of devices with lim-
ited computational and energy resources. Knowledge of over-
lap topology can help inform efficient scheduling of node
activity. Ma and Liu (2005a) estimate the correlation between
views using their previously described geometric coverage
model, to improve the efficiency of video processing in
camera networks with partially redundant views. However,
the information used is not strictly topological, and the
method applies specifically to two-camera systems. Dai and
Akyildiz (2009) address the latter issue by extending the cor-
relation problem to multiple cameras, but their model is also
not strictly topological. Kulkarni et al. (2007) construct a
vision graph using a Monte Carlo feature matching technique
with a geometric model component, and demonstrate its use
in duty cycling and triggered wake-up. Mavrinac and Chen
(2011) propose a coverage hypergraph derived directly from
their geometric coverage model, and apply it to the optimiza-
tion of load distribution using a parallel machine scheduling
algorithm.
3.3 Analysis and Comparison of Overlap Models
Table 2compares the nature and properties of a selection
of topological coverage overlap models from the literature,
grouped by application (indicated in the first column). The
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second column identifies the combinatorial structure used
(whether explicit or interpreted), and the following three
columns indicate which additional properties are exhibited:
edge weighting, k-view modeling, and modeling of par-
tial views, respectively. The remaining five columns specify
which type of data is used in constructing the model: geomet-
ric coverage information, registration results, local feature
matching, occupancy correlation, or motion correlation.
3.3.1 Combinatorial Structure
Although not all of the coverage overlap models surveyed are
explicitly formalized as graphs (or hypergraphs), they can be
cast as cases of the general model described in Sect. 3.1 with-
out loss of information. The original descriptions given by
the authors are summarized here, and instances where ancil-
lary information not captured by the graph representation is
present are highlighted.
The vision graph as described by Devarajan and Radke
(2004)—an undirected, unweighted graph with vertices rep-
resenting camera nodes and edges indicating sufficient cover-
age overlap for the purposes of the task—is the simplest and
most common combinatorial structure for models of cover-
age overlap topology seen in the literature. This is the explicit
form of the models of Cheng et al. (2007), Bajramovic et al.
(2009), Mavrinac et al. (2010), and Stauffer and Tieu (2003).
The graphs of Sawhney et al. (1998) and Kang et al. (2000)
describe temporal and spatial adjacency, but since in their
application temporally adjacent frames are assumed to be
spatially adjacent also, they are effectively describing the
vision graph structure. The graphs described by Huber (2001)
and Sharp et al. (2004) are also essentially vision graphs;
though edges are annotated with pairwise relative pose and
other relations, this information is not part of the overlap
model proper. Mandel et al. (2007) and Van Den Hengel et al.
(2006,2007) do not explicitly present graph formalisms, but
maintain sets of hypotheses about coverage overlap which
correspond to edges in the vision graph.
Some recent models extend the captured topology from
pairwise overlap to k-overlap, requiring a hypergraph or
hypergraph-like structure to accomodate the relationships.
Lobaton et al. (2009a,b,2010) partially achieve this with a
simplicial complex representation. This choice of represen-
tation, over a more abstract structure such as a hypergraph,
seems to stem from the focus being more on geometrical
properties and operations and less on combinatorial opti-
mization. They are interested in overlap topology only up
to 2-simplices (or 3-simplices in a hypothetical extension to
three dimensions), so their model does not capture general
k-overlap. Kulkarni et al. (2007) model the full k-overlap
topology of the camera network, although they do not explic-
itly formalize this model in a hypergraph representation or
use any combinatorial techniques. Mavrinac and Chen (2011)
Fig. 8 Vertex granularity in the CN-complex (Lobaton et al. 2010)—
the simplicial complex on the right more accurately describes the cov-
erage overlap topology between cameras Aand B
present an explicit hypergraph representation of k-overlap
topology, with an initial scheduling application using a com-
binatorial algorithm.
The assingment of one vertex to each camera node is sen-
sible for most combinatorial optimization purposes, but a few
models eschew this paradigm and subdivide vertex assign-
ment into coverage cells. Motivations for doing so vary. Van
Den Hengel et al. (2006,2007) subdivide views into an arbi-
trary number of windows to handle partial coverage overlap
of cameras (due to the specifics of their construction method,
discussed in Sect. 3.3.2). Mandel et al. (2007) divide views
into regions for a similar reason. In both cases, it appears
that the model of interest to the eventual application recom-
bines the coverage cells (which are each associated with a
specific camera) to the more usual granularity of one vertex
per camera. Lobaton et al. (2009a,b,2010) divide the two-
dimensional geometric coverage of each camera at rays to
occlusion events (which they call “bisecting lines”), allowing
their model to accurately capture some geometric properties
of overlap, such as static occlusions within the field of view,
as shown in Fig. 8. This increased granularity is preserved,
and shown to be beneficial in tracking applications.
Calibration and scheduling applications of overlap models
often make use of graph optimizations related to path length.
In such cases, weighting vision graph edges proportionally
to the degree of coverage overlap can yield better results.
Given an edge eAB ={A,B}linking cameras Aand B,
Kurillo et al. (2008) assign the weight w(eAB)=1
NAB , where
NAB is the number of common reference points detected by
cameras Aand B(see Sect. 3.3.2 for more on their graph
construction method). Kulkarni et al. (2007) similarly com-
pute the degree of k-overlap from the number of common
reference points of kcameras. Both describe methods of han-
dling non-uniform spatial distributions of reference points.
Mavrinac and Chen (2011) theoretically use the volume of
intersection between kcameras’ geometric coverage models
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to weight hyperedges, but in practice, the required polytope
intersection procedure is NP-hard, so they use a uniform dis-
tribution of points to compute a discrete approximation.
3.3.2 Construction from Visual Data
In theory, overlap topology is a derived property of the geo-
metric coverage of the camera network. However, since it
is often employed in applications where geometric coverage
information is unavailable, especially in calibration initial-
ization, it is often necessary to estimate it using visual data.
Finding correspondences between visual data in some form
among views is the obvious means—if camera Amatches
a piece of its own information to one from camera B, then
a hypothesis of mutual coverage between Aand Bcan be
made or strengthened.
For tasks which already make use of correspondences
between local image features, the same information, or some
subset thereof, can be used to recover overlap topology. This
is the approach originally suggested by Devarajan and Radke
(2004). Because their algorithm works in an offline central-
ized context, Kang et al. (2000) are able to directly corre-
late image features to infer topology. Cheng et al. (2007),
addressing the camera network calibration problem, attempt
to make such an approach scalable in an online distributed
context by instead sharing feature digests of SIFT (Lowe
2004) descriptors among camera nodes. Bajramovic et al.
(2009) and Brückner et al. (2009) use the pairwise joint
entropy of point correspondence probability distributions,
based on SIFT feature descriptors, as a measure of overlap.
Kurillo et al. (2008) use direct matching of a more sparse but
more accurate set of features, obtained from a structured cal-
ibration target. A similar approach is taken by Kulkarni et al.
(2007), although the structured target is only used for topol-
ogy inference and is unrelated to their application. In their
case, both the degree and geometry of coverage overlap are
estimated using a Monte Carlo technique, whereby the tar-
get is imaged at random reference points, and the k-covered
Voronoi cells around each point contribute to the estimate for
each of the kcameras covering it.
Registration-based applications are typically iterative, and
some overlap models are updated using new correspondence
data available in each iteration. Sawhney et al. (1998) infer
global overlap topology iteratively, using feedback from a
local coarse registration stage to recover graph edges, and
subsequently performing local fine registration on adjacent
views. An analogous three-dimensional process is employed
by Mavrinac et al. (2010) in a distributed calibration algo-
rithm, with coarse registration results iteratively building a
grouping graph which then informs pairwise fine registra-
tion. Huber (2001) also uses candidate registration matches
to iteratively infer overlap topology.
Camera networks often have wide baselines and large rota-
tional motion between cameras, over which local
feature detectors generally have poor repeatability and
matching performance (Mikolajczyk et al. 2006;Moreels
and Perona 2007). Fortunately, they offer the possibil-
ity of matching online motion data instead of static fea-
tures, which can be more robust under some circumstances.
Stauffer and Tieu (2003) argue that the descriptiveness, spa-
tial sparsity, temporal continuity, and linear increase in vol-
ume over time of tracking correspondences make them more
reliable in matching than static features. They correlate local
tracks between cameras over time, and infer a vision graph
edge where the expectation of a match exceeds a thresh-
old. Mandel et al. (2007) take a slightly different approach,
detecting local motion and attempting to correlate it with
motion observed in other cameras, via a distributed algo-
rithm. Lobaton et al. (2010) automatically decompose cam-
eras into coverage cells by locally finding “bisecting lines” at
which occlusion events occur (e.g., walls), then, with a dis-
tributed algorithm, globally estimate cell overlap by match-
ing concurrent occlusion events over time.
Van Den Hengel et al. (2006,2007) take the reverse
approach to those described thus far. Their exclusion algo-
rithm begins by assuming all camera nodes have overlapping
coverage, thus a complete vision graph, and eliminates edges
over time using occupancy data to rule out coverage overlap.
This method does not require any correspondence between
observations; if camera Ais occupied (currently observing
an object) and camera Bis unoccupied, this is evidence that
Aand Bdo not have mutual coverage, which through obser-
vation ratio calculations and thresholding contributes to the
final model. Partial overlaps are handled by dividing camera
coverage into an arbitrary number of coverage cells. Hill et al.
(2008) describe a number of potential shortcomings in real-
world operation, along with ways of mitigating the adverse
effects on performance. Detmold et al. (2007,2008) extend
the approach into an online distributed context for scalability
and dynamic updating of the model.
One direct route to an overlap model well-suited to the
task at hand is to use the very visual data used by the
task itself to estimate the model, if this data (or similar
data) is available. This can clearly be seen in most cases
of registration, feature-based calibration, and tracking appli-
cations in Table 2. In a distributed camera network, depend-
ing on the nature of the data and the amount of it required
to establish accurate overlap estimates, there is a poten-
tial scalability issue since, initially, the data must effec-
tively be broadcast to all other nodes. As mentioned in the
preceding section, Cheng et al. (2007) address this using
digests of the SIFT features to establish overlap topology,
then share the substantially larger full feature data pair-
wise only among cameras with sufficient overlap for cal-
ibration. Kurillo et al. (2008) also use calibration feature
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points to estimate overlap; scalability is less of an issue
because they use a structured calibration target, which yields
a set of features both sparse enough to distribute among
many cameras and robust enough to achieve accurate met-
ric calibration. In the algorithm of Stauffer and Tieu (2003),
overlap topology estimation is part of the closed-loop track-
ing correspondence task itself. The scheduling applications
of Kulkarni et al. (2007) and Mavrinac and Chen (2011)
use occupancy correlation and geometric coverage, respec-
tively, in an attempt to obtain the same fundamental informa-
tion, viz. the degree of content pertinent to the task in each
k-view.
3.4 State of the Art and Open Problems
The vision graph is a well-established concept and the-
oretical tool in the multi-camera network literature. In
the application classes of multi-view registration and cal-
ibration, which (in the surveyed cases) involve pairwise
coverage relationships exclusively, it has proven useful
in its basic form. Additional optimizations are possible
with appropriate use of edge weights and related com-
binatorial techniques, as demonstrated by Sharp et al.
(2004) and Kurillo et al. (2008) for the respective applica-
tions.
When used in direct tracking correspondence, the lim-
itations become apparent. Arbitrary subdivisions of cam-
era nodes into partial coverage cells appears to improve
performance, but this is unsatisfying from a theoretical
standpoint. Lobaton et al. (2009a,2010) present an explicit
departure from the graph model, allowing them to represent
2-coverage and 3-coverage in a simplicial complex; however,
presumably since their application does not require it, gen-
eral k-coverage modeling is absent. Kulkarni et al. (2007)
and Mavrinac and Chen (2011) use more general hyper-
graph (or equivalent) models explicitly designed for general
k-coverage, suitable for scheduling in distributed camera net-
works, but ignore the coverage subdivisions needed by track-
ing applications.
We believe that the generalized hypergraph model
presented in Sect. 3.1 includes all of the information
necessary to fit the needs of each of the applications cov-
ered here, and being a relatively straightforward combi-
nation of existing concepts from the literature, should be
backwards-compatible with all of the reviewed sources. In
the absence of task-specific geometric coverage informa-
tion, it is sensible to use the task data itself to approx-
imate the model. It remains an open question whether
the nature of the information contained in edge weights,
and the additional combinatorial optimizations they make
possible, can be incorporated into such a unified frame-
work.
4 Transition Topology
4.1 Anatomy of a Transition Model
Atransition model describes the topology of a camera net-
work in terms of the probability and/or timing of moving
agents transitioning from one region of coverage to another.
While an overlap model of the sort covered in Sect. 3cap-
tures a physical topology, a transition model captures a more
abstract functional topology of agent activity. Relationships
may exist among camera nodes with no mutual scene cover-
age (non-overlapping cameras). Since the target application
class is agent tracking, the granularity of the topology may
extend down to subsets of camera nodes’ coverage: entry
and exit points and regions of overlap are often considered
individually (Fig. 9).
In the most general form, such a model is a weighted
directed graph G=(C,A,w), where Cis a set of cover-
age cells,Ais a set of arcs, and w:A→R+is a weight
function over A. A coverage cell may represent an individual
camera node’s coverage model or some portion thereof, such
as an entry or exit zone (note that a coverage cell may be
both an entry zone and an exit zone). Cmay also include a
special source/sink node to collectively represent the uncov-
ered portion(s) of the scene. The existence of an arc a∈A
indicates that agents may transition from the tail region to
the head region. In a weighted model, w(a)is a quantitative
metric encapsulating the probability and/or duration of the
transition.
4.2 Transition Models by Application
Transition models are largely aimed at one particular applica-
tion class: predictive tracking in (generally) non-overlapping
camera networks. For a locally tracked agent leaving one cov-
erage cell, the objective is to predict in which other coverage
cell(s) the agent will reappear, possibly to inform camera
handoff. A special case occurs when the cameras have cover-
age overlap, which is addressed by several models of overlap
topology as the direct tracking correspondence problem (see
Sect. 3.2). Javed et al. (2003) show that, in the context of
non-overlapping tracking correspondence, transition prob-
abilities and durations are dependent on individual correla-
tions of entry and exit zones, of which each camera may have
a number. Their geometric counterparts are coverage cells,
and in a combinatorial transition model, they comprise the
vertex set. Various techniques have been applied to this type
of model to aid in tracking agents across non-overlapping
views (i.e., through unobserved regions).
The model presented by Ellis et al. (2003) exemplifies this
approach. Their method automatically identifies entry and
exit zones in each camera ((a problem previously addressed
by Stauffer 2003), then finds the transition topology by
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Fig. 9 Transition graph—from
2D coverage geometry (left)to
transition topology (right). Dark
ellipses denote entry and exit
zones. Intra-camera transition
arcs are shown in gray. The S/S
vertices represent external agent
source/sink
temporally correlating a large number of local trajectories
between cameras, requiring no actual tracking correspon-
dence. Makris et al. (2004) extend this method and further
develop its theoretical basis. Stauffer (2005) operates on a
closely related model, but presuming the availability of a cov-
erage overlap model—Stauffer cites his own previous work
with Stauffer and Tieu (2003)—treats cliques of overlapping
cameras (connected components in the vision graph) as the
larger coverage structure containing entry and exit zones,
on the premise that the overlapping case is better handled
by robust direct correspondences. The aforementioned meth-
ods ascribe to observations an implicit correspondence, and
assume a unimodal statistical distribution of transitions. Tieu
et al. (2005) address this with a method capable of handling
multimodal distributions.
Marinakis et al. (2005); Marinakis and Dudek (2005) con-
sider cameras with full coverage of widely-separated sections
of hallways in a building, so that transitions are constrained
to the hallway topology. Due to these constraints, the entry
and exit zone coverage cells (transition graph vertices) are
the cameras themselves, and the cameras need only be capa-
ble of detecting an agent’s presence with reasonable fidelity
for their method to successfully estimate the topology. Niu
and Grimson (2006) target a vehicle tracking application,
using appearance to match observations between, and infer
the topology of, non-overlapping cameras.
Dick and Brooks (2004) approach the predictive
tracking problem with a Markov model which captures
transition topology after a training phase, albeit not in
an explicitly combinatorial form, dividing the view into
blocks over which the topology is found. The method of
Gilbert and Bowden (2006) incrementally learns the topol-
ogy between recursively subdivided blocks of the views; their
method does not require a training phase and can adapt to
changes in the camera network. Both yield a probabilistic
topological model which can be used in conjunction with
appearance-based matching to track across disjoint views.
Zou et al. (2007) are interested in tracking humans, and
integrate appearance-based agent correspondence based on
face recognition into the inference method of Ellis, Makris,
and Black, for improved robustness in their target instance.
Nam et al. (2007) also specifically track humans, and an
appropriate appearance model is integral to their estimation
method. The method of Farrell and Davis (2008) falls within
this category as well, and is notable for its expressly distrib-
uted approach, which affords scalability to large, distributed
surveillance networks.
Finally, it should be noted that the coverage overlap model
developed by Van Den Hengel et al. (2006,2007), Detmold
et al. (2007,2008), Hill et al. (2008) can be extended, as the
authors explain, to capture non-overlapping transition topol-
ogy by adding a temporal padding window to the exclusion
method.
4.3 Analysis and Comparison of Transition Models
Table 3compares the properties of a selection of topological
transition models from the literature. Intepreting each model
as a graph, the first and second columns indicate whether
the graph is directed and weighted, respectively. The third
column indicates whether vertices of the graph represent
individual entry/exit points, of which each camera may have
several; the implication otherwise is that the granularity is
at the level of cameras only. The fourth column indicates
whether the model includes an explicit source/sink vertex,
for agents entering or leaving the scene. The fifth column
indicates whether the graph models transitions between over-
lapping cameras, thus implicitly modeling coverage overlap
to the extent described with direct tracking correspondence
applications in Sect. 3. The final two columns specify which
type of data is used in constructing the model: statistical
correlation between temporal events, or correlation via an
appearance model.
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Tabl e 3 Comparison of selected topological transition models
Model Properties Construction data
Directed Weight Entry/exit Source/sink Overlap Temporal Appearance
Ellis et al. (2003) •
Makris et al. (2004)    •
Dick and Brooks (2004)   •
Marinakis et al. (2005)•
Stauffer (2005)  •
Tieu et al. (2005)  •
Niu and Grimson (2006)•
Nam et al. (2007)   •
Zou et al. (2007)   ••
Farrell and Davis (2008)•
4.3.1 Combinatorial Structure
Relatively few of the transition models surveyed are explic-
itly presented as graphs resembling the generalized model
described in Sect. 4.1.Marinakis et al. (2005); Marinakis and
Dudek (2005) model the topology in a directed, unweighted
graph, in which vertices represent camera nodes and arcs
represent possible transitions. Transition probabilities and
durations are captured separately in an agent model. The
graph of Nam et al. (2007) also has a vertex for each camera
node, but also has intermediate vertices representing either
an overlapping or non-overlapping transition point and a
source/sink vertex; since individual entry and exit zones are
not represented, the graph is undirected. Zou et al. (2007)
use essentially the same model as Ellis et al. (2003); Makris
et al. (2004), but treat it explicitly as a weighted, directed
graph, with vertices representing entry and exit zones and
arcs indicating possible transitions. Trivially, related mod-
els, such as those of Stauffer (2005) and Tieu et al. (2005),
could be treated similarly. The transition matrix of Dick and
Brooks (2004) can be interpreted as an incidence matrix
for the transition graph. In general, it is not difficult to
apply a graph interpretation to any of the models surveyed
here.
As discussed in Sect. 3.3.1, coverage overlap models typ-
ically represent each camera node as a vertex, a structure
which offers useful combinatorial properties in most appli-
cations. Some transition models employ this structure as
well. Marinakis et al. (2005); Marinakis and Dudek (2005)
assume widely separated cameras and wish to avoid deal-
ing with complex local tracking, so this is the sensible
representation for their case. Niu and Grimson (2006) and
Farrell and Davis (2008) also consider transitions only
between strictly non-overlapping cameras. In scenes of even
moderate complexity, however, a transition topology among
individual entry and exit points is more germane to predictive
tracking. This structure is described by Ellis et al. (2003);
Makris et al. (2004) and used by a plurality of the models
surveyed (Stauffer 2005;Tieu et al. 2005;Zou et al. 2007;
Nam et al. 2007). Dick and Brooks (2004) do not automati-
cally determine entry and exit points, but do divide the cam-
eras into coverage cells, which would induce the vertices in
a graph interpretation of their model.
Makris et al. (2004) include a source/sink vertex (which
they call a “virtual node”), in addition to the entry and exit
zone vertices, to handle the probabilistic paths of agents
entering or leaving the overall coverage of the camera net-
work. Marinakis and Dudek (2005) and Nam et al. (2007)
also include such a vertex in their models.
Among explicit graph models with arc weights, the defi-
nition of the weighting function varies. Makris et al. (2004)
annotate arcs in the graphical representation of their model
according to the probability of transition, computed from the
cross-correlation of the temporal sequences of departure and
arrival events at each entry and exit zone (vertex), but do not
operate on it as a weighted graph. Zou et al. (2007) explic-
itly apply this weighting to the graph representation, with
a modified correlation function based on both identity and
appearance (as opposed to identity only). In contrast, Nam
et al. (2007) weight arcs based on the mean duration of tran-
sitions between cameras.
4.3.2 Construction from Visual Data
It is normally assumed that the camera network is uncal-
ibrated and that information about the scene and agent
dynamics is unavailable a priori. For the purposes of this dis-
cussion, we will approach the construction methods assum-
ing that entry and exit zones are known, either estimated sep-
arately (Stauffer 2003;Ellis et al. 2003;Gilbert and Bowden
2006) or specified a priori, as in the case where each camera
is a single entry/exit zone. If agents can be uniquely identified
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and reliably matched between all generally non-overlapping
views, and a sequence of their arrival and departure events
is obtained over a period of time, distributions of the proba-
bilities and durations of transitions can be established. From
this information, all of the parameters of the general transi-
tion model can be obtained.
Unfortunately, correspondence of agents of arbitrary
appearance between generally disjoint views is notoriously
difficult. Ellis et al. (2003); Makris et al. (2004) sidestep
this challenge with a method of construction based on pure
temporal correlation of otherwise unmatched observations.
Essentially, they assume implicit correspondence between
all pairs of arrival and departure events, and seek a sin-
gle mode of temporal correlation between each pair of
entry and exit zones within a time window (positive and
negative); every peak above a certain threshold induces
an arc in the transition graph between the associated ver-
tices. Stauffer (2005) employs a similar method, but con-
siders transitions between overlapping cameras separately
(see Sect. 4.3.3), so the transition time window is positive
only. Tieu et al. (2005) handle more general statistical depen-
dences, capturing richer multi-modal transition distributions
rather than simply a mean transition duration, and thus per-
mitting topology estimation from more complex agent behav-
ior. Marinakis et al. (2005); Marinakis and Dudek (2005)also
avoid direct correspondence. They assume that the dynam-
ics of an agent is a Markov process, and estimate the para-
meters of this process—the probabilities and durations of
transitions—using a Monte Carlo Expectation Maximization
method.
Methods which do rely on appearance-based agent cor-
respondence normally have a narrower application focus.
Dick and Brooks (2004) require a training phase for their
Markov model which relies on colour-based correspondence.
Niu and Grimson (2006) rely on correspondence of tracked
vehicles using an appearance model based on colour and
size. The estimation method of Nam et al. (2007) centers
around correspondence based on background subtraction and
a human appearance model. Farrell and Davis (2008)employ
an information-theoretic appearance matching process, and
infer the expected transition model from the accumulated
evidence using a modified multinomial distribution. Their
method is also notable for its distributed design: its “semi-
localized” processing yields a scalable algorithm for which
the authors demonstrate successful results in networks up to
100 nodes.
Zou et al. (2007) integrate correspondence based on face
recognition into the previously described statistical method
of Ellis, Makris, and Black, resulting in a hybrid approach
which they claim outperforms methods based purely on either
identity or appearance.
(a) (b)
Fig. 10 Possible cases of transition—dark ellipses denote entry and
exit zones, and the dotted line indicates the agent path
4.3.3 Transitions Between Overlapping Cameras
There is a question as to how transitions between cameras
with overlapping coverage should be handled in transition
models. Referring to the example agent paths in Fig. 10,itis
clear how to handle the transition between non-overlapping
cameras shown in Fig. 10a, as the surveyed methods unan-
imously agree: an arc from Aor its exit zone to Bor its
entry zone, with a positive transit duration. However, in the
transition between overlapping cameras shown in Fig. 10b,
the agent passes through the entry zone of B before passing
through the exit zone of A, and the agent is observed by one or
both cameras during the entire transition. Transitions from
one entry or exit zone to another within a single camera’s
coverage can be thought of as a special case of this scenario.
Ellis et al. (2003); Makris et al. (2004) deal with the over-
lapping case as with the non-overlapping case. For a given
departure event at time t1, they check for arrival events at time
t2∈[t1−T,t1+T], where Tis a temporal search window.
Thus, in Fig. 10a, t2>t1, whereas in Fig. 10b, t2<t1.The
advantage of this approach is that it does not require prior
estimation of overlap topology, and uses a single process to
estimate transition topology for a general-case camera net-
work with overlapping and/or non-overlapping cameras.
Stauffer (2005) argues that the overlapping case is best
handled by more robust direct tracking correspondence, and
proposes first estimating overlap topology (Stauffer and Tieu
2003), then treating connected components in the vision
graph as single “cameras”—in general, with multiple entry
and exit zone vertices—in the transition model. The advan-
tage of this approach is improved robustness in estimating
the overlapping portions of the transition topology, assum-
ing a reliable means of finding inter-camera correspondences
of agents and/or their tracks is available.
4.4 State of the Art and Open Problems
Numerous researchers have converged on the structure
described in Sect. 4.1, to varying degrees. As with cover-
age overlap models, it is safe to say that this generalized
model subsumes all existing cases; individual models have
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left out certain properties (arc directivity and weights, node
subdivision, source/sink node) either because they are unnec-
essary for the particular application case or else to facilitate
optimization. Given the clear focus on a single application
class, future optimization efforts should adopt such a unified
model, if possible, for the sake of general applicability.
Approximation of the graph from visual data is split
between statistical temporal correlation and appearance-
based correlation. Given the complementary strengths of
both methods, the way forward seems to be a hybrid approach
in the vein of Zou et al. (2007). If agent dynamics are being
modeled for the purposes of probabilistic occlusion, as by
Mittal and Davis (2008), this may also be informative for
transition model approximation.
One point of contention, to which the answer is not yet
clear, is whether the graph should model transitions strictly
between non-overlapping coverage cells, with overlapping
transitions handled separately as proposed by
Stauffer (2005), or all transitions. If the relative reliability of
the approximations for overlapping transitions is the issue,
implementation of the aforementioned hybrid approximation
approach may favor the latter unified model.
5 Conclusions and Future Directions
We have endeavoured to present a comprehensive and lucid
exposition of the theory, state of the art, and challenges of
modeling the visual coverage of camera networks. The mod-
els and estimation methods discussed in this survey represent
the efforts of researchers to develop theory to describe the
particular and unique properties of this relatively new type of
system, drawing on concepts from computer vision, sensor
networks, and other related fields.
Through an analysis of their properties in the context
of specific applications, a generalized prototype model of
each type has been derived, of which the major structure of
the models in the surveyed works can be cast as particular
instances. In addition to providing a clear overview of the
three types of models individually, it is relevant at this point
to explicitly expose the relationships between them in terms
of the information they encapsulate.
To date, camera network coverage has not been described
from the vantage of an inclusive understanding of its various
applications. Researchers have developed tools to achieve
particular objectives, adapting work from similar but not
quite identical problems elsewhere in the landscape. Over
time, this has begun to converge and evolve into a theoreti-
cal framework particular to camera networks. It is our belief
that the unique challenges involved warrant a next phase in
modeling camera network coverage, viz. the development of
a comprehensive, rigorous, and mature theory encompassing
geometric coverage as well as both notions of topology. Our
generalized models and exposition of their information hier-
archy are a first step in this direction, but a truly useful theory
will be forged in the fire of application, and many cues can be
taken from the design decisions made in the various works
surveyed here. A general, analytic understanding of coverage
will reduce duplicated effort and open up new possibilities in
solving a large cross-section of important problems in camera
networks.
Acknowledgments This research was supported in part by the Natural
Sciences and Engineering Research Council of Canada.
References
Angella, F., Reithler, L., & Gallesio, F. (2007). Optimal deployment
of cameras for video surveillance systems. In Proceedings of IEEE
conference on advanced video and signal based surveillance (pp.
388–392).
Antone, M., & Teller, S. (2002). Scalable extrinsic calibration of
omni-directional image networks. International Journal of Com-
puter Vision,49(2/3), 143–174.
Bajramovic, F., Brückner, M., & Denzler, J. (2009). Using common
field of view detection for multi camera calibration. In Proceedings
of vision, modeling, and visualization workshop.
Bodor, R., Drenner, A., Janssen, M., Schrater, P., & Papanikolopoulos,
N. (2005). Mobile camera positioning to optimize the observability
of human activity recognition tasks. In Proceedings of IEEE/RSJ
international conference on intelligent robots (pp. 4037–4042).
Bodor, R., Drenner, A., Schrater, P., & Papanikolopoulos, N. (2007).
Optimal camera placement for automated surveillance tasks. Journal
of Intelligent and Robotic Systems,50(3), 257–295.
Brand, M., Antone, M., & Teller, S. (2004). Spectral soluction of large-
scale extrinsic camera calibration as a graph embedding problem.
In Proceedings of 8th European conference on computer vision (pp.
262–273).
Brückner, M., Bajramovic, F., & Denzler, J. (2009). Geometric and
probabilistic image dissimilarity measures for common field of view
detection. In Proceedings of IEEE computer society conference on
computer vision and, pattern recognition (pp. 2052–2057).
Cerfontaine, P. A., Schirski, M., Bundgens, D., & Kuhlen, T. (2006).
Automatic multi-camera setup optimization for optical tracking. In
Proceedings of virtual reality conference (pp. 295–296).
Chen, S., & Li, Y. (2004). Automatic sensor placement for model-based
robot vision. IEEE Transactions on Systems, Man, and Cybernetics,
34(1), 393–408.
Chen, T. S., Tsai, H. W., Chen, C. P., & Peng, J. J. (2010). Object cover-
age with camera rotation in visual sensor networks. In Proceedings
of 6th international wireless communications and mobile computing
conference (pp. 79–83).
Chen, X., & Davis, J. (2008). An occlusion metric for selecting robust
camera configurations. Machine Vision and Applications,19(4),
217–222.
Cheng, Z., Devarajan, D., & Radke, R. J. (2007). Determining
vision graphs for distributed camera networks using feature digests.
EURASIP Journal on Advances in Signal Processing,2007, 1–11.
Chow, K. Y., Lui, K. S., & Lam, E. Y. (2007). Achieving 360 angle
coverage with minimum transmission cost in visual sensor networks.
In Proceedings of IEEE wireless communications and networking
conference (pp. 4112–4116).
Cowan, C. K., & Kovesi, P. D. (1988). Automatic sensor placement from
vision task requirements. IEEE Transactions on Pattern Analysis and
Machine Intelligence,10(3), 407–416.
123

Int J Comput Vis
Dai, R., & Akyildiz, I. F. (2009). A spatial correlation model for visual
information in wireless multimedia sensor networks. IEEE Transac-
tions on Multimedia,11(6), 1148–1159.
Detmold, H., Dick, A. R., Van Den Hengel, A., Cichowski, A., Hill, R.,
Kocadag, E., Falkner, K., & Munro, D. S. (2007). Topology estima-
tion for thousand-camera surveillance networks. In Proceedings of
1st ACM/IEEE international conference on distributed smart cam-
eras (pp. 195–202).
Detmold, H., Dick, A. R., Van Den Hengel, A., Cichowski, A., Hill,
R., Kocadag, E., Yarom, Y., Falkner, K., & Munro, D. S. (2008).
Estimating camera overlap in large and growing networks. In Pro-
ceedings of 2nd ACM/IEEE international conference on distributed
smart cameras.
Devarajan, D., & Radke, R. J. (2004). Distributed metric calibration of
large camera networks. In Proceedings of 1st workshop on broad-
band advanced sensor networks.
Dick, A. R., & Brooks, M. J. (2004). A stochastic approach to track-
ing objects across multiple cameras. In Proceedings Australian joint
conference on artificial intelligence (pp. 160–170).
Ellis, T. J., Makris, D., & Black, J. K. (2003). Learning a multi-camera
topology. In Proceedings of joint IEEE workshop on visual surveil-
lance and performance evaluation of tracking and surveillance (pp.
165–171).
Erdem, U. M., & Sclaroff, S. (2003). Automated placement of cam-
eras in a floorplan to satisfy task-specific constraints. Tech. Report,
Boston University.
Erdem, U. M., & Sclaroff, S. (2006). Automated camera layout to satisfy
task-specific and floor plan-specific coverage requirements. Com-
puter Vision and Image Understanding,103(3), 156–169.
Farrell, R., & Davis, L. S. (2008). Decentralized discovery of camera
network topology. In Proceedings of 2nd ACM/IEEE international
conference on distributed smart cameras.
Faugeras, O. (1993). Dimensional computer vision: A geometric view-
point. London: MIT Press.
Fiore, L., Somasundaram, G., Drenner, A., & Papanikolopoulos, N.
(2008). Optimal camera placement with adaptation to dynamic
scenes. In Proceedings of IEEE international conference on robotics
and automation (pp. 956–961).
Gilbert, A., & Bowden, R. (2006). Tracking objects across cameras by
incrementally learning inter-camera colour calibration and patterns
of activity. In Proceedings of 9th European conference on computer
vision (pp. 125–136).
González-Banos, H., & Latombe, J. C. (2001). A randomized art-gallery
algorithm for sensor placement. In Proceedings of 17th annual sym-
posium computational geometry (pp. 232–240).
Hill, R., Dick, A. R., Van Den Hengel, A., Cichowski, A., & Detmold,
H. (2008). Empirical evaluation of the exclusion approach to estimat-
ing camera overlap. In Proceedings of 2nd ACM/IEEE international
conference on distributed smart cameras.
Hörster, E., & Lienhart, R. (2006). On the optimal placement of multiple
visual sensors. In Proceedings of 4th ACM international workshop
on video surveillance and sensor, networks (pp. 111–120).
Hörster, E., & Lienhart, R. (2009). Optimal placement of multiple visual
sensors. In H. Aghajan & A. Cavallaro (Eds.), Multi-camera net-
works: Principles and applications (Chap. 5, pp. 117–138). Burling-
ton: Academic Press.
Huang, C. F., Tseng, Y. C., & Lo, L. C. (2007). The Coverage Problem
in Three-Dimensional Wireless Sensor Networks. J. Interconnection
Networks,8(3), 209–227.
Huber, D. F. (2001). Automatic 3D modeling using range images
obtained from unknown viewpoints. In Proceedings of 3rd inter-
national conference on 3D digital imaging and modeling (Vol. 7, pp.
153–160).
Javed, O., Khan, S., Rasheed, Z., & Shah, M. (2000). Camera handoff:
Tracking in multiple uncalibrated stationary cameras. In Proceedings
of workshop on human motion (pp. 113–118).
Javed, O., Rasheed, Z., Shafique, K., & Shah, M. (2003). Tracking
across multiple cameras with disjoint views. In Proceedings of 9th
IEEE international conference on computer vision (pp. 952–957).
Jiang, Y., Yang, J., Chen, W., & Wang, W. (2010). A coverage enhance-
ment method of directional sensor network based on genetic algo-
rithm for occlusion-free surveillance. In Proceedings of interna-
tional conference on computational aspects of social networks
(pp. 311–314).
Kang, E. Y., Cohen, I., & Medioni, G. G. (2000). A graph-based global
registration for 2D mosaics integrated media systems center. In Pro-
ceedings of international conference on pattern recognition (pp.
257–260).
Khan, S., & Shah, M. (2003). Consistent labeling of tracked objects
in multiple cameras with overlapping fields of view. IEEE Trans-
actions on Pattern Analysis and Machine Intelligence,25(10),
1355–1360.
Kulkarni, P., Shenoy, P., & Ganesan, D. (2007). Approximate initial-
ization of camera sensor networks. In