DTIC ADA538850: Algebraic Approach for Recovering Topology in Distributed Camera Networks

0.71 MB·English

by  Defense Technical Information Center

Algebraic Approach for Recovering Topology in
Distributed Camera Networks
Edgar J. Lobaton
Parvez Ahammad
S. Shankar Sastry
Electrical Engineering and Computer Sciences
University of California at Berkeley
Technical Report No. UCB/EECS-2009-4
http://www.eecs.berkeley.edu/Pubs/TechRpts/2009/EECS-2009-4.html
January 14, 2009
Report Documentation Page Form Approved
OMB No. 0704-0188
Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and
maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information,
including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington
VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it

does not display a currently valid OMB control number.
1. REPORT DATE 3. DATES COVERED
14 JAN 2009 2. REPORT TYPE 00-00-2009 to 00-00-2009
4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
Algebraic Approach for Recovering Topology in Distributed Camera
5b. GRANT NUMBER
Networks
5c. PROGRAM ELEMENT NUMBER
6. AUTHOR(S) 5d. PROJECT NUMBER
5e. TASK NUMBER
5f. WORK UNIT NUMBER
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION
University of California at Berkeley,Department of Electrical REPORT NUMBER
Engineering and Computer Sciences,Berkeley,CA,94720
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S)
11. SPONSOR/MONITOR’S REPORT
NUMBER(S)
12. DISTRIBUTION/AVAILABILITY STATEMENT
Approved for public release; distribution unlimited
13. SUPPLEMENTARY NOTES
14. ABSTRACT
Camera networks are widely used for tasks such as surveillance, monitoring and tracking. In or- der to
accomplish these tasks, knowledge of lo- calization information such as camera locations and other
geometric constraints about the envi- ronment (e.g. walls, rooms, and building layout) are typically
considered to be essential. How- ever, this information is not always required for many tasks such as
estimating the topology of camera network coverage, or coordinate-free ob- ject tracking and navigation.
In this paper we propose a simplicial representation (called CN-Complex) that can be constructed from
dis- crete local observations from cameras, and uti- lize this novel representation to recover the topo-
logical information of the network coverage. We prove that our representation captures the cor- rect
topological information from network cov- erage for 2.5D layouts, and demonstrate their utility in
simulations as well as a real-world experimental set-up. Our proposed approach is particularly useful in
the context of ad-hoc camera networks in indoor/outdoor urban envi- ronments with distributed but
limited computa- tional power and energy.
15. SUBJECT TERMS
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF
ABSTRACT OF PAGES RESPONSIBLE PERSON
a. REPORT b. ABSTRACT c. THIS PAGE Same as 20
unclassified unclassified unclassified Report (SAR)
Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
Copyright 2009, by the author(s).
All rights reserved.

Permission to make digital or hard copies of all or part of this work for
personal or classroom use is granted without fee provided that copies are
not made or distributed for profit or commercial advantage and that copies
bear this notice and the full citation on the first page. To copy otherwise, to
republish, to post on servers or to redistribute to lists, requires prior specific
permission.

Acknowledgement

This work was funded by the Army Research Office (ARO) Multidisciplinary
Research Initiative (MURI) program under the title "Heterogeneous Sensor
Webs for Automated Target Recognition and Tracking in Urban Terrain"
(W911NF-06-1-0076), and the Air Force Office of Scientific Research
(AFOSR) grant FA9550-06-1-0267, under a sub-award from Vanderbilt
University.
Algebraic Approach for Recovering Topology in Distributed
Camera Networks
Edgar J. Lobaton, Parvez Ahammad, S. Shankar Sastry ∗†
January 14, 2009
Abstract
is particularly useful in the context of ad-hoc
camera networks in indoor/outdoor urban envi-
Camera networks are widely used for tasks such ronments with distributed but limited computa-
as surveillance, monitoring and tracking. In or- tional power and energy.
der to accomplish these tasks, knowledge of lo-
calization information such as camera locations
1 Introduction
and other geometric constraints about the envi-
ronment (e.g. walls, rooms, and building layout)
are typically considered to be essential. How- Future generations of sensor networks are in-
ever, this information is not always required for variably going to include multiple types of sen-
many tasks such as estimating the topology of sors - including spatial sampling sensors such
camera network coverage, or coordinate-free ob- as cameras or active range scanners. Sensors
ject tracking and navigation. In this paper, like cameras will be the dominant consumers
we propose a simplicial representation (called of bandwidth and power in such heterogenous
CN-Complex) that can be constructed from dis- sensor networks. Thus, a clear understanding
crete local observations from cameras, and uti- of the constraints (such as bandwidth consump-
lizethisnovelrepresentationtorecoverthetopo- tion, power consumption, spatio-temporal sam-
logical information of the network coverage. We pling) posed by camera sensors in the context
prove that our representation captures the cor- of computation and communication will play a
rect topological information from network cov- critical role in defining the bounds for feasibil-
erage for 2.5D layouts, and demonstrate their ity of performing certain tasks in a heterogenous
utility in simulations as well as a real-world sensor network. In other words, such an under-
experimental set-up. Our proposed approach standing in the context of cameras could tell us
whether our design of the heterogenous network
∗E.J. Lobaton and S.S. Sastry are with the Electrical will be able to perform the designated task or
EngineeringandComputerSciencesDepartment,Univer- not - and what conditions are necessary in order
sity of California at Berkeley, Berkeley, CA 94720, USA
to perform such tasks.
†P.AhammadiswiththeJaneliaFarmResearchCam-
Identification of the exact location of targets
pus, Howard Hughes Medical Institute, Ashburn, VA
20147, USA and objects in an environment is essential for
1
many surveillance applications in the realm of that all of these questions can be approached us-
sensor networks. However, there are situations ing knowledge of the topology of the coverage of
in which the localization of the sensors is not the network. In particular, topology awareness
known (e.g. unavailability of GPS, or ad-hoc makesitpossibletodesignmoreefficientrouting
network setup). A common approach to over- andbroadcastingschemesasitisdiscussedbyM.
coming this challenge has been to determine the Li et al [9]. This knowledge in turn can also aid
exact localization of the sensors and reconstruc- withcontrolmechanismformoreenergy-efficient
tion of the surrounding environment. Neverthe- usage.
less, we will provide evidence supporting the hy- Figure 1 serves as a didactic tool to under-
pothesisthatmanyofthetasksathandmaynot stand the information required for our approach
require exact localization information. For in- to coordinate free tracking and navigation prob-
stance, when tracking individuals in an airport, lems. Observe that the complete floor plan
we may want to know whether they are in the (left) and corresponding abstract representation
vicinity of a specific gate. In this scenario, it is (right) serves an equivalent purpose. The ab-
not absolutely necessary to know their exact lo- stract representation allows us to track a target
cation. Another example is navigation through andnavigatethroughtheenvironment. Ourgoal
an urban environment. This task can be accom- in this context is to use the continuous observa-
plished by making use of target localization and tionsfromcameranodestoextractthenecessary
a set of directions such as where to turn right, symbols to create this representation.
and when to keep going straight. In both situa-
tions, a general description of our surroundings
and the target location is sufficient. The type
of information that we desire is a topological de-
scription of the environment that captures the
appropriate structure of the environment.
One of the fundamental questions in the con-
text of camera networks is whether a network is
limited to perform only tasks that a single cam- Figure 1: A physical (left) and an abstract
era can perform but at a larger scale, or if the (right) layout of an environment are compared.
total network is “greater” than the sum of the In both cases we observe a target and the corre-
parts. Imagineacameranetworkwherenointer- sponding path for its motion.
relationship between the cameras is known. It
is natural to ask what the spatial relationship In this paper, we consider a camera network
between cameras is. For an application such as where each camera node can perform local com-
surveillanceinwhichmultipleviewsarecertainly putations,extractsomesymbolic/discreteobser-
useful, we investigate how object tracking infor- vations to be transmitted for further processing.
mationfrommultiplecamerascanbeaggregated This conversion to symbolic representation alle-
and analyzed. A related and important ques- viates the communication overhead for a wire-
tion here is as to how we manage the processing less network. We then use these discrete ob-
and flow of data between the cameras. We note servations to build a model of the environment
2
withoutanypriorlocalizationinformationofob- ence encountered in the context of camera net-
jects or the cameras themselves. Once such non- works is not handled in traditional sensor net-
metric reconstruction of the camera network is work literature.
accomplished,thisrepresentationcanbeusedfor
Connectivity between overlapping camera
tasks such as coordinate-free navigation, target-
views by determining the correspondence mod-
tracking, and path identification.
els between cameras and extracting homogra-
The rest of the discussion is as follows. We
phy models has been approached by Stauffer
first start with a brief and informal discussion
and Tieu [15]. Cheng et al [5] build a vi-
aboutdifferentapproachestocapturingtopolog-
sion graph in a distributed manner by exchang-
ical information in sensor networks and discuss
ing feature descriptors from each camera view.
the related work in this domain. We then in-
In their work, each camera encodes a spa-
troduce the algebraic topological tools used for
tially well-distributed set of distinctive, approx-
constructing our model. Next, we discuss how
imately viewpoint-invariant feature points into
the topological recovery (or non-metric recon-
a fixed-length “feature digest” that is broadcast
struction) of the camera network can be done
throughout the network to establish correspon-
in 2.5D along with simulation and experiments.
dence between cameras. Yeo et al [16] utilize a
Appendix A provides a brief introduction to the
distributedsourcecodingframeworktoexchange
algebraic topological tools and terminology used
compact feature descriptors in a rate-efficient
for this work.
mannertoestablishcorrespondencebetweenvar-
ious camera views.
2 Related Work Marinakis et al [11] work on finding connec-
tivity between non-overlapping coverage of cam-
Finding the topology of a domain embedded in eras by using only reports of detection and no
R2 is closely related to detecting holes. There description of the target. They use a Markov
has been much work on the detection and re- model for modeling the transition probabilities
covery of holes by topological methods for sen- and minimize a functional using Markov Chain
sor networks, most of which consider symmetric Monte Carlo Sampling. They also present a
coverage(explicitlyorimplicitly)orhighenough different formulation of the same problem with
density of sensors in the field. In particular, Vin “timestamp free” observation with only order-
de Silva and Ghrist [6] obtain the Rips complex ing available (still no target description) [12].
based on the communication graph of the net- Other approaches to solving the same problem
work and compute homologies using this repre- with target identification have been explored by
sentation. These methods assume some symme- Zou et al [17]. Camera network with overlaps
try in the coverage of each sensor node (such have been studied using the statistical consis-
ascircularcoverage), however, suchassumptions tencyoftheobservationdatabyMakrisetal[10].
are not valid for camera networks. Spatial sam- Rahimi et al [14] describe a simultaneous cali-
pling of plenoptic function [2] from a network of bration and tracking algorithm (with a networks
cameras is rarely i.i.d. (independent and identi- of non-overlapping sensors) by using velocity ex-
cally distributed). The notion of spatial coher- trapolation for a single target.
3
3 The Environment Model
The Target in 2.5D : A target will have the
following properties:
3.1 The Problem in 2.5D
• The target will be a line segment perpen-
Our problem will be defined in terms of the de-
dicular to the bounding planes of our do-
tection of a target moving through an environ-
main which connects the points (x,y,0) to
ment. For the sake of mathematical clarity, we
(x,y,h ), where x and y are arbitrary and
t
first focus on the case of a single target mov-
h ≤ h is the height of the target. The
t max
ing through the environment. Let us start by
target is free to move along the domain as
describing our setup:
long as it does not intersect any of the ob-
The Environment in 2.5D : We consider a jects in the environment.
domain in 3D with the following constraints:
• A target is said to be detected by camera
• All objects and cameras in the environ- α if there exists a point p := (x,y,z) in
mentwillbewithinthespacedefinedbythe the target such that p ∈ F and o3Dp does
α α
planesz = 0(the“floor”)andz = hmax(the not interscect any of the objects in the en-
“ceiling”). vironment.
• Objectsintheenvironmentconsistsofstatic
Note that these assumptions may seem very
“walls” erected perpendicular to our plane
restrictive,buttheyaresatisfiedbymostcamera
from z = 0 to z = h . The perpendicular
max networks in indoor and outdoor environments.
projection of the objects to the plane z = 0
Also, some of these choices in our model (such
must have a piecewise linear boundary. Ob-
as the vertical line target and polygonal objects)
jects must enclose a non-zero volume.
are made in order to simplify our analysis. We
Cameras in 2.5D : A camera α has the fol- will see that our methods work in real-life sce-
lowing properties: narios through our experiments.
The example in figure 2 shows a target and a
• It is located at position o3D with an arbi-
α
camera with its corresponding FOV.
trary 3D orientation and a local coordinate
frame Ψ3D.
α Problem 1 (2.5D Case): Given the camera
• Its camera projection in 3D , Π3D : and environment models in 2.5D , our goal is to
α
F → R2, is given by obtain a representation that captures the topolog-
α
ical structure of the detectable set for a cam-
Π3D(p) = (p /p ,p /p ),
α x z y z eranetwork(i.e., theunionofthesetsinwhich
where p is given in coordinate frame Ψ3D, a target is detectable by a camera). The con-
α
and Fα ⊂ ({(x,y,z)|z > 0}), referred to as struction of this representation should not rely
the field of view (FOV) of the camera, is on camera or object localization.
an open convex set such that its closure is
a convex cone based at o3D. The image of The formulation of the problem is very
α
this mapping, i.e. Π3D(F ), will be called generic. We are choosing a simplicial represen-
α α
the image domain Ω3D. tation because we are after a combinatorial rep-
α
4
the space between z ≥ 0 and z ≤ h .
target
Since the latter is an intersection of con-
vex sets, and orthogonal projections pre-
serve convexity, then D is convex. We can
α
also check that D will be open.
α
Figure 2: Mapping from 2.5D to 2D : A cam- • Also, we can give a 2D description of the
era and its field of view (FOV) are shown from coverage of a camera. A point (x,y) is in
multipleperspectives(leftandmiddle). Thecor- the coverage Cα of camera α if the target
responding mapping of this configuration to 2D locatedat(x,y)isdetectablebythecamera.
is shown on the right. For the 2.5D configura-
tion, the planes displayed bound the space that 3.3 The Problem in 2D
can be occupied by the target.
We now proceed by characterizing our problem
after mapping the original configuration from a
resentation that does not contain metric infor- 2.5D space to 2D . The following definitions are
mation. We are also after a distributed solution, presented to formalize our discussion.
i.e. processing information at local nodes.
The Environment: The space under consid-
eration is similar to the one depicted in figure
3.2 Mapping from 2.5D to 2D
1 (left), where cameras are located in the plane,
The structure of the detectable set for a camera andonlysetswithpiecewise-linearboundaries
network becomes clear through an identification areallowed(includingobjectandpaths). Weas-
of our 2.5D problem to a 2D problem. Since sume a finite number of objects in our environ-
the target is constrained to move along the floor ment.
plane, it is possible to map our problem to a 2D
Cameras: A camera object α is specified by:
problem. In particular:
itsposition o intheplane; andanopenconvex
α
• Cameras located at locations (x,y,z) are domain Dα, referred to as the camera domain.
mapped to location (x,y) in the plane.
The camera domain D can be interpreted as
α
the set of points visible from camera α when no
• Objects in our 2.5D domain are mapped to
objects occluding the field of view are present.
objects with piecewise linear boundaries in
The convexity of this set will be essential for
the plane.
some of the proofs. Some examples of camera
• We can also do a simple identification be- domains are shown in figure 4.
tween the FOV of a camera to a domain D
α
of a camera in 2D . A point (x,y) in the Definition 1 The subset of the plane occupied
plane is in D if the target located at that by the i-th object, which is denoted by O , is a
α i
point intersects the FOV F . The set D connected closed subset of the plane with non-
α α
is the orthogonal projection (onto the xy- empty interior and piecewise linear boundary.
plane) of the intersection between F , and The collection {O }No , where N < ∞ is the
α i i=1 o
5
number of objects in the environment, will be re-
ferred to as the objects in the environment.
Definition 2 Given a camera α, a point p ∈ R2
is said to be visible from camera α if p ∈
D and o p∩ No O = ∅, where o p is the
α α (cid:16)Si=1 i(cid:17) α
line between the camera location o and p. The
α
set of visible points is called the coverage C of
α
camera α.
Figure 3: Examples illustrating nerve complexes
We consider the following problem:
obtained using the collection of camera coverage
Problem 2 (2D Case): Given the camera and {C }. One complex captures the correct topo-
α
environment models in 2D , our goal is to ob- logical information (left) but the other does not
tain a simplicial representation that captures the (right).
topological structure of the coverage of the
camera network (i.e., the union of the cov-
appendixA)isnotsatisfied(inparticular,C ∩C
erage of the cameras). The construction of this 1 2
is not contractible). From the physical layout of
representation should not rely on camera or ob-
the cameras and the objects in the environment,
ject localization.
itisclearhowwecandivideC inordertoobtain
1
Observation 1 Note that the camera network contractible intersections. We are after a a de-
coverage has the same homology (i.e. topological compositionofthecoveragethatcanbeachieved
information) as the domain (R2− O ) if these without knowing the exact location of objects in
i
S
twosetsarehomotopic(i.e., wecancontinuously the environment.
deform one into the other).
4.1 The Decomposition Theorem
4 The CN-Complex
Before we proceed let us consider the following
useful definitions:
Our goal is the construction of a simplicial com-
plex that will capture the homology of the union Definition 3 Given the objects {O }No , a
i i=1
of camera coverage Cα. One possible approach piecewise linear path Γ : [0,1] → R2 is said to
S
foraccomplishingthistaskistoobtainthenerve be feasible if Γ([0,1])∩( O ) = ∅.
i
S
complex (see appendix A) using the set of cam-
era coverage {C }. However, this approach will Definition 4 Given camera α with camera do-
α
only work for simple configurations without ob- main Dα and corresponding boundary ∂Dα, a
jects in the domain. An example illustrating our line Lα is a bisecting line for the camera if:
claim is shown in figure 3.
• L goes through the camera location o .
α α
The reason figure 3 (right) does not capture
the topological structure of the coverage is be- • There exists a feasible path Γ : [0,1] → R2
cause the hypothesis of the Cˇech Theorem (see such that for any ǫ > 0 there exists a δ such
6
that 0 < δ < ǫ, Γ(0.5−δ) ∈ C , Γ(0.5+δ) ∈/ individual camera. This construct will capture
n
C , Γ(0.5) ∈ L , and Γ(0.5) ∈/ ∂D . the correct topological structure of the coverage
α α α
of the network.
If we imagine a target traveling through the Figure 5 displays examples of CN-complexes
pathΓ,wenotethatthelastconditioninthedef- obtained after decomposing the coverage of each
inition of a bisecting line identifies when an oc- camerausingtheircorrespondingbisectinglines.
clusion event is detected (i.e., the target tran- The CN-complex captures the correct topolog-
sitions from visible to not visible, or viceversa). ical information, given that we satisfy the as-
However, we will ignore the occlusion events due sumptions made for the model described in sec-
to the target leaving through the boundary of tion 3. The following theorem, which proof can
the camera domain Dα. be found in appendix B, states this fact.
Definition 5 Let {L }NL be a finite collection
α,i i=1
of bisecting lines for camera α. Consider the set
ofadjacentconesintheplane {K }NC bounded
α,j j=1
by these lines, where NC = 2NL, then the de-
composition of C by lines {L } is the col-
α α,i
lection of sets
C := K ∩C .
α,j α,j α
Figure 5: Examples of CN-complexes. On the
Note that the decomposition of C is not a
α left, camera 1 is decomposed into three regions,
partition since the sets C are not necessarily
α,j each of which becomes a different vertex in our
disjoint.
complex. Ontheright, cameras1and2areboth
decomposed into three regions.
Theorem 1 (Decomposition Theorem)
Let {C }N be a collection of camera coverage
α α=1
where each C is connected and N is the number
α
of cameras in the domain. Let {C } be
α,k (α,k)∈AD
Figure 4: Three examples of camera domains the collection of decomposed sets by all possible
D . Note that cameras can be inside or out- bisecting lines, where A is the set of indices in
α D
side these sets. Our camera model spans projec- the decomposition. Then, any finite intersection
tion models from perspective camers to omni- C ,whereAisafinitesetofindices,
T(α′,k′)∈A α′,k′
directional cameras. Examples of decomposi- is contractible.
tions are shown for each set C .
α
Hence, the hypothesis of the Cˇech Theorem
The construction of the camera network is satisfied if we have connected coverage which
complex (CN-complex) is based on the identi- are decomposed by all of their bisecting lines.
ficationofbisectinglinesforthecoverageofeach Thisimpliesthatcomputingthehomologyofthe
7