Distributed target localization via spatial sparsity

Abstract

We propose an approximation framework for distributed target localization in sensor networks. We represent the unknown target positions on a location grid as a sparse vector, whose support encodes the multiple target locations. The location vector is linearly related to multiple sensor measurements through a sensing matrix, which can be locally estimated at each sensor. We show that we can successfully determine multiple target locations by using linear dimensionality-reducing projections of sensor measurements. The overall communication bandwidth requirement per sensor is logarithmic in the number of grid points and linear in the number of targets, ameliorating the communication requirements. Simulations results demonstrate the performance of the proposed framework.

Full text

DISTRIBUTED TARGET LOCALIZATION VIA SPATIAL SPARSITY
Volkan Cevher∗, Marco F. Duarte, and Richard G. Baraniuk
Department of Electrical and Computer Engineering
Rice University, Houston, TX 77005
ABSTRACT
We propose an approximation framework for distributed tar-
get localization in sensor networks. We represent the un-
known target positions on a location grid as a sparse vec-
tor, whose support encodes the multiple target locations.
The location vector is linearly related to multiple sensor
measurements through a sensing matrix, which can be lo-
cally estimated at each sensor. We show that we can suc-
cessfully determine multiple target locations by using lin-
ear dimensionality-reducing projections of sensor measure-
ments. The overall communication bandwidth requirement
per sensor is logarithmic in the number of grid points and
linear in the number of targets, ameliorating the communica-
tion requirements. Simulations results demonstrate the per-
formance of the proposed framework.
1. INTRODUCTION
Target localization using a set of sensors presents a
quintessential parameter estimation problem in signal pro-
cessing. Many design challenges arise when the sensors are
networked wirelessly due to the limited resources inherent to
the sensor network. For example, any inter-sensor commu-
nication exerts a large burden on the sensor batteries. Since
sufficient statistics are often non-existent for the localization
problem, accurate localization requires the full collection of
the network sensing data. Thus, the choice of algorithms is
usually steered away from those achieving optimal estima-
tion. To increase the lifetime of the sensor network and to
provide scalability, low dimensional data statistics are often
used as inter-sensor messages, such as local range or bearing
estimates at the sensors. Hence, the sensor network localiza-
tion performance is sacrificed so that the sensors can live to
observe another day.
To improve the estimation performance and robustness
of the sensor network in the presence of noise over classical
maximum likelihood and subspace methods, sparsity based
localization have been slowly gaining popularity [1–5]. The
main idea in these papers is that under specific conditions [6],
the localization estimates can be obtained by searching for
the sparsest solution of under-determined linear system-of-
equations that frequently arise in localization. In this context,
∗Corresponding author. This work was supported by the grants
DARPA/ONR N66001-06-1-2011 and N00014-06-1-0610, NSF CCF-
0431150 and DMS-0603606, ONR N00014-07-1-0936, AFOSR FA9550-
07-1-0301, ARO W911NF-07-1-0502, ARO MURI W911NF-07-1-0185,
and the Texas Instruments Leadership University Program. E-mail:
{volkan,duarte,richb}@rice.edu. Web: dsp.rice.edu/cs.
a vector is called sparse if it contains only a small number of
non-zero components in some transform domain, e.g, Fourier
or wavelets. The ℓ0-norm is the appropriate measure of the
sparsity, which simply counts the number of non-zero ele-
ments of a vector. Unfortunately, minimizing the ℓ0-norm is
NP-hard and becomes prohibitive even at moderate dimen-
sions. At the cost of slightly more observations, it has been
proven that ℓ1-norm minimization results in the same solu-
tion and has computational complexity on the order of the
vector dimensions cubed [7, 8].
We formulate the localization problem as the sparse ap-
proximation of the measured signals in a specific dictionary
of atoms. The atoms of this dictionary are produced by dis-
cretizing the space with a localization grid and then syn-
thesizing the signals received at the sensors from a source
located at each grid point. We show how this localization
dictionary can be locally constructed at each sensor. Within
this context, the search of the sparsest approximation to the
received signals that minimizes the data error implies that
the received signals were generated by a small number of
sources located within the localization grid. Hence, our algo-
rithm performs successful source localization by exploiting
the direct relationship between the small number of sources
present and the corresponding sparse representation for the
received signals. We assume that the individual sensor loca-
tions are known a priori; however, the number of sources
need not be known. The resulting sparse approximation
problem can be solved using greedy methods such as orthog-
onal matching pursuit [9] or other solvers such as fixed point
continuation methods [10].
Since we are interested in distributed estimation over
wireless channels where minimizing communications is cru-
cial, we discuss how to solve the localization problem when
lower dimensional projections of the sensor signals are
passed as inter-sensor messages. To preserve the informa-
tion content of these messages, a projection matrix must be
chosen so that it is incoherent with the sparsifying basis, i.e.,
the localization dictionary. Fortunately, a matrix with inde-
pendent and identically distributed (i.i.d.) Gaussian entries
satisfies the incoherence property with any fixed basis with
high probability [11]. Based on results from compressive
sensing (CS) [7, 8], we show that the total number of sam-
ples that are needed for recovering the locations of Ktargets
is O(Klog(N/K)), where Nis the number of grid points. We
also show that the total number of bits encoding the sensor
measurements that must be communicated can be made quite
small with graceful degradation in performance.
First published in the Proceedings of the 16th European Signal Processing Conference (EUSIPCO-2008) in 2008, published by EURASIP

Given that (i) each sensor has a localization dictionary,
(ii) we would like to localize the target locations within a
certain resolution as defined by N, and (iii) the total num-
ber of targets is much less than the number of grid points
(the sparsity assumption), an estimate of multiple target lo-
cations within our framework can be done when each sen-
sor receives at least O(Klog (N/K)) samples. This implies
that the resolution of the grid and the expected number of
targets rather than the number of sensors define the commu-
nication bandwidth, hence the proposed localization frame-
work is scalable and is suitable for distributed estimation.
Compared to the other distributed estimation methods such
as belief propagation [12–14], our approach does not require
a refinement process where local message passing is contin-
ued until the sensor network reaches convergence. However,
such scheme can still be used to improve the localization ac-
curacy within our framework. Compared to the decentralized
data fusion [15], our approach does not have data association
problems, which is combinatorial in the number of targets K.
Moreover, due to the democratic nature of the measurements
within our framework, our approach has built in robustness
against packet drops commonly encountered in practice in
sensor networks. In contrast, when low dimensional data
statistics, such as local range and bearing estimates, are used
in distributed estimation algorithms, package drops result in
significant performance degradation.
Similar to our paper, other sparse approximation ap-
proaches to source localization have been proposed be-
fore [1–5]. In [1], spatial sparsity is assumed to improve
localization performance; however, the computational com-
plexity of the algorithm is high, since it uses the high-
dimensional received signals. Dimensionality reduction
through principal components analysis was proposed in [2];
however, this technique is contingent on knowledge of the
number of sources present for acceptable performance and
also requires the transmission of all the sensor data to a cen-
tral location to perform singular value decomposition. Sim-
ilar to [2], we do not have incoherency assumptions on the
source signals. In [3], along with the spatial sparsity as-
sumption, the authors assume that the received signals are
also sparse in some known basis and perform localization in
near and far fields; however, similar to [1], the authors use the
high-dimensional received signals and the proposed method
has high complexity and demanding communication require-
ments. CS was employed for compression in [4, 5], but the
method was restricted to far-field bearing estimation. In con-
trast, this paper extends the CS-based localization setting to
near-field estimation, and examines the constraints necessary
for accurate estimation in the number of measurements and
sensors taken, the allowable amount of quantization, the spa-
tial resolution of the localization grid, and the conditions on
the source signals.
The paper is organized as follows. Section 2 lays down
the theoretical background for CS, which is referred in the
ensuing sections. Construction of the sensor localization
dictionaries is described within the localization framework
in Sect. 3. Section 4 describes the spatial estimation lim-
its of the proposed approach such as minimum grid spac-
ing or maximum localization grid aperture. Section 5 dis-
cusses communication aspects of the problem including the
message passing details and the bandwidth requirements. Fi-
nally, simulation results demonstrating the performance of
the localization framework are given in Sect. 6.
2. COMPRESSIVE SENSING BACKGROUND
CS provides a framework for integrated sensing and com-
pression of discrete-time signals that are sparse or compress-
ible in a known basis or frame. Let zdenote a signal of in-
terest, and Ψdenote a sparsifying basis or frame, such that
z=Ψ
θ
, with
θ
∈RNbeing a K-sparse vector, i.e. k
θ
k0=K.
Transform coding compression techniques acquire first zin
its entirety, and then calculate its sparse representation
θ
in
order to encode its nonzero values and their locations. CS
aims to preclude the full signal acquisition by measuring a
set yof linear projections of zinto vectors
φ
i, 1 ≤i≤M. By
stacking these vectors as rows of a matrix Φ, we can repre-
sent the measurements as y=Φz =ΦΨ
θ
. The main result
in CS states that when the matrix ΦΨ holds the restricted
isometry property (RIP) [8], then the original sparse repre-
sentation
θ
is the unique solution to the linear program
b
θ
=arg min
θ
∈RNk
θ
k1s.t. y=ΦΨ
θ
,(1)
known as Basis Pursuit [6]. Thus, the original signal zcan
be recovered from the measurement vector yin polynomial
time. Furthermore, choosing Φto be a matrix with inde-
pendent gaussian-distribtued entries satisfies the RIP for ΦΨ
when Ψis a basis or tight frame and M=O(Klog(N/K)).
Recovery from noisy measurements can be performed using
Basis Pursuit Denoising (BPDN), a modified algorithm with
relaxed constraints. We employ a fixed point continuation
method [10] to solve the BPDN optimization efficiently.
3. LOCALIZATION VIA SPATIAL SPARSITY
In a general localization problem, we have L+2 parame-
ters for each of the targets at each estimation period: the 2D
coordinates of the source location and the source signal it-
self, which has length L. In general, the estimation of the
these parameters are entangled: the source signal estimate
depends on the source location, and viceversa. Our formu-
lation can localize targets without explicitly estimating the
source signal, therefore reducing computation and commu-
nication bandwidth.
Assume that we have Ksources in an isotropic medium
with Psensors with known positions
ζ
i= [
ζ
xi,
ζ
yi]′(i=
1,...,P) on the ground plane. We do not assume that the
number of sources Kis known. Our objective is to deter-
mine the multiple target locations
χ
i= [
χ
xi,
χ
yi]′using the
sensor measurements. To discretize the problem, we only al-
low the unknown target locations to be on a discrete grid of
points
ϕ
={
ϕ
n|n=1,...,N;
ϕ
n= [
ϕ
xn,
ϕ
yn]′}. By perform-
ing this discretization and limiting the number of sources to
be localized, the localization problem can be cast as a sparse

approximation problem of the received signal, where we ob-
tain a sparse vector
θ
∈RNthat contains the amplitudes of
the sources present at the Ntarget locations. Thus, this vec-
tor only has Knonzero entries. We refer to this framework
as localization via spatial sparsity (LVSS).
Define a linear convolution operator for signal propaga-
tion, denoted as L
χ
→
ζ
, which takes the continuous signal for
a source at a location
χ
and outputs the Lsamples recorded
by the sensor at location
ζ
, by taking into account the physics
of the signal propagation and multipath effects. Similarly,
define the pseudoinverse operator L†
ζ
→
χ
that takes an ob-
served signal at a location
ζ
and deconvolves to give the
source signal, assuming that the source is located at
χ
. A
simple example operator that accounts for propagation atten-
uation and time delay can be written as
L
χ
→
ζ
(x) = "1
d
α
χ
,
ζ
xl
Fs
−dik
c#L
l=1
,
where d
χ
,
ζ
is the distance from source
χ
to sensor
ζ
,cis the
propagation speed,
α
is the propagation attenuation constant,
and Fsis the sampling frequency for the Lsamples taken.
Additionally, denote the signal from the kth source as xk
We can then express the signal received at sensor ias zi=
Xi
θ
, where
Xi=L
χ
1→
ζ
i(x1)L
χ
2→
ζ
i(x2)... L
χ
N→
ζ
i(xN)
is called the ith sensor’s source matrix. Similarly, we can ex-
press the signal ensemble as a single vector Z= [zT
1... zT
P]T;
by concatenating the source matrixes into a single dictionary
Ψ= [XT
1XT
2... XT
1]T,(2)
the same sparse vector
θ
used for each signal generates the
signal ensemble as Z=Ψ
θ
.
An estimate of the jth sensor’s source matrix Xjcan be
determined using the received signal at a given sensor i. If
we assume that the signal ziobserved at sensor iis originated
from a single source location, we can then write
b
Xj|i=hL
χ
1→
ζ
j(L†
ζ
i→
χ
1(zi)) ... L
χ
N→
ζ
j(L†
ζ
i→
χ
N(zi))i.
Furthermore, we can obtain an estimate b
Ψiof the signal en-
semble sparsity dictionary by plugging in the source matrices
estimates into (2).
Thus, by having each sensor transmit its own received
signal zito all other sensors in the network (or to a central
processing unit), we can then apply a sparse approximation
algorithm to Zand b
Ψito obtain an estimate of the sparse
location indicator vector b
θ
iat sensor i. By using CS theory,
we can reduce the amount of communication by having each
sensor transmit M=O(Klog(N/K)) random projections of
ziinstead of the L-length signal.
4. RESOLUTION OF THE GRID
The dictionary obtained in this fashion must meet the condi-
tions for successful reconstruction using sparse approxima-
tion algorithms. A necessary condition was posed in [16]:
Theorem 1 [16] Let Ψ∈RL×Nbe a dictionary and
ψ
jde-
note its jth column. Define its coherence
µ
(Ψ)as
µ
(Ψ) = max
1≤j,k≤T,j6=k
ψ
j,
ψ
k


ψ
j
k
ψ
kk,
Let K ≤1+1/16
µ
and let Φ∈RM×Lbe a matrix with i.i.d.
Gaussian-distributed entries, where M ≥O(Klog(N/K)).
Then with high probability, any K-sparse signal
θ
can be
reconstructed from the measurements y =ΦΨ
θ
through the
ℓ1minimization (1).
Thus, the coherence of the dictionary used by the sensor
controls the maximum number of localizable sources. Define
the normalized cyclic autocorrelation of a signal zas
Rz[m] = ∑L
n=1z(tn)z(tmod[(n+m),L])
kzk2.
Then
µ
(Ψi)depends on Rz[m], since

ψ
i,j,
ψ
i,k


ψ
i,j


ψ
i,k
=
∑P
p=1
RzihFs
c(d
χ
j,
ζ
p−d
χ
k,
ζ
p−d
χ
j,
ζ
i+d
χ
k,
ζ
i)i
d
χ
j,
ζ
pd
χ
k,
ζ
p
α
r∑P
p=1d−
α
χ
j,
ζ
p∑P
p=1d−
α
χ
k,
ζ
p.
The coherence
µ
will thus depend on the maximum value
attained by RzihFs
c(d
χ
j,
ζ
p−d
χ
k,
ζ
p−d
χ
j,
ζ
i+d
χ
k,
ζ
i)i; we as-
sume that the cyclic autocorrelation function is inversely pro-
portional to its argument’s absolute value. The coherence
then depends on the minimum value of the function’s argu-
ment. In the location grid setting, this minimum is approxi-
mately ∆/2D, with ∆denoting the grid spacing, and Ddenot-
ing the maximum distance between a grid point and a sensor.
Such maximum distance Dis dependent on both the exten-
sion of the grid and the diameter of the sensor deployment.
In summary, to control the maximum coherence, it will
be necessary to establish lower bounds for the localization
resolution – determined by the grid spacing – and upper
bounds for the extension of the grid and the diameter of the
sensor deployment.
5. INTER-SENSOR COMMUNICATIONS
Compared to distributed estimation algorithms that use a sin-
gle low dimensional data statistic from each sensor, the spar-
sity based localization algorithms [1–3] require the collection
of the observed signal samples to a central location. Hence,
for a sensor network with single sensors, a total of P×L
numbers must be communicated as opposed to, for example,
Preceived signal strength (RSS) estimates. Since Lis typ-
ically a large number, the lifetime of a wireless sensor net-
work would be severely decreased if such a scheme is used.

Considering the lifetime extension, the performance degra-
dation in target localization is considered a fair tradeoff.
Starting with the knowledge of the localization dictionary
Ψat any given sensor, CS results state that to perfectly re-
cover a Ksparse vector in Ndimensions, O(Klog(N/K))
random projections of Zare needed. This can easily be
achieved at each sensor by multiplying the sensed signal by
a pre-determined random projection matrix before communi-
cation, effectively resulting in a block diagonal measurement
matrix structure [17]. Thus, the dominant factor of the com-
munication bandwidth becomes the number of grid points,
as opposed to the number of sensors. As an example, con-
sider L=1000, N=1002,K=5, P=100: P×L=105
vs. Klog (N/K)≈38. When compared to distributing the
full sensor network data, there is a significant reduction;
however, LVSS is still not competitive with sending an RSS
estimate per sensor.1However, when RSS estimates are sent
in the presence of multiple targets, signal interference effects
and data association issues decrease the localization perfor-
mance. In general, the estimated localization dictionary is
noisy; hence, a larger number of measurements is needed.
Another way of understanding the minimum required
inter-sensor communications is to use information theoretic
arguments to determine the minimum number of bits re-
quired to localize K-coordinates in an N-dimensional space:
we need Klog2Nbits to encode this information. Since we
process the received signals to obtain
θ
, we can only lose
entropy. Thus, the resulting Klog2Nnumber of bits of our
analysis presents a lower bound the number of bits that each
sensor needs to receive for localization over a grid size N
to determine Ktarget locations. Even when quantization is
considered for the O(Klog (N/K)) measurements needed by
LVSS, there is an evident gap between the lower limit and the
LVSS requirement, since LVSS recovers both the location of
the nonzero coefficients and their values.
The aforementioned gap can be explored via quanti-
zation of the CS measurements. It is known within the
CS framework that compressive measurements are robust
against quantization noise as the CS reconstruction is robust
against additive noise [18]. Thus, we obtain two degrees of
freedom to determine the message size required in LVSS. In
practice with simulated and field data, we have found that
assigning 1-bit to encode the mean of the absolute values of
the compressive measurements is effective in recovery (see
also [4]). In this quantization scheme, the sensors pass the
sign of the compressive measurements as well as the mean
of their absolute values. Hence, the inter-sensor messages
would incorporate one additional number, which also needs
to be quantized, encoding the quantization level, along with
1-bit messages encoding the sign of the measurements.
6. SIMULATIONS
Our objectives in this section are two fold. We first demon-
strate the distributed estimation capabilities of the proposed
1We assume that there is no communication overhead. If there is some
overhead in sending messages, then LVSS becomes competitive.
(a) Sensor 30 (b) Sensor 25 (c) Sensor 21
(d) Sensor 18 (e) Sensor 12 (f) Sensor 7
Figure 1: Distributed estimation results: each sensor obtains lo-
calization estimates independently from random measurements re-
ceived from all sensors in the network. The results are similar for
most sensors.
framework. We then examine the effects of the inter-sensor
communication message sizes and signal-to-noise (SNR) ra-
tio on the performance of the algorithm.
Our simulation setup consists of P=30 sensor nodes
sensing two coherent targets that transmit a standard signal-
ing frame in MSK modulation with a random phase shift.
The sent signals have length L=512 and a unit grid of
N=30 ×30 points is used for localization, where the speed
of propagation c=1. For each simulation, a fixed point con-
tinuation solver [10] was used for the sparse approximations.
The algorithm employs a parameter
µ
that weights the good-
ness of fit of the solution against its sparsity; this parameter
is fixed for all simulations at all sensors. We note that when
the number of compressive measurements change, adjusting
this parameter can improve the localization results.
In the first experiment, we study the dependence of the
localization performance on the choice of sensor. We fix the
number of measurements per sensor M=30 and set the SNR
to 20dB. Figure 1 illustrates the sparse approximation results
at a representative subset of the sensors. In the figure, the
sensors are represented by filled stars at the ceiling, and the
ground truth for the source locations is represented by the
yellow asterisks. The surface plots show the output of the
sparse approximation, each normalized so that they sum up
to 1, defining a PDF of the multitarget posterior. The figure
shows consistent localization PDFs at the different sensors.
Within the 30 sensor network, a few sensors miss one of the
targets (e.g., sensor 25). Note that these PDFs are calculated
at each sensor independently after receiving the compressive
measurements from the network.
In the second experiment, we study the dependence of
the localization performance on the number of measurements
per sensor Mand the SNR. For each combination of these
parameters, we performed a Monte Carlo simulation involv-
ing 100 realizations of a uniformly random sensor deploy-
ment, as well as 50 realizations of Gaussian noise per de-
ployment. The location estimates in each Monte Carlo run
are obtained using K-means clustering on the estimated
θ
,

(a) RMS=0.34, Div.=16% (b) RMS=0.28, Div.=21% (c) RMS=0.24, Div.=23%
(d) RMS=0.26, Div.=0% (e) RMS=0.19, Div.=0% (f) RMS=0.17, Div.=0%
Figure 2: Results from Monte Carlo simulations. Top row: M=2,
bottom row: M=10. From left to right, SNR = 0dB, 5dB, 30dB.
with the number of clusters equal to the number of targets.
Figure 2 shows scatter plots for the localization estimates
for the different setups, together with the root mean square
(RMS) error and the likelihood of divergence (Div.) in the
sparse approximation algorithm. Intuitively, the figure shows
improvement in performance as the SNR or the number of
measurements increases. Reducing the number of measure-
ments, however, increases the likelihood of divergence in the
sparse reconstruction. For Fig. 2(a-c) each sensor receives
58 >Klog(N/K)≈12 measurements for localization. In
general, this increase is due to the noisy localization dic-
tionary estimates in the presence of multiple targets and the
block-diagonal nature of the measurement matrix.
7. CONCLUSIONS
Our fusion of existing sparse approximation techniques for
localization and the CS framework enables the formulation
of a communication-efficient distributed algorithm for target
localization. LVSS exhibits tolerance to noise, packet drops
and quantization, and provides a natural distributed estima-
tion framework for sensor networks. The performance of
the algorithm is dependent on both the number of measure-
ments and the SNR, as well as the observed signal, the sen-
sor deployment and the localization grid. Furthermore, the
algorithm performance can be improved by increasing the
number of measurements taken at each of the sensors, pro-
viding a tradeoff between the communication bandwidth and
the accuracy of estimation. Future work will investigate the
fundamental limits of localization within the sparsity frame-
work and compare the sparsity based localization algorithms
with other state-of-the-art distributed localization algorithms
to provide a Pareto frontier of the localization performance
as a function of communications. We also plan to study the
inclusion of signal sparsity into our framework.
REFERENCES
[1] I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction
from limited data using FOCUSS: A re-weighted minimum
norm algorithm,” IEEE Transactions on Signal Processing,
vol. 45, no. 3, pp. 600–616, 1997.
[2] D. Malioutov, M. Cetin, and A. S. Willsky, “A sparse signal
reconstruction perspective for source localization with sensor
arrays,” IEEE Transactions on Signal Processing, vol. 53, no.
8, pp. 3010–3022, 2005.
[3] D. Model and M. Zibulevsky, “Signal reconstruction in sensor
arrays using sparse representations,” Signal Processing, vol.
86, no. 3, pp. 624–638, 2006.
[4] V. Cevher, A. C. Gurbuz, J. H. McClellan, and R. Chel-
lappa, “Compressive wireless arrays for bearing estimation,”
in IEEE Int. Conf. on Acoustics, Speech and Signal Processing
(ICASSP), Las Vegas, NV, Apr. 2008.
[5] A. C. Gurbuz, V. Cevher, and J. H. McClellan, “A compressive
beamformer,” in IEEE Int. Conf. on Acoustics, Speech and
Signal Processing (ICASSP), Las Vegas, NV, 2008.
[6] S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic
Decomposition by Basis Pursuit,” SIAM Journal on Scientific
Computing, vol. 20, pp. 33, 1998.
[7] D. L. Donoho, “Compressed sensing,” IEEE Trans. Info. The-
ory, vol. 52, no. 4, pp. 1289–1306, September 2006.
[8] E. J. Cand `es, “Compressive sampling,” in Int. Congress of
Mathematicians, Madrid, Spain, 2006, vol. 3, pp. 1433–1452.
[9] J. Tropp and A. C. Gilbert, “Signal recovery from partial in-
formation via orthogonal matching pursuit,” IEEE Trans. Info.
Theory, vol. 53, no. 12, pp. 4655–4666, Dec. 2007.
[10] E. T. Hale, W Yin, and Y. Zhang, “Fixed-point continuation
for ℓ1minimization: Methodology and convergence,” Tech.
Rep. TR07-07, Rice University Department of Computational
and Applied Mathematics, Houston, TX, 2007.
[11] E. J. Cand `es and J. Romberg, “Sparsity and incoherence in
compressive sampling,” Inverse Problems, vol. 23, no. 3, pp.
969–985, June 2007.
[12] K. Murphy, Y. Weiss, and M. I. Jordan, “Loopy belief propa-
gation for approximate inference: An empirical study,” Pro-
ceedings of Uncertainty in AI, pp. 467–475, 1999.
[13] J. Yedidia, W. T. Freeman, and Y. Weiss, “Generalized be-
lief propagation,” Advances in Neural Information Processing
Systems, vol. 13, pp. 689–695, 2001.
[14] A. T. Ihler, Inference in Sensor Networks: Graphical Models
and Particle Methods, Ph.D. thesis, Massachusetts Institute of
Technology, 2005.
[15] J. Manyika and H. Durrant-Whyte, Data Fusion and Sen-
sor Management: A Decentralized Information-Theoretic Ap-
proach, Prentice Hall, Upper Saddle River, NJ, 1995.
[16] H. Rauhut, K. Schnass, and P. Vandhergheynst, “Compressed
sensing and redundant dictionaries,” IEEE Trans. Info. The-
ory, vol. 54, no. 5, pp. 2210–2219, May 2008.
[17] D. Baron, M. B. Wakin, M. F. Duarte, S. Sarvotham, and R. G.
Baraniuk, “Distributed compressed sensing,” Available at
http://www.dsp.rice.edu/cs, 2005.
[18] E. J. Cand `es and J. Romberg, “Encoding the ℓpball from lim-
ited measurements,” in Proc. IEEE Data Compression Con-
ference (DCC), Snowbird, UT, March 2006.