Adaptive High Dimensional Data Fusion and Sensing
for Dynamic Target Detection and Tracking
Dr. Ruixin Niu
Department of Electrical and Computer EngineeringVirginia Commonwealth University
Richmond, VA, USA
AFOSR PI’s meeting, DDDAS Program
September 19, 2018
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 1 / 21
Summary
Summary of EffortAF Relevance: information fusion, target detection/tracking are crucial forsurveillance and situational awarenessAFRL POC: Peter Zulch
Key Focus of Scientific ResearchDevelop efficient approaches for high dimensional data fusion for targettracking, joint sequential detection & tracking, and adaptive sensing.
RF Sensors
Sequential Detection
/Tracking
Acoustic
Sensors
Video
Cameras
Data Stream
Fusion.
.
.
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 2 / 21
Summary
Challenges:
highly nonlinear systems/measurementsheterogeneous/asynchronous sensor datalimited resources in distributed networked systems
AccomplishmentsNew Theory/Results
Joint sequential detection and trackingPreliminary results on sparsity based data fusionPreliminary results on fault tolerant source localization
Transition - DOD/Industry/Toolboxes
Collaboration/Summer Faculty Program with AFRL (mentors: Zulchand Huie)Collaboration with a company (IFT) on an Air Force STTR projectResearch Collaboration with Army Research Lab
Other performers on project
will recruit a student/post-doc
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 3 / 21
THEORY and Results: Joint Sparsity Based Data Fusion
Novelty: a new heterogeneous data-level fusion approach
Targets are sparse in discretized state space −→ joint sparse representationsof targets
Does not require knowledge of number of targets
Non-parametric and data driven: flexible and requires minimum priorinformation
Can handle non-linearity and discrete-time data (RF sensor and image data)
RF
Signals Joint Sparsity
Based Grid
Computation
for
Heterogeneous
Data Fusion
Joint Target Detection
& Estimation
FFT
Video
Images
Background
Subtraction
Intensity
Normalization
& Vectorization
Dim. Reduction
via Random
Projection
Magnitude
Normalization
Figure: Overview of the JSDLF fusion approach
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 4 / 21
THEORY and Results: Joint Sparsity Based Data Fusion
At a reference time, target’s state is denoted as
x = [x0 y0 vx0 vy0 ]T
State space is discretized into NG grid points, and each grid point: a hypothesizedtarget stateHeterogeneous data are linked through target state grids:θ: RF signal amplitude in frequency domain; β: intensity of background-subtractedimage
Figure: Hypothesized target states and corresponding measurement representations. k:time, l : index for RF sensors.
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 5 / 21
THEORY and Results: Joint Sparsity Based Data Fusion
Frequency domain signal amplitude and image intensity correspond to the sametargets −→ joint sparse signals
Θ = {θlk ,βk} (k = 1, · · · ,K ; l = 1, · · · , L− 1), a NG × (KL) matrix, has the
same non-zero support or is jointly sparse
Reconstruct Θ or its support (S-OMP, AMP, l1-l2 minimization, etc.)
Θ = {θlk ,βk} =
0 0 0 · · · 0∅ ∅ ∅ · · · ∅0 0 0 · · · 00 0 0 · · · 0∅ ∅ ∅ · · · ∅0 0 0 · · · 0...
......
. . ....
0 0 0 · · · 0
NG×(KL)
(There are 2 targets with 2nd and 5th hypothesized states.)
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 6 / 21
THEORY and Results: Joint Sparsity Based Data Fusion
rlk : sampled signal at l-th RF sensor at time k (N × 1 vector)
rlk = F
−1B
lkθ
lk + n
lk
Blk : N × NG frequency selection matrix
Blk(i , j) = 1 if Doppler shift for the j-th hypothesized state corresponds to
i-th DFT frequency binF−1: inverse DFT matrix
βk : vectorized image data at time k (M × 1 vector)
yk = Akβk + wk
image data are large −→ use a N ×M random compression matrix
zk = Φkyk = ΦkAkβk +Φkwk
Based on {rlk , zk} over l and k, recover Θ or its support
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 7 / 21
Joint Sparsity Based Data Fusion: Numerical Results
Numerical examples (Niu et al., SPIE’17, Aerospace’18)
100 150 200 250 300 350 400 450 500
−5
−4
−3
−2
−1
0
1
2
3
4
Frame Number
Po
sit
ion
Err
or
Ea
st
(m)
100 150 200 250 300 350 400 450 500
−4
−2
0
2
4
6
8
Frame Number
Po
sit
ion
Err
or
No
rth
(m
)
(a) (b)
Figure: (a)A testing scenario with 4 RF sensors and 1 video camera (b) Targetposition estimation errors.
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 8 / 21
Joint Sparsity Based Data Fusion: Future Work
Extend from non-Bayesian parameter estimation to Bayesian target tracking
Instead of using uniform grid, adaptively quantize state space with aparticle/particle flow filter
Find analytical performance for proposed approach
RF Sensors
Target State
Information
Acoustic
Arrays
Video
Camera
Data Fusion
via JSDLF.
.
.
Particle
Filter
Propagated
Particles
Figure: Adaptive state space discretization based on particle filtering.
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 9 / 21
Joint Sequential Detection and Tracking: Motivation
For moving objects with very weak SNRs, the detector based on a single samplecannot deliver acceptable detection performance
The joint object detection and tracking approach has the potential to significantlyimprove the detection of extremely weak moving objects
Different from the existing algorithms, the proposed algorithm works in continuousstate space, without the knowledge of possible object trajectories, and does notrequire very informative prior knowledge
Sequential detector vs. fixed sample size (FSS) detector
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 10 / 21
Joint Sequential Detection and Tracking: Related Work
Joint object detection and tracking problem can be viewed as a special case of thegeneral joint detection and estimation problem.
Middleton and Esposito, IEEE T-IT, 1968.Baygun and Hero, IEEE T-IT, 1995.Fredriksen, Middleton, and VandeLinde, IEEE T-IT, 1972.Moustakides, Jajamovich, Tajer, and Wang, IEEE T-IT, 2012.Jajamovich, Tajer, and Wang, IEEE T-SP, 2012.
Existing joint detection and tracking approaches:
Track-before-detect algorithms, along-track integration, probabilityhypothesis density (PHD) filter and Bernoulli filter.
Our previous work on joint detection and tracking:
Niu, WPMC, 2013, which provides the optimal FSS detector.
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 11 / 21
Problem Formulation
Under hypothesis H1, state sequence xk is a first-order Markov process
xk+1 = Fxk + Γvk
F: state transition matrix, vk : independent white process noise with zero meanand variance Q, Γ: gain matrix for vk .
Observations are obtained according to the measurement equation
zk = Hxk + wk
H: measurement matrix, wk : independent white measurement noise with zeromean and variance Rw .
Under hypothesis H0, the measurement is purely noise
zk = uk
uk : i.i.d. Gaussian distributed with mean µ and covariance Ru.
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 12 / 21
Likelihood Ratio
Using chain rule, the likelihood function p(z1:K |H1) is
p(z1:K |H1) = p(z1|H1)
K−1∏
k=1
p(zk+1|z1:k ,H1)
If z1, · · · , zk are independent under H0, the likelihood function p(z1:K |H0) is
p(z1:K |H0) =
K∏
k=1
p(zk |H0)
The optimal test statistic, the likelihood ratio, is
Λ(z1:K ) =p(z1:K |H1)
p(z1:K |H0)=
p(z1|H1)∏K−1
k=1 p(zk+1|z1:k ,H1)∏K
k=1 p(zk |H0)
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 13 / 21
Log-likelihood Ratio
In the Kalman filter, p(zk+1|z1:k ,H1) is:
p(zk |z1:k−1,H1) = N (Hx̂k|k−1,Sk)
where Sk is the observation residue covariance provided by the KF
Under hypothesis H0,p(zk |H0) = N (µ,Ru)
The log-likelihood ratio can be written in the summation form
Λ(z1:K ) =K∑
k=1
logp(zk |z1:k−1,H1)
p(zk |H0)=
1
2t(z1:K )
in which
t(z1:K ) =
K∑
k=1
[
log|Ru|
|Sk |+ (zk − µ)TR−1
u (zk − µ)− (zk −Hx̂k|k−1)TS−1k (zk −Hx̂k|k−1)
]
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 14 / 21
Wald’s SPRT
Since the observations are dependent over time under hypothesis H1, in this casethe optimum detector is in the form of a generalized sequential probability ratiotest (GSPRT) 1.
Λ(z1:K )
≥ AK stop and decide H1
≤ BK stop and decide H0
otherwise continue
where AK and BK : thresholds that are functions of K . However, the determinationof AK and BK is still an open problem.
1Cochlar and Vrana, Kybernetika, 1978; Eisenberg, Ghosh, and Simons, The Annals of Statistics, 1976
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 15 / 21
Expected Values of the Test Statistic
Proposition
The expectation of test statistic t(z1:K ) under hypothesis H1 when using the KF is
provided as follows
E [t(z1:K )|H1]
=
K∑
k=1
{log|Ru|
|Sk |+ tr[R−1
u HFkP0|0(HF
k)T
+ R−1u
k−1∑
i=0
HFiΓQ(HF
iΓ)T + R
−1u Rw ]
+(
HFkx̂0|0 − µ
)T
R−1u
(
HFkx̂0|0 − µ
)
− nz}
(1)
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 16 / 21
Expected Values of the Test Statistic
Proposition
The expectation of test statistic t(z1:K ) under H0 when using the KF is
E [t(z1:K )|H0]
=K∑
k=1
{
log|Ru|
|Sk |+ nz
− tr
[
S−1k Ru + S
−1k
k−1∑
i=1
Bk,iWk−iRu(Bk,iWk−i )T
]
−
[(
I−k−1∑
i=1
Bk,iWk−i
)
µ− Bk,k x̂0|0
]T
S−1k
·
[(
I−k−1∑
i=1
Bk,iWk−i
)
µ− Bk,k x̂0|0
]}
(2)
where Bk,i = HF∏i−1
j=1 [(I−Wk−jH)F].
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 17 / 21
SPRT detector vs. FSS detector
10−4
10−3
10−2
10−1
100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Probability of false alarm
Pro
ba
bili
ty o
f d
ete
ctio
n
Figure: ROC Curves for FSS detector (Meng and Niu, Fusion’15).
When SNR is −20dB, Pfa = 6.1× 10−4 and Pm = 1.4× 10−4.The ASN|H1 ≈ 10 and ASN|H0 ≈ 5 by using SPRT detector.To achieve the same or better performance, FSS detector needs 19 samples.
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 18 / 21
Conclusion
A new joint object detection and tracking algorithm based on Wald’s SPRT andthe Kalman filter has been proposed.
The first and second moments of the test statistic under H1 and H0 have beenderived, respectively.
The expected values of the test statistic are monotone functions of the number ofsamples, and they cross the threshold in a few samples.
The sequential detection approach detects a moving object with a small ASN andlow probabilities of error even under low SNR conditions, and it outperforms theoptimal FSS detector significantly.
A terminative joint sequential object detection and tracking approach: twohypothesis testing statistics cooperate with each other to guarantee that theWald’s SPRT procedure will eventually terminate
Future Work: extend to nonlinear settings
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 19 / 21
Accomplishments
Accomplishments
A journal paper on joint sequential detection and tracking, nearcompletion, to be submitted to IEEE Trans. on Information Theory
Transitions
Theories and algorithms will be transitioned to STTR program, AFRL,and ARL
Coordination/Synergy
Collaboration with Other PIs (Drs. Varshney and Chen from SyracuseUniversity)Ongoing collaboration with AFRL, ARL, and a small company
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 20 / 21
Appendix: Fault Tolerant Source Localization
Ongoing work, preliminary results published in Fusion’18
Ruixin Niu (VCU) Adaptive High Dimensional Data Fusion September 19, 2018 21 / 21
Source Location with Quantized Sensor Data Corruptedby False Information
Maitham Al-Salman and Ruixin Niu
Department of Electrical and Computer EngineeringVirginia Commonwealth University
Richmond, VA, USA
FUSION’18, University of Cambridge, UK
July 11, 2018
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 1 / 20
1 Motivation
2 Related Work
3 System Model
4 MLE-QRSS-GMM
5 CRLB Derivation
6 Simulation ResultsRMSE vs. Attack ProbabilityRMSE vs. Number of SensorsMLE-QRSS-GMM with Parameter Mismatch
7 Conclusion
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 2 / 20
Motivation
Sensor network measurement reliability issues
Crucial role of sensor networks’ applications
Secure state estimation in sensor networks under false informationinjection attacks (spoofing attack) is an important topic, which hasbeen studied recently
Sensor data can be corrupted due to natural interference, sensorfailure, or intentional false information injection by an adversary tomislead the fusion center (FC)
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 3 / 20
Related Work
Quantized received signal strength (Q-RSS)Niu and Varshney, IEEE T-SP, 2006.
Gaussian mixture model (GMM) in localizationYin et al., IEEE T-SP, 2015.Pfaf, Plagemann, and Burgard, IEEE ICRA , 2008.Zhang et al., IEEE Communications Letters, 2017.
False information injection attacks on state estimation (spoofingattacks)
Niu and Lu, CISS’15Lu and Niu, Fusion’14
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 4 / 20
System Model
We assume an isotropic signal attenuation model:
a2i =P0
(di )n(1)
ai : signal amplitude at the ith sensor.P0: radiated power by the target at a reference distance d0n: attenuation exponentdi : Euclidean distance.
di =√
(xi − xt)2 + (yi − yt)2 (2)
(xi , yi ) and (xt , yt): coordinates of the ith sensor and the targetrespectively.
Corrupted measurements of the attacked sensor:
ri = ai + bi (3)
bi : false information injected at the ith sensor.M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 5 / 20
System Model
We assume bi follows a GMM:
bi ∼2∑
k=1
ωi ,kN (bi ; 0, σ2k)
ωi ,1 = 1− pa: probability of not being attackedωi ,2 = pa: attack probability, and σ22 > σ21
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 6 / 20
MLE-QRSS-GMM
Quantization process:
Di =
0 −∞ ≤ ri < ηi1
1 ηi1 ≤ ri < ηi2
: :
: :
L− 1 ηL−1 ≤ ri <∞
(4)
Quantized data vector D = [D1, · · · ,DN ]T .Di ∈ {0, · · · , 2M − 1}.M: number of quantization bits (L = 2M)
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 7 / 20
MLE-QRSS-GMM
Denoting the parameter vector as θ = [P0, xt , yt ]T , the MLE
problem is:θ̂ = arg max
θp(D|θ)
According to GMM :
pil(ηi ,θ) =2∑
k=1
ωi ,k
[Q
(ηil − aiσk
)− Q
(ηil+1 − ai
σk
)](5)
Joint probability:
p (D|θ) =N∏i=1
L−1∏l=0
[pil(ηi ,θ)]δl, Di (6)
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 8 / 20
MLE-QRSS-GMM
Log-likelihood :
log p(D|θ) =N∑i=1
L−1∑l=0
δl ,Dilog pil(ηi ,θ) (7)
where δi ,j is Kronecker delta function:
δi ,j =
{1 if j = i0 o.w.
MLE-QRSS-GMM:
θ̂ = arg maxθ
log p(D|θ) (8)
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 9 / 20
CRLB Derivation
Theorem
For an unbiased estimator θ̂(D), the CRLB is given by
E{
[θ̂(D)− θ][θ̂(D)− θ]T}≥ J−1 (9)
where J is the 3× 3 Fisher information matrix (FIM).
FIM elements:
j11 =N∑i=1
βid−2ni a−2
i
j12 = j21 = nN∑i=1
βid−(n+2)i (xi − xt)
j13 = j31 = nN∑i=1
βid−(n+2)i (yi − yt)
(10)
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 10 / 20
CRLB Derivation
j22 = n2N∑i=1
βia2i d
−4i (xi − xt)
2
j23 = j32 = n2N∑i=1
βia2i d
−4i (xi − xt)(yi − yt)
j33 = n2N∑i=1
βia2i d
−4i (yi − yt)
2
(11)
βi =1
8π
L−1∑l=0
1
pil(ηi ,θ)
[K∑
k=1
wi ,kγi ,l ,kσk
]2
γi ,l ,k = e− (ηil−ai )
2
2σ2k − e
− (ηil+1−ai )2
2σ2k
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 11 / 20
RMSE of MLE-QRSS-GMM vs. pa
10−2
10−1
100
2000
4000
6000
8000
10000
12000
14000
Attack probability pa
RM
SE
of P
0
nominal MLE
Q−RSS−GM
CRLB
10−2
10−1
100
2
4
6
8
10
12
14
16
18
Attack probability pa
RM
SE
of x
t in m
nominal MLE
Q−RSS−GM
CRLB
Figure: RMSE of MLE-QRSS-GMM vs. pa
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 12 / 20
RMSE of MLE-QRSS-GMM vs. pa
10−2
10−1
100
2
4
6
8
10
12
14
16
18
Attack probability pa
RM
SE
of y
t in m
nominal MLE
Q−RSS−GM
CRLB
Figure: RMSE of MLE-QRSS-GMM vs. pa
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 13 / 20
RMSE of MLE-QRSS-GMM vs. Number of Sensors(pa = 0.03)
8 10 12 14 16 18 201000
1500
2000
2500
3000
3500
4000
4500
5000
Square Root of Sensor Number
RM
SE
of P
0
nominal MLE
Q−RSS−GM
CRLB
8 10 12 14 16 18 202
3
4
5
6
7
8
9
Square Root of Sensor numberR
MS
E o
f x
t in m
ete
r
nominal MLE
Q−RSS−GM
CRLB
Figure: RMSE of MLE-QRSS-GMM vs. Number of Sensors (pa = 0.03)
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 14 / 20
RMSE of MLE-QRSS-GMM vs. Number of Sensors(pa = 0.03)
8 10 12 14 16 18 202
3
4
5
6
7
8
9
Square Root of Sensor number
RM
SE
of y
t in m
ete
r
nominal MLE
Q−RSS−GM
CRLB
Figure: RMSE of MLE-QRSS-GMM vs. number of sensors (pa = 0.03)
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 15 / 20
RMSE of MLE-QRSS-GMM vs. Number of Sensors(pa = 0.3)
8 10 12 14 16 18 201500
2000
2500
3000
3500
4000
4500
5000
5500
6000
Square Root of Sensor Number
RM
SE
of P
0
nominal MLE
Q−RSS−GM
CRLB
8 10 12 14 16 18 202
3
4
5
6
7
8
9
10
11
12
Square Root of Sensor numberR
MS
E o
f x
t in m
ete
r
nominal MLE
Q−RSS−GM
CRLB
Figure: RMSE of MLE-QRSS-GMM vs. number of sensors (pa = 0.3)
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 16 / 20
RMSE of MLE-QRSS-GMM vs. Number of Sensors(pa = 0.3)
8 10 12 14 16 18 202
3
4
5
6
7
8
9
10
11
12
Square Root of Sensor number
RM
SE
of y
t in m
ete
r
nominal MLE
Q−RSS−GM
CRLB
Figure: RMSE of MLE-QRSS-GMM vs. number of sensors (pa = 0.3)
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 17 / 20
MLE-QRSS-GMM with Parameter Mismatch
Mismatches exist between true parameters and nominal parameters.MLE-QRSS-GMM is designed assuming pan = 0.05, but a different pa(pat ) is used by the attacker.
Table: RMSE of P0 for the Q-RSS-GM estimator with a mismatched pa
N pat = 0 pat = 0.01 pat = 0.05 pat = 0.1
144 2505.1 2505.6 2594.9 2714.0
256 1827.2 1847.0 1912.2 2026.6
400 1449.6 1460.3 1493.2 1571.1
Table: RMSE of xt for the Q-RSS-GM estimator with a mismatched pa
N pat = 0 pat = 0.01 pat = 0.05 pat = 0.1
144 4.0614 4.1942 4.4723 4.6697
256 3.0083 3.0859 3.3337 3.4781
400 2.3707 2.4047 2.7200 2.8098M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 18 / 20
MLE-QRSS-GMM with Parameter Mismatch
Table: RMSE of yt for the Q-RSS-GM estimator with a mismatched pa
N pat = 0 pat = 0.01 pat = 0.05 pat = 0.1
144 4.1767 4.1357 4.4543 4.7472
256 3.0306 3.0286 3.2758 3.5819
400 2.4028 2.4592 2.4864 2.7359
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 19 / 20
Conclusion
1 Target localization in WSNs based on quantized RSS in the presenceof false information injection attacks studied.
2 Using Gaussian mixture model, we developed a maximum likelihoodestimator.
3 The proposed estimator is much more robust and efficient comparedto nominal MLE which ignores the possible attacks.
4 The corresponding CRLB has been derived to evaluate the estimationperformance.
5 Proposed method is robust to injected false information andparameter mismatch, and its performance reaches the CRLB as thenumber of sensors increases.
6 Ongoing work: an estimator that works with much less knowledgeabout the attacks.
M. Al-Salman and R. Niu (VCU) Source Location with Quantized Sensor Data Corrupted by False InformationJuly 11, 2018 20 / 20