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1
Easy Does It: User Parameter FreeDense and Sparse Methods
for Spectral Estimation
Jian Li
Department of Electrical and Computer Engineering University of FloridaGainesville, Florida
USA
2
Spectral Estimation
The goal of spectral estimation is to determine how power distributes over frequency from a finite number of data samples.
Diverse Applications
For example: synthetic aperture radar (SAR) imaging.
Data-Independent ApproachesFFT, Matched Filter, Delay-
and-Sum (DAS)
Poor resolution
High sidelobe levels, especially with missing data.
x (m)y
(m
)
FFT; HH
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A SAR imaging example using FFT.
3
Data-Adaptive Spectral Estimation
Data-Adaptive Approaches Examples: APES, Capon
Multiple snapshots needed to form reliable sample covariance matrices – fails for single or few snapshots, irregularly sampled data
High computational complexities
High resolution
Low sidelobe levels
Recent Development Iterative Adaptive Approach (IAA)
Applicable to single snapshot scenario
High computational complexities
High resolution
Low sidelobe levels
Dense and accurate
WFFT
IAA
4
Iterative Adaptive Approach (IAA)
Each iteration of IAA includes two steps (user parameter
free):
Estimate coefficients:
Update covariance matrix estimate
5
IAA-R (IAA with Regularization)
Noise effect taken into account explicitly:
Still user parameter free!
6
Active Sensing Example
s
t
y
TnJ s
TnJ s
Clutter ReturnTarget Return
sMatched Filter: z = sTy
Transmitter
Receiver
Active sensing (radar, sonar, etc.) Received signal decomposition:
-10 -5 0 5 100
0.2
0.4
0.6
0.8
1
sTy
Range-Doppler Imaging
Matched Filter Initialization
Movies Are Nice
Local Quadratic Convergence of IAA Proven.
Radar GMTI Example
9
Terrain map
The goal of ground moving target indication (GMTI) is to detect slow moving targets in the stationary background.
yellow or green dots: moving vehicles
STAP
10
An
t en
na
Ele
me
nt
s
Pulsesslowtime
1 M
N
1
MN samples for fixed range bin
Range
bin
s
fast
time
(J. Ward ’94)
STAP: space-time adaptive processing
Datacube:
Adaptive Processing
11
Space-Time Adaptive Processor
(Guerci et al. ’06)
12
Angle-Doppler Imaging in STAP
dB
IAADAS
Clutter power distribution over angle-Doppler for a fixed range
13
Target angle: 195
A total of 200 targets with constant power
Average SCNR over range: -18.94 dB
Ground truth denoted by x
o
Simulated Ground Truth
Target Detection for Fixed Angle
14
Range-Doppler Images
Ideal (total knowledge)
Prior (wrong knowledge)
IAA
dB
GLC (partial knowledge)
15
ROC Curves
Median CFAR algorithm applied to target detection
GLC detector
Automatic diagonal loading
Sample Number N = 20
Prior detector
Wrong prior knowledge of the clutter-and-noise covariance matrix
16
KASSPER DataSet
Main-beam width: 5 target angles: 190 - 200 (3-D target detection)
A total of 246 targets with varying power
Slow-moving targets and/or weak targets present
o
o o
o195azimuth =
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ROC Curves (KASSPER Data)
Median CFAR algorithm applied for target detection
18
Sparse Approaches Related work:
is replaced by to yield a convex optimization problem. LASSO: The least absolute shrinkage and selection operator. BP: Basis pursuit, very similar to LASSO FOCUSS: Focal underdetermined system solution SBL: Sparse Bayesian learning L1-SVD: L1 – singular value decomposition, similar to BP CoSaMP: Compressive Sampling Matching Pursuit
Most existing algorithms require Large computation times User parameters
Hard to decide Performance sensitive to choice of user parameter
Minimize such that is satisfied.
19
Kragh et al. Approach
Kragh et al. uses optimization transfer technique to obtain an iterative procedure:
A recent paper on SAR imaging states:
“
’’
This is FOCUSS.
20
SLIM
Sparse Learning via Iterative Minimization (SLIM) Solves the User Parameter Problem! (Tan, Roberts, Li, and Stoica, 2010)
SLIM Assumes the Following Hierarchical Bayesian Model:
SLIM is a MAP Approach:
21
SLIM Iterations
SLIM Iterates the Following Steps (Starting with DAS):
Given q, SLIM is User Parameter Free– Easy to Use!
Regularized Minimization in SLIM
22
Cyclic approach with majorization minimization employed to minimize cost function.
Conjugate gradient + FFT can be used for efficient implementation of SLIM.
For fixed noise variance (i.e., making it a user parameter), SLIM becomes FOCUSS/Kragh et al. Approach.
23
FFT for GOTCHA
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SLIM for GOTCHA
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SLIM for GOTCHA
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26
IAA (Dense) vs. SLIM (Sparse)
IAA is dense; SLIM is sparse.
IAA is more accurate; SLIM tends to bias downward.
IAA has high resolution; SLIM has higher resolution.
IAA’s fast implementation is trickier, especially for non-uniformly sampled data; SLIM is faster and its fast implementation is more straightforward.
27
Concluding Remarks
We need to devise dense and sparse methods that are user parameter free – easy to use in practice,
And accurate, And with high resolution, And computationally
efficient.
28
Thank you!