adaptive optimal illumination in clutter george papanicolaou
TRANSCRIPT
Adaptive Optimal Illumination in Clutter
George Papanicolaou
Department of Mathematics, Stanford University
http://georgep.stanford.edu
In collaboration with:
L. Borcea (CAAM Rice) and C. Tsogka (Univ. of Chicago).
$Id: ARO/DARPA Workshop June 28, 2006 $
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Outline
• Brief review of Coherent Interferometry (CINT) based on the
decoherence frequency and length
• Adaptive CINT.
• Simulations with CINT for imaging structures in clutter with
passive and active arrays.
• Optimal illumination and waveform design.
• Summary of work on imaging in clutter
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Numerical simulations imaging underground structures
λ 0
λ 0λ 0
λ 0
100
100Source
absorbing medium
arra
y
L=90
d=6
da L~
Computational domain 100λ0 × 100λ0 with central wavelength
λ0 = 3m (at central frequency f0 = 1KHz and with c0 =
3km/sec), surrounded by a perfectly matched layer (pink).
The array has 185 receiving elements λ0/2 apart, for an aperture
of 92λ0.
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Setup for numerical simulations
The three sources are ultrawideband (more than 100% band-
width) at a distance of 90λ0 from the array. The near ones are
6λ0 apart and the far one is 3λ0 behind the near ones.
The random fluctuations of the propagation speed, on the right,
have 3% STD and a Gaussian correlation (monoscale) function.
The correlation length is equal to the central wavelength λ0 =
3m.
We use a FINITE ELEMENT TIME DOMAIN CODE (2D or
3D), resolving all scales with 30 − 40 points per wavelength. It
takes about two hours on a workstation to produce a set of (2D)
synthetic data.
We also have well established and tested 3D elastic and electro-
magnetic FETD codes.
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Kirchhoff migration or travel time imaging (passive array)
Down: Homogeneous, 1%, 2%, 3%STD. Across: different realizations.
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Critique of Kirchhoff migration imaging
How does time reversal imaging or Kirchhoff Migration (KM) (or
Synthetic Aperture Imaging) work?
It does not work well in clutter.
It is statistically unstable in clutter.
The reason is that KM tries to cancel the random phase of
the signals arriving at the array with a deterministic phase using
travel times.
The true Green’s function for the random medium is not known
and so cannot be used for imaging, which would result is huge
resolution enhancement as in physical TR.
KM is inherently unstable statistically, which is a very serious
flaw.
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Decoherence distance Xd and decoherence frequency Ωd
Time traces at xr and xr′ decorrelate rapidly in a random mediumas the distance between xr and xr′ increases. That is, the tracecross-correlation
P (xr, ·) ∗t P (xr′,−·)(t)
goes to zero as |xr − xr′| increases.
The decoherence distance Xd is related to the effective aperture
in time reversal, which is essentially independent of any physical
aperture. In fact Xd ∼ λLae
and 0 ≤ Xd ≤ a.
The phases of P (xr, ω1) and P (xr, ω2) decorrelate when |ω1−ω2|increases. The decoherence frequency Ωd is inversely propor-
tional to the delay spread (reverberation) in the traces due to
the random inhomogeneities and 0 ≤ Ωd ≤ B.
Both Xd and Ωd can be ESTIMATED from the array data di-
rectly.
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Coherent interferometric imaging
To combine good random phase cancellation, which KM does
not have but matched field methods have, with exploitation of
residual coherence we introduce the Coherent Interferometric
imaging functional:
ICINT (yS;Xd,Ωd) =∫ ∫
|ω1−ω2|≤Ωd
dω1dω2
∑ ∑|xr−x′
r|≤Xd
P (xr, ω1)P (x′r, ω2)e
−i(ω1τ(xr,yS)−ω2τ(x′r,y
S))
The smoothing or thresholding parameters are the Decoherence
Length Xd, 0 < Xd ≤ a and the Decoherence Frequency Ωd,
0 < Ωd ≤ B. Here a is the array size and B the bandwidth. They
are determined Adaptively as the image is formed.
If we take Xd = a and Ωd = B, which means that there is
no smoothing, then the CINT functional is just the Kirchhoff
migration functional squared: ICINT = (IKM)2. The case Ωd =
0, suitably interpreted, corresponds to the Matched Field imaging
functional.
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Adaptive Selection of Xd and Ωd
For CINT to be effective we need to be able to determine the
decoherence length and frequency adaptively. We do this by min-
imizing the bounded variation norm of the (random) functional
ICINT
X∗d ,Ω∗
d = argminXd,Ωd||ICINT (·;Xd,Ωd)||BV
where
||f ||BV =∫
|f(y)|dy + α∫
|∇f(y)|dy
The bounded variation norm is used because it is smoothing
on small scales but is respectful of large scale features (discon-
tinuities). The smoothing is limited by the L1 norm. Other
sparcity-type norms can be used, such as entropy norms.
If the background velocity is not known then we need to do
a velocity estimation. This is particularly relevant in seismic
imaging and elsewhere.
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Coherent interferometric imaging results
Pass
ive
arra
yA
ctiv
e ar
ray
Coherent Interferometry iimages in random media with s = 3%.
Left Figures: Xd = a, Ωd = B (Kirchhoff Migration, no
smoothing)
Middle Figures: Xd = X∗d, Ωd = Ω∗
d (Adaptively selected optimal
smoothing)
Right Figures: Xd < X∗d, Ωd < Ω∗
d (Too much smoothing)
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Resolution theory
• The main results are: (a) the resolution in range is c0/Ωd with
c0 the homogeneous propagation speed, and (b) the cross
range resolution is λ0L/Xd We assume here that the SNR
at the array is high (essentially infinite) (Inverse Problems,
21 (2005), pp. 1419-1460, and another paper to appear in
Inverse Problems in July 2006.)
• Note that in a deterministic medium Ωd = B, the bandwidth,
and Xd = a, the array aperture. Loss of resolution is directly
tied to space and frequency decoherence. (0 ≤ Ωd ≤ B, 0 ≤
Xd ≤ a.)
• We estimate Ωd and Xd optimally as the image is formed
using a Bounded Variation criterion, as explained above.
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Summary: Resolution limits
Determin Known Rand (TR) Unknown Rand (CINT)
Rangec0
B
c0
B
c0
Ωd
Cross Rangec0L
ωa
c0L
ωae∼ Xd
c0L
ωXd
∼ ae
Resolution limits in deterministic media and in random media,
when the random medium is known as in physical time rever-
sal, and when the random medium is not known as in coherent
interferometry.
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Role of coherence in array imaging
Coherence is essential in array imaging. A more physical, and
more conventional, way to measure coherence is through the
transport mean free path l⋆. In the regime where waves energy
propagates by diffusion, the diffusion coefficient is given by
D =c0l⋆
3.
If the transport mean free path l⋆ is small compared to the range
L of the object to be imaged then migration methods, including
CINT, will not work. In our numerical simulations l⋆ is of the
order of L, which is the regime where we expect CINT to be
effective.
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Optimal illumination and waveform design
Optimal illumination in array imaging is up to now aimed at
DETECTON. That is, if f(xs, ω) is the signal in the frequency
domain that is emitted at xs then the signal received at xr is
given by
Ns∑
s=1
Π(xr,xs, ω)f(xs, ω).
The total power received at the array is given by
P =∫
|ω−ω0|≤Bdω
Nr∑
r=1
∣∣∣∣∣∣
Ns∑
s=1
Π(xr,xs, ω)f(xs, ω)
∣∣∣∣∣∣
2
.
We want to maximize this functional over all illumination signals
f(xs, ω) with the normalization
∫
|ω−ω0|≤Bdω
Ns∑
s=1
∣∣∣f(xs, ω)∣∣∣2
= 1
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Optimal illumination and waveform design II
In general, Optimal Illumination for Detection produces images
with bad resolution.
Just like in Adaptive Coherent Interferometry we can, however,
introduce an optimal illumination objective that is tied to the
resolution of the image itself.
We now show the results of some simple numerical simulations
using this approach to optimal illumination. Full details are in:
Adaptive interferometric imaging in clutter and optimal illumi-
nation available at http://georgep.stanford.edu.
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Optimal Illumination, Random Medium
20 40 60 80 100 120 140 160 1800
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
Source index1.5 2 2.5 3 3.5 4
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
Frequency in MHz
20 40 60 80 100 120 140 160 1800
0.01
0.02
0.03
0.04
0.05
Source index1.5 2 2.5 3 3.5 4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Frequency in MHz
20 40 60 80 100 120 140 160 1800
0.01
0.02
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0.05
0.06
Source index1.5 2 2.5 3 3.5 4
0
0.01
0.02
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0.04
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0.06
Frequency in MHz
20 40 60 80 100 120 140 160 1800
0.01
0.02
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0.06
Source index1.5 2 2.5 3 3.5 4
0
0.01
0.02
0.03
0.04
0.05
0.06
Frequency in MHz
Left: image; Center, weights; Right: pulse.
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Optimal Illumination, Deterministic Medium
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range in λ0
cros
s−ra
nge
in λ
0
20 40 60 80 100 120 140 160 1800
0.005
0.01
0.015
0.02
0.025
0.03
Source index1.5 2 2.5 3 3.5 4
0
0.005
0.01
0.015
0.02
Frequency in MHz
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range in λ0
cros
s−ra
nge
in λ
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20 40 60 80 100 120 140 160 1800
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Source index1.5 2 2.5 3 3.5 4
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
Frequency in MHz
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range in λ0
cros
s−ra
nge
in λ
0
20 40 60 80 100 120 140 160 1800
0.005
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0.015
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Source index1.5 2 2.5 3 3.5 4
0
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0.02
Frequency in MHz
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range in λ0
cros
s−ra
nge
in λ
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20 40 60 80 100 120 140 160 1800
0.005
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0.015
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0.04
Source index1.5 2 2.5 3 3.5 4
0
0.002
0.004
0.006
0.008
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0.012
0.014
0.016
Frequency in MHz
Left: image; Center: weights; Right: pulse
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Optimal Illumination, Iterative TR
20 40 60 80 100 120 140 1600
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10
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frequency in kHz
λ1
λ2
λ3
λ4
λ5
The first five singular values of the
response matrix as a function of
frequency (in kHz).
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range in λ0
cros
s−ra
nge
in λ
0
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range in λ0
cros
s−ra
nge
in λ
0
Illuminating with the singular vec-
tor of the maximal singular value.
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Optimal Illumination, Full SVD (DORT)
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range in λ0
cros
s−ra
nge
in λ
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range in λ0
cros
s−ra
nge
in λ
0
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range in λ0
cros
s−ra
nge
in λ
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range in λ0
cros
s−ra
nge
in λ
0
Kirchhoff migration images obtained by illu-
minating the targets with the singular vec-
tors corresponding to the significant singular
values (here the first four) of the response
matrix.
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The algorithm
We compute first the weighted average of the migrated traces
m(xr,yS, ω) =
Ns∑
s=1
wsfB(ω − ω0)Π(xr,xs, ω)e−iω[τ(xr,yS)+τ(xs,yS)
].
We then cross correlate these migrated traces
ICINT(yS;w, f) =∫
|ω−ω0|≤Bdω
∫
|ω′ − ω0| ≤ B|ω′ − ω| ≤ Ωd
dω′
Nr∑
r=1
Nr∑
r′ = 1
|xr − x′r| ≤
2c0
(ω+ω′)κd
m(xr,yS, ω)m(xr′,y
S, ω′),(1)
where w = (w1, ..., wNs).
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The algorithm, continued.
We determine w and f by minimizing an objective function O(w, f)
that quantifies the quality of the image. We take it to be the L1
norm. We then determine the weights w = (w1, . . . , wNs) and the
waveform f(ω) as minimizers of O(w, f), subject to the following
constraints.
The weights should be nonnegative and sum to one
Ns∑
s=1
ws = 1, ws ≥ 0, s = 1, . . . , Ns. (2)
The support of f(ω) = fB(ω − ω0) is restricted to the fixed
frequency band [ω0 − B, ω0 + B], we ask that
fB(ω − ω0) ≥ 0, for all ω ∈ [ω0 − B, ω0 + B] (3)
and its integral is one.
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The SNR issue
We also impose a quadratic constraint on the waveforms with a
lower bound N on the received power at the array
Nr∑
r=1
∫
|ω−ω0|≤Bdω
∣∣∣P (xr,xs, ω)∣∣∣2≥ N , for all s = 1, . . . , Ns. (4)
This sets a lower level N for the received power at the array.
It must be less than the maximum received total power at the
array, 0 < N < P. Setting the level N close to P is appropriate
when the signals received at the array are weak and instrument
noise affects them significantly. In this case the image that we
obtain with the optimal waveforms is somewhat similar to the
one we obtain with the optimal power illumination.
The ratio P/N plays the role of a signal to noise ratio.
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Comments on the algorithm
• Why is f(xs, ω) taken to be a non-negative function?
• Why a product?
This is based on the analysis of imaging a single point scatterer
in a homogeneous medium. In that case the optimal illumination
has the form: Illuminate from the two ends of the array with a
waveform f that is proportional to ω2.
No such result is known in random media.
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Summary and concluding remarks
• Coherent interferometry, which is the backpropagation of local cross-correlations of traces, is a good way to deal with cluttered environmentsin array imaging.
• Adaptive estimation of the space-frequency decoherence in the array dataaddresses well the difficult issue of implementing coherent interferometryand makes the CINT algorithm more robust.
• The key parameters Ωd and Xd, which characterize the clutter as far asimaging is concerned, play a triple role: they are smoothing or threshold-ing parameters for CINT, they determine its resolution, and characterizethe coherence of the data.
• In Optimal Illumination a new algorithm has been introduced that usesthe CINT functional and optimizes the illumination for best resolution,rather than maximum received power.
• Implementation of the optimal illumination algorithms, and their theory,need further development.
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