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 $

1

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

2

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.

3

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.

4

Kirchhoff migration or travel time imaging (passive array)

Down: Homogeneous, 1%, 2%, 3%STD. Across: different realizations.

5

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.

6

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.

7

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.

8

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.

9

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)

10

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.

12

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

0.03

0.04

0.05

0.06

Source index1.5 2 2.5 3 3.5 4

0

0.01

0.02

0.03

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

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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

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nge

in λ

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20 40 60 80 100 120 140 160 1800

0.005

0.01

0.015

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0.02

Frequency in MHz

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Frequency in MHz

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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

5

10

15

20

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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).

86 88 90 92 94

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Illuminating with the singular vec-

tor of the maximal singular value.

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Optimal Illumination, Full SVD (DORT)

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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.

19

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.

21

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.

22

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.

23

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.

24