a fast inverse solver and tunnel imaginglcarin/liu8.3.05.pdfaug 03, 2005 · • inversion of...

A Fast Inverse Solver and Tunnel Imaging

Qing H. LiuDepartment of Electrical and Computer Engineering

Duke University

August 3, 2005

Outline• Introduction• Diagonal tensor approximation (DTA)• A fast rigorous inversion method with DTA-

CSI• Inversion of experimental data• Imaging of tunnels• Summary

Introduction: The Problem

yz

x12

i

N-1N

D3D inhomogeneous object

Layered Media in Sensing and Imaging Problems

• Applications are ubiquitous:– Landmine detection– Underground tunnel imaging– Biomedical imaging

• Key issues:– Optimal sensing configuration– How to obtain best resolution for a given data

set?

Conflicting Factors in EM Imaging: Resolution and Probing Volume

• Resolution is limited by– Frequency (wavelength)– Aperture size

• High resolution requires high frequency– Shallow penetration

depth because of larger attenuation

• Probing volume is limited by– Attenuation– Signal-to-noise ratio

(SNR)

• Large probing volume requires low frequency because attenuation increases with frequency

Maxwell’s Equations for Layered Media

• We will reconstruct permittivity and conductivity of the 3D target:

Electrical Field Integral Equation• Forward problem: Inside the target domain D

where and is thedyadic Green’s function of the layered medium.

• Numerical solution of E requires inversion of a large matrix. This is time consuming.• Previously, we used BCGS-FFT method to speed up this solution. To further improve the speed, we propose a new scattering approximation.

Scattering Approximations to Avoid Matrix Inversion

• Born Approximation,– Valid for the case of weak scattering

• The Extended Born Approximation (EBA) by Habashy et al (1993)

• The Quasi-Linear (QL) approximationZhdanov and Fang (1996)

• The Quasi-Analytical (QA) approximation by Zhdanov et al (2000)

The Diagonal Tensor Approximation (DTA)

Introducing a scattering tensor

Then the integral equation becomes

Approximate the scattering tensor as a diagonal tensor

Now noting the Green’s function is highly peaked at the source point, inside the integrand we can approximate

•Note: The first term of the RHS is the scattered field in Born approximation

•The equation can be solved in closed form

• Closed form solution

where

• After is found, fields

everywhere can be obtained by Green’s function.

Observations

• Cross polarization is included• The DTA is a source-dependent approximation• Born approximation and quasi-analytic

approximation are special cases of DTA• No need to invert a large system matrix to obtain

the field solution.• The cost is almost as inexpensive as the Born

approximation.

100 MHz

Concentriccubes

4

8 161.6 m

400 MHz 100 MHz 400 MHzDTA 3.52% 4.01%

QA 9.22% 9.15%

EBA 23.77% 24.16%

Inverse Solver Based on DTA

• Use the Born iterative method• During each iteration, Green’s operation is

accelerated by FFT (a nice property of the Green’s function even in layered media)

• For high contrasts, DTA inversion result can serve as the initial solution of full nonlinear inversion methods (CSI and DBIM) to speed up the solution

Example of Hybrid DTA/BCGS-FFT Method for Inversion

800 MHz; 45 sources and 45 receivers on five faces

(32,0.8)

(48, 0.4)

• Dielectric constant and conductivity have different contrast patterns

DTA/BCGS

Imaging from Measured Data

D

Circular array

Air

200 mm

200 mm

ε = 1.45r

ε = 3.0r

30 mm

ε = 3.0r

20 mm 60 mm

30 mm

18 x 241

(Data collected by K. Belkebir and M. Saillard, Institut Fresnel)

Imaging Results for Dielectric Constant

x (mm)

z (

mm

)

−100 −50 0 50 100−100

−50

0

50

100

1

1.5

2

2.5

2 GHz

x (mm)

z (

mm

)

−100 −50 0 50 100−100

−50

0

50

100

1

1.5

2

2.5

3

3.5

2 GHz to 3 GHz

2 cm

x (mm)

z (

mm

)

−100 −50 0 50 100−100

−50

0

50

100

1

1.5

2

2.5

3

3.5

4 GHz to 5 GHz x (mm)

z (

mm

)

−100 −50 0 50 100−100

−50

0

50

100

1

1.5

2

2.5

3

3.5

3 GHz to 4 GHz

x (mm)

z (

mm

)

−100 −50 0 50 100−100

−50

0

50

100

1

1.5

2

2.5

3

3.5

6 GHz to 7 GHz x (mm)

z (

mm

)

−100 −50 0 50 100−100

−50

0

50

100

1

1.5

2

2.5

3

5 GHz to 6 GHz

x (mm)

z (

mm

)

−100 −50 0 50 100−100

−50

0

50

100

1

1.5

2

2.5

3

3.5

8 GHz to 9 GHz

x (mm)

z (

mm

)

−100 −50 0 50 100−100

−50

0

50

100

1

1.5

2

2.5

3

3.5

7 GHz to 8 GHz

x (mm) z

(m

m)

−100 −50 0 50 100−100

−50

0

50

100

1

1.5

2

2.5

3

3.5

9 GHz to 10 GHz

• Observation: Super-resolution is achieved. (At 3 GHz, the well-resolved wall thickness of the outer cylinder is less than 1/4 wavelength.)

NUFFT Imaging of 3D Objects

y

x

3D Configuration (W. Scott, Georgia Tech)

3D Imaging Results (GT plywood)

Raw Data

GEOMETRY OF TUNNEL

1.25m

0.75m

1.5m

xx x x xx……………

1.5m0.02m

D

16 T/R

earth

1 14, 0.01rε σ= =

2 220, 0.02rε σ= =

1rε =4.0m

1.0m

1.0m

4mx4m

Single-Frequency Imaging

−2 −1 0 1 2

1

2

3

4

5 1

1.5

2

2.5

3

3.5

4

−2 −1 0 1 2

1

2

3

4

5 1

1.5

2

2.5

3

3.5

4

True Profile 20 MHz

50 MHz 200 MHz(Inadequate Sampling)

−2 −1 0 1 2

1

2

3

4

5 1

1.5

2

2.5

3

3.5

4

−2 −1 0 1 2

1

2

3

4

5 1

1.5

2

2.5

3

3.5

4

Multi-Frequency Imaging(Low-Frequency Results as Initial Solution

of High-Frequency Inversion)

−2 −1 0 1 2

1

2

3

4

5 1

1.5

2

2.5

3

3.5

4

−2 −1 0 1 2

1

2

3

4

5 1

1.5

2

2.5

3

3.5

4

True profile 50 MHz

−2 −1 0 1 2

1

2

3

4

5 1

1.5

2

2.5

3

3.5

4

−2 −1 0 1 2

1

2

3

4

5 1

1.5

2

2.5

3

3.5

4

50 MHz with 100 MHz 100 MHz with 150 MHz

Multi-Frequency Imaging (Cont.)

−2 −1 0 1 2

1

2

3

4

5 1

1.5

2

2.5

3

3.5

4

−2 −1 0 1 2

1

2

3

4

5 1

1.5

2

2.5

3

3.5

4


−2 −1 0 1 2

1

2

3

4

5 1

1.5

2

2.5

3

3.5

4

−2 −1 0 1 2

1

2

3

4

5 1

1.5

2

2.5

3

3.5

4


THEORETICAL DATA ERROR

0 100 200 30010

−3

10−2

10−1

100

Iteration Number

Dat

a E

rror

50 MHz100 MHz150 MHz200 MHz

Acoustic Imaging of a Tunnel3: 1.168 /

340 /

air kg m

v m s

ρ ==

6m

2m2m

1.5m

8m

3 1: 1400 /

2000 /

Soil kg m

v m s

ρ ==

3 2: 2700 /

3000 /

Soil kg m

v m s

ρ ==

0.1m

SourceReceiver

Wave Propagation for One Source6m

2m2m

1.5m

8m

0.4m

Movie of Wave PropagationAir

Top soil

Bottom soil

Waveforms at Different Locations

0 1 2 3 4 5 6

x 10-3

-0.05

0

0.05

time

x=-1

m

0 1 2 3 4 5 6

x 10-3

-0.05

0

0.05

time

x=0m

0 1 2 3 4 5 6

x 10-3

-0.05

0

0.05

time

x=1m

B-Scan

-2 -1 0 1 2

0

1

2

3

4

5

Lateral dis tance [m]

Dep

th [m

]

-2 -1 0 1 2

0

1

2

3

4

5

Summary

• A novel diagonal tensor approximation has been developed to accelerate forward and inversion solutions.

• Experimental data inversion confirms high fidelity and super-resolution of the method.

• Electromagnetic and acoustic imaging of tunnels has been simulated.

• Optimal configuration for tunnel imaging is under investigation.

A Fast Inverse Solver and Tunnel ImagingOutlineIntroduction: The ProblemLayered Media in Sensing and Imaging ProblemsConflicting Factors in EM Imaging: Resolution and Probing VolumeMaxwell’s Equations for Layered MediaElectrical Field Integral EquationOutlineScattering Approximations to Avoid Matrix InversionThe Diagonal Tensor Approximation (DTA)Approximate the scattering tensor as a diagonal tensorNow noting the Green’s function is highly peaked at the source point, inside the integrand we can approximateObservationsOutlineInverse Solver Based on DTAExample of Hybrid DTA/BCGS-FFT Method for InversionOutlineImaging from Measured DataImaging Results for Dielectric ConstantNUFFT Imaging of 3D Objects3D Imaging Results (GT plywood)OutlineGEOMETRY OF TUNNELSingle-Frequency ImagingMulti-Frequency Imaging(Low-Frequency Results as Initial Solution of High-Frequency Inversion)Multi-Frequency Imaging (Cont.)THEORETICAL DATA ERRORWave Propagation for One SourceMovie of Wave PropagationWaveforms at Different LocationsB-ScanSummary

a fast inverse solver and tunnel imaginglcarin/liu8.3.05.pdfaug 03, 2005 · • inversion of...

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