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GPR IMAGING USING COMPRESSED MEASUREMENTS A presentation by Prof. James H. McClellan School of Electrical and Computer Engineering Georgia Institute of Technology 26-Feb-2009 Compressive Sensing Compressive Subsurface Imaging with GPR

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GPR IMAGING USING COMPRESSED MEASUREMENTS

A presentation by Prof. James H. McClellan

School of Electrical and Computer Engineering Georgia Institute of Technology

26-Feb-2009

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

Acknowledgements

Dr. Ali Cafer Gürbüz
Prof. Waymond Scott

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

Motivation

Currently more than 110 million active landmines worldwide.

http://www.un.org/Depts/dha/mct/facts.htm

Multiple sensing modes

Metal Detectors: Electro-Magnetic Induction (EMI) Ground Penetrating Radar (GPR) Arrays Seismic Arrays that sense Surface Waves

Imaging over small areas (a few m2) Would be good to have...

lower data acquisition times to move faster

mobile (robotic) sensor systems

cheaper (analog) hardware and processing

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

CS Promises More Efficient Data Acquisition

We wish to form an “image” of the subsurface, b ∈ ℜn, from m measurements taken by a scanning sensor. yk =< x, φk >, k = 1, ..., m y = Φx (measurements) x = Ψb (sensor model, x ∈ ℜs) CS tells us: Possible with m ≪ n, if the image is sparse Use inner products, which could be random sampling Reconstruction requires computation

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

Robust Compressive Sensing

Signals are generally noisy. A realistic model for the measurements y = Φx + z zk i.i.d N(0, σ2) Dantzig Selector (Candes and Tao) If the Restricted Isometry Property holds ˆ b = argmin b1 s.t. AT(y − Ab)∞ < ǫNσ. where A = ΦΨ Selecting ǫN = √2 log N makes the true b feasible with high probability.

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results Compressive Imaging Theory

GPR Antenna System

Lab System Geometry of GPR Acquisition

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results Compressive Imaging Theory

GPR Imaging via BackProjection (BP)

Raw Data Surface Removed BackProjected (BP) Image

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results Compressive Imaging Theory

GPR Data Model (for the Ψ Matrix)

A point target model is assumed. Targets don’t interact so superposition is valid. Time Domain

ζi(t) = A s(t − τi(p))

Stepped Frequency system

ζi(ℓ) =

Aσ S(R(p))ej2πfℓ(t−τi(p))

where fℓ = f0 + ℓ∆f with ℓ = 0, 1, 2, ...L−1 when Ae−j2π(f0+ℓ∆f )t is transmitted.

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results Compressive Imaging Theory

Creating Dictionary for GPR Data at i-th Scan Position

A discrete inverse operator can be created by discretizing the spatial domain target space and synthesizing the GPR model data for each discrete spatial position. ζi =

P

b(k) exp [−jω(t − τi(πk))] ζi(f ) = Ψib where [Ψi]j = exp [−jω(t − τi(πj))]

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results Compressive Imaging Theory

Compressive Sensing Data Acquisition

In CS, linear projections of ζi onto a second set of basis vectors φim, m = 1, 2, ...M are measured. In matrix form for the ith aperture point is βi = Φiζi = ΦiΨib

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results Compressive Imaging Theory

GPR Imaging with CS (the Ψ Matrix)

We use L scan positions and form a “super problem” with the matrices Ψ = [ΨT

1 , . . . , ΨT L ]T , and Φ = diag{Φ1, . . . , ΦL},

and the measurements β = [βT

1 , . . . , βT L ]T

ˆ b = argmin b1 s.t. β = ΦΨb For noisy data we might solve either ˆ b = arg min b1 s.t. AT(β − Ab)∞ < ǫ1

min b1 s.t. β − Ab2 < ǫ2 where A = ΦΨ.

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

List of Simulation/Experimental Results

Simulation Results

Comparison of standard backprojection to CS imaging in terms

f generated image quality and number of measurements used

for both time/frequency domains. Effect of random spatial sampling on the generated images Performance in varying noise levels Ability to resolve closely spaced targets

Experimental Results using Lab Data

Time-Domain and Stepped Frequency-Domain Imaging of a 1" metal sphere in air Imaging of multiple buried targets

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

Time Domain Imaging: Measurements

X Depth 5 10 15 20 25 30 5 10 15 20 25 30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Target space

X Time Index 5 10 15 20 25 30 100 200 300 400 500 −35 −30 −25 −20 −15 −10 −5

Space-Time domain data (SNR = −5 dB) 512 × 30

X Measurement Index 5 10 15 20 25 30 5 10 15 20 −4 −2 2 4 6 x 10

−5

Compressive Measurements 20 × 30

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

Time Domain Imaging: Images

X Depth 5 10 15 20 25 30 5 10 15 20 25 30 −35 −30 −25 −20 −15 −10 −5

Compressive Sensing (20/512 of the data)

X Z 5 10 15 20 25 30 5 10 15 20 25 30 −35 −30 −25 −20 −15 −10 −5

Backprojection (All the data)

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

Frequency Domain Imaging: Measurements

5 10 15 20 25 30 5 10 15 20 25 30

Target space

X(cm)

Freq. Index

5 10 15 20 25 30 10 20 30 40 50 60 70 80 90 100 0.5 1 1.5 2 2.5 3 3.5 x 10

−3

Space-frequency data (SNR = 0 dB) 100 × 30

X(cm)

Freq. Index

5 10 15 20 25 30 10 20 30 40 50 60 70 80 90 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Measured Frequencies (black) 20 × 30

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

Frequency Domain Imaging: Images

X(cm) Depth(cm) 5 10 15 20 25 30 5 10 15 20 25 30 −35 −30 −25 −20 −15 −10 −5

BP w/ all freq. data

X(cm) Depth(cm) 5 10 15 20 25 30 5 10 15 20 25 30 −35 −30 −25 −20 −15 −10 −5

BP w/ randomly selected data (20%)

X(cm) Depth(cm) 5 10 15 20 25 30 5 10 15 20 25 30 −35 −30 −25 −20 −15 −10 −5

CS w/ randomly selected data (20%)

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

Random Spatial Sampling & Random Frequencies

X(cm)

Freq. Index

5 10 15 20 25 30 10 20 30 40 50 60 70 80 90 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Measured (20 × 10) Space-frequency data

X(cm) Depth(cm) 5 10 15 20 25 30 5 10 15 20 25 30 −35 −30 −25 −20 −15 −10 −5

BP Result

X(cm) Depth(cm) 5 10 15 20 25 30 5 10 15 20 25 30 −35 −30 −25 −20 −15 −10 −5

CS Result

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

Increased Resolution for CS

−20 20 40 50 60 0.5 1 X(cm) Height(cm) −20 20 40 50 60 0.5 1 X(cm) Height(cm)

Backprojection

−20 20 40 50 60 0.5 1 X(cm) Height(cm) −20 20 40 50 60 0.5 1 X(cm) Height(cm)

Compressive Sensing Frequency Range: 3–5 GHz and resolution limit is 7.5 cm in air

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

Behavior in Noise (One Reflector)

Variance of target positions vs. SNR

−30 −20 −10 10 20 −60 −50 −40 −30 −20 −10 10 20 30 SNR(dB) Variance(dB) M=5 M=10 M=20 M=30 BP

Variance of created images vs. SNR

−30 −20 −10 10 20 0.2 0.4 0.6 0.8 1 1.2 1.4 SNR(dB) Image Variance M=5 M=10 M=20 M=30 M=50 SBP

BP uses M = 220 measurements

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

Behavior vs. Number of Reflectors

Success vs. # Targets

10 20 30 40 50 60 0.2 0.4 0.6 0.8 1 M PSR P=1 P=2 P=3 P=4 P=6 P=8 P=10

Success vs. SNR

−30 −20 −10 10 20 0.2 0.4 0.6 0.8 1 SNR(dB) PSR M=5 M=10 M=20 M=30 M=50 20 / 29

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

List of Simulation/Experimental Results

Experimental Results using Lab Data

Time-Domain and Stepped Frequency-Domain Imaging of a 1" metal sphere in air Imaging of multiple buried targets

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

GPR Air Experiments

Measurement Setup

X(cm) Time(s) −50 50 50 100 150 200 −50 −40 −30 −20 −10

Space-Time domain data (220 × 70) Space-Time Compressive Measurements (10 × 70)

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

Imaging of 1" Metal Sphere: Time-Domain

Compressive Sensing (10 × 70) Backprojection (220 × 70)

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

1-inch Metal Sphere: Stepped Frequency

X(cm) Frequency (GHz) −50 50 1 2 3 4 5 6 7 8 0.5 1 1.5 2 2.5 x 10

−3

Space-frequency data

Z(cm) −30 −20 −10 10 20 30 30 40 Z(cm) −30 −20 −10 10 20 30 30 40 Z(cm) X(cm) −30 −20 −10 10 20 30 30 40 −35 −30 −25 −20 −15 −10 −5

Results for random freq. measurements 30 out of 379 frequencies are measured

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

Buried Target Experiments

Picture of Buried Targets Burial Map

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

2D Slice Imaging: Frequency Domain

X(cm) Z(cm) −50 50 −20 −15 −10 −5 −25 −20 −15 −10 −5 X(cm) Frequency (GHz) −60 −40 −20 20 40 60 1 2 3 4 5 6 7 8 x 10

X(cm) Z(cm) −50 50 −20 −15 −10 −5 −25 −20 −15 −10 −5

Backprojection image uses all 379 frequencies. CS uses only 100 randomly selected frequencies.

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

2D Slice Imaging: Time Domain

Backprojection image uses 128 samples per scan point; CS uses only 15 projections.

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

3D Subsurface Imaging Results

Backprojection

Y(cm) X(cm) −50 50 −60 −40 −20 20 40 60 −25 −20 −15 −10 −5

Compressive Sensing

Y(cm) X(cm) −50 50 −60 −40 −20 20 40 60 −25 −20 −15 −10 −5

3D iso-surface image of the selected region

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Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results

Questions

jim.mcclellan@ece.gatech.edu

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