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 Simulation & Experimental Results Acknowledgements Dr. Ali Cafer Gürbüz Prof. Waymond Scott 2 / 29
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 m 2 ) Would be good to have... lower data acquisition times to move faster 1 mobile (robotic) sensor systems 2 cheaper (analog) hardware and processing 3 3 / 29
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 ∈ ℜ n , from m measurements taken by a scanning sensor. y k = < x , φ k >, k = 1 , ..., m y = Φ x (measurements) (sensor model, x ∈ ℜ s ) x = Ψ b CS tells us: Possible with m ≪ n , if the image is sparse Use inner products, which could be random sampling Reconstruction requires computation 4 / 29
Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results Robust Compressive Sensing Signals are generally noisy. A realistic model for the measurements z k i . i . d N ( 0 , σ 2 ) y = Φ x + z Dantzig Selector (Candes and Tao) If the Restricted Isometry Property holds ˆ � A T ( y − Ab ) � ∞ < ǫ N σ. b = argmin � b � 1 s . t . where A = ΦΨ Selecting ǫ N = √ 2 log N makes the true b feasible with high probability. 5 / 29
Compressive Sensing Compressive Subsurface Imaging with GPR Compressive Imaging Theory Simulation & Experimental Results GPR Antenna System Lab System Geometry of GPR Acquisition 6 / 29
Compressive Sensing Compressive Subsurface Imaging with GPR Compressive Imaging Theory Simulation & Experimental Results GPR Imaging via BackProjection (BP) Raw Data Surface Removed BackProjected (BP) Image 7 / 29
Compressive Sensing Compressive Subsurface Imaging with GPR Compressive Imaging Theory Simulation & Experimental Results GPR Data Model (for the Ψ Matrix) A point target model is assumed. Targets don’t interact so superposition is valid. Time Domain ζ i ( t ) = A s ( t − τ i ( p )) Stepped Frequency system S ( R ( p )) e j 2 π f ℓ ( t − τ i ( p )) A σ ζ i ( ℓ ) = where f ℓ = f 0 + ℓ ∆ f with ℓ = 0 , 1 , 2 , ... L − 1 when Ae − j 2 π ( f 0 + ℓ ∆ f ) t is transmitted. 8 / 29
Compressive Sensing Compressive Subsurface Imaging with GPR Compressive Imaging Theory Simulation & Experimental Results 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. P � ζ i = b ( k ) exp [ − j ω ( t − τ i ( π k ))] k = 1 ζ i ( f ) = Ψ i b where [Ψ i ] j = exp [ − j ω ( t − τ i ( π j ))] 9 / 29
Compressive Sensing Compressive Subsurface Imaging with GPR Compressive Imaging Theory Simulation & Experimental Results Compressive Sensing Data Acquisition In CS, linear projections of ζ i onto a second set of basis vectors φ im , m = 1 , 2 , ... M are measured. In matrix form for the i th aperture point is β i = Φ i ζ i = Φ i Ψ i b 10 / 29
Compressive Sensing Compressive Subsurface Imaging with GPR Compressive Imaging Theory Simulation & Experimental Results GPR Imaging with CS (the Ψ Matrix) We use L scan positions and form a “super problem” with the L ] T , and Φ = diag { Φ 1 , . . . , Φ L } , matrices Ψ = [Ψ T 1 , . . . , Ψ T and the measurements β = [ β T 1 , . . . , β T L ] T ˆ b = argmin � b � 1 β = ΦΨ b s . t . For noisy data we might solve either ˆ � A T ( β − Ab ) � ∞ < ǫ 1 b = arg min � b � 1 s.t. or min � b � 1 s . t . � β − Ab � 2 < ǫ 2 where A = ΦΨ . 11 / 29
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 of 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 12 / 29
Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results Time Domain Imaging: Measurements −5 x 10 6 0.9 −5 5 0.8 100 5 4 −10 Measurement Index 0.7 10 Time Index −15 2 200 0.6 Depth 15 10 0.5 −20 0 300 0.4 −25 20 −2 0.3 15 −30 400 0.2 25 −4 −35 0.1 500 20 30 5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30 X X X Target space Space-Time domain Compressive data (SNR = − 5 dB) Measurements 512 × 30 20 × 30 13 / 29
Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results Time Domain Imaging: Images −5 −5 5 5 −10 −10 10 10 −15 −15 Depth 15 15 −20 −20 Z −25 −25 20 20 −30 −30 25 25 −35 −35 30 30 5 10 15 20 25 30 5 10 15 20 25 30 X X Compressive Sensing Backprojection (20/512 of the data) (All the data) 14 / 29
Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results Frequency Domain Imaging: Measurements −3 x 10 10 3.5 10 0.9 5 20 20 0.8 3 30 30 0.7 10 2.5 Freq. Index 40 Freq. Index 40 0.6 15 50 2 50 0.5 60 60 0.4 1.5 20 70 70 0.3 1 80 80 0.2 25 0.5 90 90 0.1 30 100 100 5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30 X(cm) X(cm) Target space Space-frequency data Measured Frequencies (SNR = 0 dB) (black) 100 × 30 20 × 30 15 / 29
Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results Frequency Domain Imaging: Images −5 −5 −5 5 5 5 −10 −10 −10 10 10 10 −15 −15 −15 Depth(cm) Depth(cm) Depth(cm) 15 15 15 −20 −20 −20 −25 −25 −25 20 20 20 −30 −30 −30 25 25 25 −35 −35 −35 30 30 30 5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30 X(cm) X(cm) X(cm) BP w/ all freq. data BP w/ randomly CS w/ randomly selected data (20%) selected data (20%) 16 / 29
Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results Random Spatial Sampling & Random Frequencies 10 0.9 −5 −5 5 5 20 0.8 −10 −10 30 0.7 10 10 −15 −15 Freq. Index 40 0.6 Depth(cm) Depth(cm) 15 15 50 0.5 −20 −20 60 0.4 −25 −25 20 20 70 0.3 −30 −30 80 0.2 25 25 −35 −35 90 0.1 30 30 100 5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30 X(cm) X(cm) X(cm) Measured (20 × 10) BP Result CS Result Space-frequency data 17 / 29
Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results Increased Resolution for CS 1 1 0.5 0.5 0 0 60 60 20 20 50 50 0 0 40 −20 40 −20 Height(cm) Height(cm) X(cm) X(cm) 1 1 0.5 0.5 0 0 60 60 20 20 50 50 0 0 40 40 Height(cm) −20 Height(cm) −20 X(cm) X(cm) Backprojection Compressive Sensing Frequency Range: 3–5 GHz and resolution limit is 7.5 cm in air 18 / 29
Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results Behavior in Noise (One Reflector) Variance of target positions vs. SNR Variance of created images vs. SNR 30 1.4 M=5 M=5 20 M=10 M=10 1.2 M=20 M=20 10 M=30 M=30 1 Image Variance BP M=50 0 Variance(dB) SBP 0.8 −10 −20 0.6 −30 0.4 −40 0.2 −50 −60 0 −30 −20 −10 0 10 20 −30 −20 −10 0 10 20 SNR(dB) SNR(dB) BP uses M = 220 measurements 19 / 29
Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results Behavior vs. Number of Reflectors Success vs. # Targets Success vs. SNR 1 1 M=5 M=10 0.8 M=20 0.8 M=30 M=50 0.6 0.6 PSR PSR P=1 0.4 0.4 P=2 P=3 P=4 0.2 0.2 P=6 P=8 P=10 0 0 −30 −20 −10 0 10 20 0 10 20 30 40 50 60 SNR(dB) M 20 / 29
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 21 / 29
Compressive Sensing Compressive Subsurface Imaging with GPR Simulation & Experimental Results GPR Air Experiments −10 50 −20 Time(s) 100 −30 −40 150 −50 200 −50 0 50 X(cm) Measurement Setup Space-Time domain Space-Time data (220 × 70) Compressive Measurements (10 × 70) 22 / 29
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