GPR IMAGING USING COMPRESSED MEASUREMENTS

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1 GPR IMAGING USING COMPRESSED MEASUREMENTS A presentation by Prof. James H. McClellan School of Electrical and Computer Engineering Georgia Institute of Technology 26-Feb-09

2 Acknowledgements Dr. Ali Cafer Gürbüz Prof. Waymond Scott 2 / 29

3 Motivation Currently more than 110 million active landmines worldwide. 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... 1 lower data acquisition times to move faster 2 mobile (robotic) sensor systems 3 cheaper (analog) hardware and processing 3 / 29

4 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 = Φx (measurements) x = Ψb (sensor model, x R s ) 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

5 Robust Compressive Sensing Signals are generally noisy. A realistic model for the measurements y = Φx + z z k i.i.d N(0,σ 2 ) Dantzig Selector (Candes and Tao) If the Restricted Isometry Property holds ˆb = argmin b 1 s.t. A T (y Ab) <ǫ N σ. where A = ΦΨ Selectingǫ N = 2 log N makes the true b feasible with high probability. 5 / 29

6 GPR Antenna System Lab System Compressive Imaging Theory Geometry of GPR Acquisition 6 / 29

7 Compressive Imaging Theory GPR Imaging via BackProjection (BP) Raw Data Surface Removed BackProjected (BP) Image 7 / 29

8 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 ζ i (t) = A s(t τ i (p)) Stepped Frequency system ζ i (l) = Aσ S(R(p)) ej2πf l(t τ i(p)) where f l = f 0 +l f withl = 0, 1, 2,...L 1 when Ae j2π(f0+l f )t is transmitted. 8 / 29

9 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. P ζ i = b(k) exp [ jω(t τ i (π k ))] k=1 ζ i (f ) = Ψ i b where [Ψ i ] j = exp[ jω(t τ i (π j ))] 9 / 29

10 Compressive Imaging Theory Compressive Sensing Data Acquisition In CS, linear projections ofζ 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

11 Compressive Imaging Theory GPR Imaging with CS (the Ψ Matrix) We use L scan positions and form a super problem with the matrices Ψ = [Ψ T 1,...,ΨT L ]T, and Φ = diag{φ 1,...,Φ L }, and the measurementsβ= [β T 1,...,βT L ]T ˆb = argmin b 1 For noisy data we might solve either or where A = ΦΨ. s.t. β = ΦΨb ˆb = arg min b 1 s.t. A T (β Ab) <ǫ 1 min b 1 s.t. β Ab 2 <ǫ 2 11 / 29

12 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

13 Time Domain Imaging: Measurements Depth X Target space Time Index X Space-Time domain data (SNR = db) Measurement Index X Compressive Measurements x / 29

14 Time Domain Imaging: Images Depth X Compressive Sensing (/512 of the data) Z X Backprojection (All the data) / 29

15 Frequency Domain Imaging: Measurements x Freq. Index Freq. Index Target space Space-frequency data (SNR = 0 db) 100 Measured Frequencies (black) 15 / 29

16 Frequency Domain Imaging: Images Depth(cm) Depth(cm) Depth(cm) BP w/ all freq. data BP w/ randomly selected data (%) CS w/ randomly selected data (%) 16 / 29

17 Random Spatial Sampling & Random Frequencies Freq. Index Measured ( 10) Space-frequency data Depth(cm) BP Result Depth(cm) CS Result / 29

18 Increased Resolution for CS Height(cm) 40 Height(cm) Height(cm) 40 Backprojection Height(cm) 40 Compressive Sensing Frequency Range: 3 5 GHz and resolution limit is 7.5 cm in air 18 / 29

19 Behavior in Noise (One Reflector) Variance of target positions vs. SNR Variance of created images vs. SNR Variance(dB) M=5 M=10 M= M= BP SNR(dB) BP uses M = 2 measurements Image Variance M=5 M=10 M= M= M=50 SBP SNR(dB) 19 / 29

20 Behavior vs. Number of Reflectors PSR Success vs. # Targets P=1 0.4 P=2 P=3 P=4 0.2 P=6 P=8 P= M PSR Success vs. SNR M=5 M=10 M= M= M= SNR(dB) / 29

21 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

GPR Air Experiments 50 Time(s) 100 150 40 0 0 Measurement Setup 0 0 50

22 GPR Air Experiments 50 Time(s) Measurement Setup Space-Time domain data (2 70) Space-Time Compressive Measurements (10 70) 22 / 29

23 Imaging of 1" Metal Sphere: Time-Domain Compressive Sensing (10 70) Backprojection (2 70) 23 / 29

24 1-inch Metal Sphere: Stepped Frequency Z(cm) 40 x 10 3 Z(cm) Frequency (GHz) Space-frequency data Z(cm) Results for random freq. measurements out of 379 frequencies are measured 24 / 29

25 Buried Target Experiments Picture of Buried Targets Burial Map 25 / 29

26 2D Slice Imaging: Frequency Domain 1 2 x 10 9 Frequency (GHz) Z(cm) Z(cm) Backprojection image uses all 379 frequencies. CS uses only 100 randomly selected frequencies / 29

27 2D Slice Imaging: Time Domain Backprojection image uses 128 samples per scan point; CS uses only 15 projections. 27 / 29

3D Subsurface Imaging Results 60 Backprojection 60 Compressive Sensing 40 40 0 0 40 25

28 3D Subsurface Imaging Results 60 Backprojection 60 Compressive Sensing Y(cm) Y(cm) 3D iso-surface image of the selected region 28 / 29

29 Questions 29 / 29

Compressive Sensing for GPR Imaging

Compressive Sensing for GPR Imaging Compressive Sensing for GPR maging Ali Cafer Gurbuz James H. McClellan Waymond R. Scott Jr. School of Electrical and Computer Engineering Georgia nstitute of Technology Atlanta, GA 30330 AbstractThe theory