Distributed Sensor Imaging

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1 Distributed Sensor Imaging George Papanicolaou Department of Mathematics - Stanford University URL: In collaboration with: Gregoire Derveaux, INRIA, France Chrysoula Tsogka, University of Crete Liliana Borcea, Rice University G. Papanicolaou. Distributed sensor imaging. August 2, 27 1

2 Distributed Sensors for Structural Health Monitoring Objective: to monitor constantly a complex structure, for example an airplane or a bridge, with distributed sensors embeded in it, in order to detect and image defects efficiently sensor 1 p q We assume some knowledge of the structure without damage in the following sense: For a given set of N distributed sensors, the Response Matrix of the healthy structure is gathered: Sensor p is firing Echos are measured at all sensors q = 1,... N 9 = ; P pq(t) G. Papanicolaou. Distributed sensor imaging. August 2, 27 2

3 Structural Health Monitoring: Formulation of the problem Suppose now that some damage appears in this structure: 2... sensor defect 1 p q The Response Matrix of the damaged structure P d pq(t) is obtained with the same set of N distributed sensors The difference Response Matrix P pq (t) = P d pq(t) P pq(t) contains the echos coming from the defects only. G. Papanicolaou. Distributed sensor imaging. August 2, 27 3

4 Structural Health Monitoring: Formulation of the problem Goal: Image the defects - Detect their presence and their number - Find their location - Further, find their size, shape, etc... Some issues: - The background: Complexity: homogeneous, some inhomogeneities (scatterers), random medium Physical properties: dispersion (thin elastic structure), dissipation What do we know about the background? Assume nothing. - Sensor deployment: number, location, distance between them, properties,... - The defects: their number, location, size, shape,... G. Papanicolaou. Distributed sensor imaging. August 2, 27 4

5 Numerical setup y absorbing medium (PML) sensor small object (λ/2) defect bigger object (3λ) λ (,) x 5λ Wave equation in 2 dimensions, in free space. Computational domain 5λx5λ, with central wavelength λ = 1cm the reference wave speed c = 35m.s 1 The healthy structure contains 25 scatterers with Dirichlet Boundary conditions 12 sensors (4 rows of 3) located at x p, p = 1,... N We want to image 4 pointlike defects with Dirichlet Boundary conditions G. Papanicolaou. Distributed sensor imaging. August 2, 27 5

6 Numerical setup y absorbing medium (PML) sensor small object (λ/2) defect bigger object (3λ) λ 2 (9, 18) (8, 16) (12,17) (13,15) (,) x 5λ Wave equation in 2 dimensions, in free space. Computational domain 5λx5λ, with central wavelength λ = 1cm the reference wave speed c = 35m.s 1 The healthy structure contains 25 scatterers with Dirichlet Boundary conditions 12 sensors (4 rows of 3) located at x p, p = 1,... N We want to image 4 pointlike defects with Dirichlet Boundary conditions. G. Papanicolaou. Distributed sensor imaging. August 2, 27 6

7 Numerical setup y absorbing medium (PML) sensor small object (λ/2) defect bigger object (3λ) λ (,) x 5λ Wave equation in 2 dimensions, in free space. Computational domain 5λx5λ, with central wavelength λ = 1cm the reference wave speed c = 35m.s 1 The healthy structure contains 25 scatterers with Dirichlet Boundary conditions 12 sensors (4 rows of 3) located at x p, p = 1,... N We want to image 4 pointlike defects with Dirichlet Boundary conditions. G. Papanicolaou. Distributed sensor imaging. August 2, 27 7

8 About the numerical simulations We use a Finite Element Time Domain code to solve the 2D wave equation. The space resolution is 32 points per wavelength. Perfectly matched layeres (PML) are used to simulate the propagation in free space. It takes about 1:3 hour on a workstation to produce a synthetic Response Matrix. The probing pulse is ultrawideband: second derivative of a gaussian with central frequency ν = 35kHz and 13% bandwidth: f(t) = (2( 2πνt t ) 2 1) exp( ( 2πνt t ) 2 ) 1 Probing pulse 1 Fourier Transform of the probing pulse Time (relative to the central period) Frequency (Hz) x 1 5 The probing pulse in the time domain (left) and its Fourier Transfom (right) G. Papanicolaou. Distributed sensor imaging. August 2, 27 8

9 Traces Number of sensor Number of sensor Number of sensor Time (relative to central period) Healthy Time (relative to central period) Damaged Time (relative to central period) Differences Sensor # 6 is firing. Normalized traces measured at all sensors #1 to #12. X-axis: time, Y-axis: number of sensor G. Papanicolaou. Distributed sensor imaging. August 2, 27 9

10 Singular values.12 The singular values of the response matrix det3. 4 targets The first 5 singular values of the response matrix b P (ω) versus frequency Singular vectors and singular valuse of the response matrix bπ(ω)bv j (ω) = σ j (ω)bu j (ω), j = 1,..., N The singular value representation of the response matrix bπ(ω) = NX σ j (ω)bu j (ω)bv j (ω) j=1 G. Papanicolaou. Distributed sensor imaging. August 2, 27 1

11 Imaging algorithms: Travel time migration I KM ( y S ) = = Z dω ω ω o B/2 NX e iωτ( x r, y S ) r=1 s=1 NX NX P ( x r, x s, τ( x r, y S ) + τ( x s, y S )). r=1 s=1 NX bp ( x r, x s, ω)e iωτ( x s, y S ) Here y S is a search point in the region where we form the image and τ( x r, y S ) is the travel time of the waves from the array element x r to y S, in the background medium with sound speed c o. Since we assume a constant c o, τ( x r, y S ) = x r y S /c o. G. Papanicolaou. Distributed sensor imaging. August 2, 27 11

12 Imaging algorithms: Coherent interferometry (CINT) I CINT ( y S ; Ω d, b f) = Z Z dω dω ω ω o B ω 2 ω o B 2, ω ω Ω X X d fs b fs b Q( x b r, x s, ω; y S ) Q( x b r, x s, ω ; y S ), r,r s,s where the bar means complex conjugate and b Q( x r, x s, ω; y S ) is the Fourier transform of the trace P ( x r, x s, t) migrated to y S The b f s are the source weights. bq( x r, x s, ω; y S ) = b P ( x r, x s, ω)e iω [τ( x s, y S )+τ( x r, y S )]. (1) G. Papanicolaou. Distributed sensor imaging. August 2, 27 12

13 Performance of the CINT algorithm L 1, 1 and BV norms of the cint image versus frequency correlation width (Ω d ) 14 l1 norms vs Wd 8 g1 norms vs Wd 2 TV norms vs Wd Ω d = (INT) optimal Ω d (BV) Ω d = (KM) G. Papanicolaou. Distributed sensor imaging. August 2, 27 13

14 The DORT method In the travel time migration algorithm, replace the response matrix by D[ b Π(ω); d] = NX d j (ω)σ j (ω)bu j (ω)bv j (ω) j=1 where the d j (ω) are successively or 1 (Fink+Prada 1993). G. Papanicolaou. Distributed sensor imaging. August 2, 27 14

15 The DORT method : Singular vectors # 1 et # 2 Ω d = (INT) optimal Ω d (BV) Ω d = (KM) G. Papanicolaou. Distributed sensor imaging. August 2, 27 15

16 The DORT method : Singular vectors # 3 et # 4 Ω d = (INT) optimal Ω d (BV) Ω d = (KM) G. Papanicolaou. Distributed sensor imaging. August 2, 27 16

17 Optimal subspaces and optimal illumination Select successively the subspace weights d j (ω) and the illumination weights b f s. Then, I CINT ( y S ; Ω d, {d j }, { b f s }) G. Papanicolaou. Distributed sensor imaging. August 2, 27 17

18 Adaptive imaging. Starting with sing. vector # 1. 1st iterations Step 1: Optimal illumination. Right: optimal weights Optimal Illuminations vec 1 vec 2 vec 3 vec relative frequency Step 2 : Optimal subspace - FIRST ITERATION. Right: optimal set of waveforms. G. Papanicolaou. Distributed sensor imaging. August 2, 27 18

19 Adaptive imaging. Starting with sing. vector # 1. Following iterations Step 3. Optimal illumination. Right: optimal weights. G. Papanicolaou. Distributed sensor imaging. August 2, 27 19

20 Adaptive imaging. Starting with sing. vector # 1. Following iterations Optimal Illuminations vec 1 vec 2 vec 3 vec relative frequency Step 4 : Optimal subspace - SECOND ITERATION. Right: optimal set of waveforms Step 5. Optimal illumination. SECOND ITERATION. Right: optimal weights. G. Papanicolaou. Distributed sensor imaging. August 2, 27 2

Adaptive imaging. Starting with sing. vector # 4. 1st iterations 5 45 4 35 3 25 2 15 1 5 1 2 3 4 5 Step 1: Optimal illumination. Right: optimal weights..35.3.25 Optimal Illuminations vec 1 vec 2 vec 3 vec 4.

21 Adaptive imaging. Starting with sing. vector # 4. 1st iterations Step 1: Optimal illumination. Right: optimal weights Optimal Illuminations vec 1 vec 2 vec 3 vec relative frequency Step 2 : Optimal subspace - FIRST ITERATION. Right: optimal set of waveforms. G. Papanicolaou. Distributed sensor imaging. August 2, 27 21

22 Adaptive imaging. Starting with sing. vector # 4. Following iterations Step 3. Optimal illumination. Right: optimal weights. G. Papanicolaou. Distributed sensor imaging. August 2, 27 22

23 Adaptive imaging. Starting with sing. vector # 4. Following iterations Optimal Illuminations vec 1 vec 2 vec 3 vec relative frequency Step 4 : Optimal subspace - SECOND ITERATION. Right: optimal set of waveforms Step 5. Optimal illumination. SECOND ITERATION. Right: optimal weights. G. Papanicolaou. Distributed sensor imaging. August 2, 27 23

24 Summary and conclusions - Migration imaging (modern form of triangulation practiced widely today) does not work well in complex environments - DORT (unpotimized use of SVD) with travel time migration does not work well because it is unstable - CINT stabilizes the image by thresholding out clutter. Makes optimization possible - Combined optimal subspace and optimal illumination gives very stable images - Computational complexity of new algorithms that deal with environmental complexity is huge G. Papanicolaou. Distributed sensor imaging. August 2, 27 24

Imaging with Ambient Noise III

Imaging with Ambient Noise III George Papanicolaou Stanford University http://math.stanford.edu/ papanico Department of Mathematics Richard E. Phillips Lectures April 23, 2009 With Liliana Borcea (Rice