inverse wave scattering problems: fast algorithms, resonance and applications · 2016. 11. 22. ·...
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Inverse wave scattering
problems: fast algorithms,
resonance and applications
Wagner B. Muniz
Department of Mathematics
Federal University of Santa Catarina (UFSC)
w.b.muniz@ufsc.br
III Coloquio de Matematica da Regiao Sul
2014
Inverse scattering (acoustics, EM)
iu
su
D
ui(x) = known incident wave
us(x) = measured scattered wave
incident ui + scattered us = total field u
Time-harmonic assumption: ω = frequency
acoustics: p(x, t) = ℜeu(x)e−iωt
,
EM: (E,H)(x, t) = ℜe(E,H)(x)e−iωt
1
Inverse scattering (acoustics, EM)
iu
su
D
ui(x) = known incident wave
us(x) = measured scattered wave
Direct problem: Given D (and its physical
properties) describe the scattered field us
Inverse ill-posed problem : Determine the
support (shape) of D from the knowledge of
us far away from the scatterer (far field region)
2
Outline
1. Approaches for inverse scattering:
− Traditional methods
− Qualitative sampling methods
2. Forward scattering
− Radiating (outgoing) solutions
− Rellich’s lemma
3. Elements of inverse scattering theory
− Far field operator
− Herglotz wave function
4. Sampling formulation
− Fundamental solution
− Linear sampling method
− Factorization method
5. Resonant frequencies
− Modified Jones/Ursell far-field operator
− Object classification algorithm
6. Applications
− Real experimental data
− Buried obstacles detection
3
1. Approaches for inverse scattering
Qualitative/sampling schemesGoal: try to• recover shape as opposed to physical properties
• recover shape and possibly some extra info
Fixed frequency of incidence ω:
iu
su
D
Sampling: Collect the far field data u∞ (or the near
field data us) and solve an ill-posed linear integral
equation for each sample point z
4
Inverse Scattering MethodsNonlinear optimization methods Kleinmann,
Angell, Kress, Rundell, Hettlich, Dorn, Weng Chew, Ho-
hage, Lesselier ...
• need some a priori information− parametrization, # scatterers, etc
• flexibility w.r.t. data• need forward solver (major concern)• full wave model• inverse crimes not uncommon!
Asymptotic approximations (Born, iterated-Born, geometrical optics, time-reversal/mi-gration, ...) Bret Borden, Cheney, Papanicolaou, ...
• need a priori information so linearizationsbe applicable (not for resonance region)
• (mostly) linear inversion schemes• radar imaging with incorrect model?
Qualitative methods (sampling, Factoriza-tion, Point-source, Ikehata’s, MUSIC?...)Colton, Monk, Kirsch, Hanke-Bourgeois, Cakoni, Pot-
thast, Devaney, Hanke, Ikehata, Ammari, Haddar, ...
• no forward solver• no a priori info on the scatterer• no linearization/asymptotic approx.:
– full nonlinear multiple scattering model• need more data• do not determine EM properties (σ, ϵr)
5
2. Forward wave propagation 101
Wave equation
(pressure p = p(x, t), velocity c)
∂2
∂t2p− c2p = 0
Time-harmonic dependency:
ω = frequencyp(x, t) = ℜe
u(x)e−iωt
Helmholtz (reduced wave) equation:
(−iω)2u− c2u = 0 ⇒ −u− k2u = 0
where k = ω/c is the wavenumber.
Plane wave incidence
’Plane wave’ in the direction d, |d| = 1,
p(x, t) = cos k(x · d− c0t) = ℜeeikx·de−iωt
Plane wave ui(x) = eikx·d satisfies
−ui − k2ui = 0 em R3, where k = ω/c0
6
Forward scattering
Incident field (say plane wave or point source)
−ui − k2ui = f in R3, where k = ω/c0
Helmholtz equation for the total field
−u− k2u = 0 in R3 \D,Bu = 0 on ∂D,
Total field u = ui+ us,
us perturbation due to D
Boundary condition (impenetrable)
Bu := ∂νu+ iλu impedance (Neumann λ = 0)
= u Dirichlet/PEC
Analogous to Maxwell with
∇×∇×E − k2E = F in R3 \D
7
Sommerfeld/Silver-Muller conditions
Exterior boundary value problem for us
Uniqueness: us travels away from the obstacle
−us − k2us = 0 in R3 \D,Bus = f := −Bui on ∂D,
limR→∞
∫r:=|x|=R
∣∣∣∣ ∂∂rus − ikus∣∣∣∣2 ds(x) = 0
(Sommerfeld radiation condition)
Here x = |x|x = rx, x ∈ Ω
Notation: Ω unit sphere
Sommerfeld: ”... energy does not propagate
from infinity into the domain ...”
8
Radiating solutions II
Sommerfeld radiation condition on us
−us − k2us = 0 in R3 \D,Bus = f := −Bui on ∂D,
limR→∞
∫r:=|x|=R
∣∣∣∣ ∂∂rus − ikus∣∣∣∣2 ds(x) = 0
Asymptotic behavior of radiating solutions
Def. us is radiating if it satisifies– Helmholtz outside some ball and– Sommerfeld radiation condition
Theor. If us is radiating then
us(x) =eik|x|
|x|u∞(x) +O
(1
|x|2
)
0.5
1
1.5
30
210
60
240
90
270
120
300
150
330
180 0
9
Rellich’s lemma [1943]
Key tool in scattering theory:
Identical far field patterns
⇓Identical scattered fields
(in the domain of definition)
Rellich’s lemma (fixed wave number k > 0)
If v1∞(x) = v2∞(x) for infinitely many x ∈ Ω then
vs1(x) = vs2(x), x ∈ R3 \D.
That is, if v1∞(x) = 0 for x ∈ Ω then
vs1(x) = 0, x ∈ R3 \D.
Remark: R >> 1,∫|x|=R
|vs(x)|2ds(x) ≈∫Ω|v∞(x)|2ds(x)
10
3. Inverse Scattering Theory
Inverse problem: ill-posed and nonlinear
Given several incident plane waves with dir. d
ui(x, d) = eikx·d,
measure the corresponding far-field pattern
u∞(x, d), x ∈ Ω
and determine the support of D
Re
100 200 300
50
100
150
200
250
300
350
Im
100 200 300
50
100
150
200
250
300
350
11
Far field operator (data operator):
F : L2(Ω) → L2(Ω)
(Fg)(x) :=∫Ωu∞(x, d)g(d)ds(d)
Remark 1: F is compact (smooth kernel u∞)
Remark 2: F is injective and has dense range
whenever k2 = interior eigenvalue
Proof: Fg = 0 implies (Rellich)∫Ωus(x, d)g(d)ds(d) = 0, x ∈ R3 \D
−B∫Ωui(x, d)g(d)ds(d) = 0, x ∈ ∂D
that is, − Bvg(x) = 0, x ∈ ∂D
where Herglotz wave function:
vg(x) :=∫Ωeikx·dg(d)ds(d), kernel g ∈ L2(Ω)
so that vg satisfies the interior e-value problem
−vg−k2vg = 0 in D, Bvg = 0 on ∂D and
vg = 0, g = 0, if k2 = eigenvalue 12
Far field operator (data operator): ( )
F : L2(Ω) → L2(Ω)
(Fg)(x) :=∫Ωu∞(x, d)g(d)ds(d)
Obs.: F normal in the Dirichlet, Neumann and
non-absorbing medium cases
13
Herglotz wave function
Superposition with kernel g∫Ω
eikx·dg(d)ds(d) ;
∫Ω
us(x, d)g(d)ds(d) ;
∫Ω
u∞(x, d)g(d)ds(d)
∥ ∥ ∥
vg(x) ; vs(x) ; (Fg)(x)
By superposition the incident Herglotz func-tion vg(x) induces the far field pattern (Fg)(x)
The fundamental solution (R3):
Φ(x, z) :=eik|x−z|
4π|x− z|, x = z,
is radiating in R3 \ z.
Fixing the source z ∈ R3 as a parameter, thenΦ(·, z) has far field pattern
Φ(x, z) :=eik|x|
|x|Φ∞(x, z) +O
(1
|x|2
),
withΦ∞(x, z) =1
4πe−ikx·z
14
4. Linear Sampling Method (LSM)
Far field equation Let z ∈ R3. Consider
Fgz(x) = Φ∞(x, z)
It is solvable only in special cases, if z = z0 and
D is a ball centered at z0. In general a solution
doesn’t exist.
Ex. 2D Neumann obstacle: (k = 3.4, k = 4) k =3.4
−2 0 2
−3
−2
−1
0
1
2
3
10
20
30
40
50
60
k =4
−2 0 2
−3
−2
−1
0
1
2
3
10
20
30
40
50
60
z inside D, ||gz|| remains bounded
z outside D, ||gz|| becomes unbounded
Nevertheless the regularized algorithm is nu-
merically robust and the following approxima-
tion theorem holds
15
LSM theorem
( )
Theorem If −k2 = Dirichlet eigenvalue for
the Laplacian in D then
(1) For any ϵ > 0 and z ∈ D, there exists a gz ∈ L2(Ω)such that
- ∥Fgz −Φ∞(·, z)∥L2(Ω) < ϵ, and
- limz→∂D ∥gz∥L2(Ω) = ∞, limz→∂D ∥vgz∥H1(D) = ∞.
(2) For any ϵ > 0, δ > 0 and z ∈ R3 \ D, there exists agz ∈ L2(Ω) such that
- ∥Fgz −Φ∞(·, z)∥L2(Ω) < ϵ+ δ and
- limδ→0 ∥gz∥L2(Ω) = ∞, limδ→0 ∥vgz∥H1(D) = ∞
where vgz is the Herglotz function with kernel gz.
16
LSM motivation (Dirichlet)
• Assume u∞(x, d) known for x, d ∈ Ω corresponding to
ui(x, d) = eikx·d
• Let z ∈ D and g = gz ∈ L2(Ω) solve Fg = Φ∞(·, z):∫Ωu∞(x, d)g(d)ds(d) = Φ∞(x, z)
• Rellich’s lemma:∫Ωus(x, d)g(d)ds(d) = Φ(x, z), x ∈ R3 \D
• Boundary condition us(x, d) = −eikx·d on ∂D implies:
−∫Ωeikx·dg(d)ds(d) = Φ(x, z), x ∈ ∂D, z ∈ D.
If z ∈ D and z → x ∈ ∂D then ||g||L2(Ω) → ∞
since |Φ(x, z)| → ∞
Same analogy: Neumann, impedance, inho-
mogeneous medium
17
Factorization method (Dirichlet)
Generalized scattering problem: f ∈ H1/2(∂D)
∆v+ k2v = 0 in R3 \D,v = f on ∂D,
v radiating
Data to far-field operator: takes f into v∞
G : H1/2(∂D) → L2(Ω), f ; Gf := v∞
Theorem z ∈ D iff Φ∞(·, z) ∈ Range(G)
Proof: Rellich + singularity of Φ(·, z) at z.
18
Factorization: characterizes range of G (and
therefore D by the previous theorem) in terms
of the data operator F , i.e. in terms of the
singular system of F
Theorem Let k2 =Dirichlet e-value of −∆ in
D. Let σj, ψj, ϕj be the singular system of F .
Then
z ∈ D iff∞∑1
|(Φ∞(·, z), ψj)|2
σj<∞
( )
Factorization method (Dirichlet) II
Factorization of the far field operator:
F = −GS∗G∗
where S is the adjoint of the single layer po-
tential
Obs. This corresponds to solving in L2(Ω)
(F ∗F )1/4g = Φ∞(·, z)
i.e.
Range(G) = Range(F ∗F )1/4
5. And resonant frequencies?
2 Dirichlet eigenvalues (peanut)
Lack of injectivity of F
k =1.6805 k =2.6 k =3.0418
• Is it a true failure?
• Can we get some extra info about the
scatterer at eigenfrequencies?
• First an algorithm that works for all k.
19
Modified far field operator ( )
Back to Jones, Ursell (1960s), Kleinman & Roach
and Colton & Monk (1988, 1993)
Find a ball BR(0) of radius R > 0, BR ⊂ D.
Define amn, n = 0,1, ..., |m| ≤ n, such that
(1) |1+2amn| > 1 for all n = 0,1, . . . , , |m| ≤ n
(2)∞∑n=0
n∑m=−n
(2n
keR
)2n|amn| <∞,
R
O
D
Define a series of far field patterns
u0∞(x, d) :=4π
ik
∞∑n=0
n∑m=−n
amnYmn (x)Y mn (d),
where Y mn = spherical harmonics
20
Modified far field operator ( )
(F0 g)(x) :=∫Ω
(u∞(x, d)− u0∞(x, d)
)g(d)ds(d)
Each term of the series of far field patterns
4π
ikamnY mn (d)Y mn (x)
corresponds to radiating Helmhotz solutions of
the form
us,0mn(x) = 4πinamnY mn (d) h(1)n (k|x|)Y mn (x)
21
Modified LSM valid for all k > 0
Theor. F0 : L2(Ω) → L2(Ω) is
injective with dense range.
Theor. (as before with F0, without restriction on k)
Jones/Ursell modification F0: k =1.6805 k =2.6 k =2.8971
Before: k =1.6805 k =2.6 k =3.0418
22
Object classification at e-frequencies
Claim: at eigenfrequencies, imaging ||gz|| in-
dicates the zeros of the corresponding eigen-
functions (easy to see in the 2D/3D spherical
case)
Corollary: Given the far field data for
k ∈ [k0, k1] (containing e-freq.)
then one can classify a scatterer as either a
PEC (Dirichlet) or not.
Dirichlet k =4.3934 k =5 k =5.3551
Neumann k =2.7096 k =3 k =3.3694
23
6. Applications
Landmine detection: near field inversions
Real far-field 2D data inversions
24
Landmine detection
Carl Baum:
”... we detect everything,
we identify nothing! ”
Metal detectors : high rate of false alarms
(non landmine artifacts)
?sand
air
• high cost (due to false alarms) :
USD 3 to buy, USD 200–1000 to clear
• requires high level of detection accuracy (deminers
safety) as opposed to military demining
≈ 100 million landmines world-wide
≈ 2000 victims per month
25
Humanitarian Demining Project
(HuMin/MD: http://www.humin-md.de)
Our goal: Decrease the number of falsealarms through fast new imaging algorithms.
1. Local 3D imaging: Karlsruhe, Mainz,Cologne, Gottingen, & des Saarlandes2. Signal analysis3. Hardware and soil
Our frequency domain approach:• Factorization Method
(Kirsch, Grinberg, Hanke-Bourgeois)• Linear Sampling Method
(Colton, Kirsch, Monk, Cakoni)
(Multi-static/array data setting)
26
3D EM inversions: synthetic dataMulti-static measurement on 12 x 12 grid (40 x 40 cm)
Frequency 1 kHz, k− = k+ ≈ 2.1 · 10−5, PEC objects
Reconstruction in perspective
Zoomed reconstruction
27
2D inversions: synthetic data
Two-layered background. Frequency 10 kHz.
Soil EM properties: σ− = 10−3 S/m, ϵ−r = 10
k− ≈ 0.0063(1 + i) (δ = O(100m))k+ ≈ 2.1 · 10−4
30 meas./source points along Γ = [−0.4,0.4]× 0.05,
Two penetrable obstacles
−0.4 −0.2 0 0.2 0.4−0.3
−0.2
−0.1
0
−0.4 −0.2 0 0.2 0.4−0.3
−0.2
−0.1
0
σD = 105 (high), ϵDr = 8
U-shape metal
Linear sampling
−0.4 −0.2 0 0.2 0.4−0.3
−0.2
−0.1
0Factorization
−0.4 −0.2 0 0.2 0.4−0.3
−0.2
−0.1
0
σD = 106 (high) ϵr = 2.
28
Plastic only mine.
Linear sampling
−0.4 −0.2 0 0.2 0.4−0.3
−0.2
−0.1
0Factorization
−0.4 −0.2 0 0.2 0.4−0.3
−0.2
−0.1
0
σout = σin = 10−1 (weakly conductive)
ϵinr = 3, ϵoutr = 3 (plastic/TNT)
Metal trigger.
Linear sampling
−0.4 −0.2 0 0.2 0.4−0.3
−0.2
−0.1
0Factorization
−0.4 −0.2 0 0.2 0.4−0.3
−0.2
−0.1
0
Further multiple PEC scatterers
−0.4 −0.2 0 0.2 0.4−0.3
−0.2
−0.1
0
−0.4 −0.2 0 0.2 0.4−0.3
−0.2
−0.1
0
−0.4 −0.2 0 0.2 0.4−0.3
−0.2
−0.1
0
−0.4 −0.2 0 0.2 0.4−0.3
−0.2
−0.1
0
−0.4 −0.2 0 0.2 0.4−0.3
−0.2
−0.1
0
−0.4 −0.2 0 0.2 0.4−0.3
−0.2
−0.1
0
−0.4 −0.2 0 0.2 0.4−0.3
−0.2
−0.1
0
−0.4 −0.2 0 0.2 0.4−0.3
−0.2
−0.1
0
29
Experimental 2D far-field data
Free-space parameters
Frequency 10 GHz, λ = 3 cm, L = 15 cm
Ipswich data (US Air Force Research Lab)
Multi-static setting: 32 incident and measurement dir.
Aluminum triangle Plexiglas triangle
FM
−15 −10 −5 0 5 10 15−15
−10
−5
0
5
10
15FM
−15 −10 −5 0 5 10 15−15
−10
−5
0
5
10
15
CavityFM
−15 −10 −5 0 5 10 15−15
−10
−5
0
5
10
15
30
RemarkSuperposition of the array data via∫
Ωu∞(x, d)g(d)ds(d)
allows us to devise a criterion to determine whether asampling point z belongs to the scatterer.
• This is done by testing the data against the back-ground Green’s function (or dyadic in 3D)
Φ(x, z)
through a linear equation for each point z.
• Scattering data from an obstacle D is compatiblewith the field due to a point source when z is in-side D and not compatible when z is outside D(ranges...)
References:
The factorization method for inverse problems(2008), Kirsch and Grinberg, Springer
Qualitative methods in inverse scattering the-ory (2007), Cakoni and Colton , Springer
Inverse acoustic and EM scattering theory (2013),3rd ed., Colton and Kress, Springer
Stream of papers in Inverse problems journal
31
Recapping
Sampling methods
• No forward solver
• No a priori info on the scatterer
• No asymptotic approximation (full EM)
• Potentially fast
• Eigenfrequencies exploitable
• Robust within various settings
Drawbacks
• Too much data – multi-static setup
• Cannot easily incorporate extra info
• Does’t determine scatterer properties
• Needs background Green’s function
− Approximately
− Greens tensor in 3D
− Hankel transforms in the layered case
32
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