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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)
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