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

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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.

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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

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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