Scattered multi-static array imaging and eigenvalue problems - - PDF document

SLIDE 1

Scattered multi-static array imaging and eigenvalue problems

Wagner B. Muniz Department of Mathematics Federal University of Santa Catarina (UFSC) w.b.muniz@ufsc.br Brazilian Mathematics Colloquium 2013

SLIDE 2

Inverse scattering (acoustics, EM)

i

u

s

u

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) = ℜe

{

u(x)e−iωt} , EM: (E, H)(x, t) = ℜe

{

(E, H)(x)e−iωt}

1

SLIDE 3

Inverse scattering (acoustics, EM)

i

u

s

u

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

SLIDE 4

Outline

• 1. Approaches for inverse scattering:

− Traditional methods − Qualitative sampling methods

• 2. Forward scattering

− Radiating (outgoing) solutions − Rellich’s lemma

• 3. The inverse scattering problem

− Far field operator − Herglotz wave function

• 4. Sampling formulation

− Fundamental solution − Linear sampling method − Factorization method

• 5. Failure (?) at eigenfrequencies

− Modified Jones/Ursell far-field operator − Object classification at eigenfrequencies

• 6. Applications

3

SLIDE 5

• 1. Approaches for inverse scattering

Qualitative/sampling schemes Goal: try to

• recover shape as opposed to physical properties
• recover shape and possibly some extra info

Fixed frequency of incidence ω:

i

u

s

u

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

SLIDE 6

Inverse Scattering Methods

Nonlinear 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 linearizations

be 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

SLIDE 7

• 2. Forward wave propagation 101

Wave equation (pressure p = p(x, t), velocity c) ∂2 ∂t2 p − c2△p = 0 Time-harmonic dependency: ω = frequency p(x, t) = ℜe

{

u(x)e−iωt} Helmholtz (reduced wave) equation: (−i ω)2u − c2 △u = 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)} = ℜe

{

eikx·de−iωt} Plane wave ui(x) = eikx·d satisfies −△ui − k2ui = 0 em R3, where k = ω/c0

6

SLIDE 8

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

SLIDE 9

Sommerfeld/Silver-M¨ uller 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, lim

R→∞

∫

r:=|x|=R

• ∂

∂rus − ikus

• 2

ds(x) = 0 (Sommerfeld radiation condition) Here x = |x|ˆ x = rˆ x, ˆ x ∈ Ω Notation: Ω unit sphere Sommerfeld: ”... energy does not propagate from infinity into the domain ...”

8

SLIDE 10

Radiating solutions II

Sommerfeld radiation condition on us −△us − k2us = 0 in R3 \ D, Bus = f := −Bui on ∂D, lim

R→∞

∫

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∞(ˆ x) + O

(

1 |x|2

)

0.5 1 1.5 30 210 60 240 90 270 120 300 150 330 180

9

SLIDE 11

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

∞(ˆ

x) = v2

∞(ˆ

x) for infinitely many ˆ x ∈ Ω then vs

1(x) = vs 2(x), x ∈ R3 \ D.

That is, if v1

∞(ˆ

x) = 0 for ˆ x ∈ Ω then vs

1(x) = 0, x ∈ R3 \ D.

Remark: R >> 1,

∫

|x|=R |vs(x)|2ds(x) ≈

∫

Ω |v∞(ˆ

x)|2ds(ˆ x)

10

SLIDE 12

• 3. The Inverse Scattering Problem

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∞(ˆ x, d), ˆ 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

SLIDE 13

Far field operator (data operator): F : L2(Ω) → L2(Ω) (Fg)(ˆ x) :=

∫

Ω u∞(ˆ

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)

∫

Ω us(x, d)g(d)ds(d) = 0, x ∈ R3 \ D

−B

∫

Ω ui(x, d)g(d)ds(d) = 0, x ∈ ∂D

that is, − Bvg(x) = 0, x ∈ ∂D where Herglotz wave function: vg(x) :=

∫

Ω eikx·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

SLIDE 14

Far field operator (data operator): (SKP) F : L2(Ω) → L2(Ω) (Fg)(ˆ x) :=

∫

Ω u∞(ˆ

x, d)g(d)ds(d) Obs.: F normal in the Dirichlet, Neumann and non-absorbing medium cases

13

SLIDE 15

Herglotz wave function

Superposition with kernel g

∫

Ω

eikx·dg(d)ds(d) ❀

∫

Ω

us(x, d)g(d)ds(d) ❀

∫

Ω

u∞(ˆ x, d)g(d)ds(d) ∥ ∥ ∥ vg(x) ❀ vs(x) ❀ (Fg)(ˆ x)

By superposition the incident Herglotz func- tion vg(x) induces the far field pattern (Fg)(ˆ 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, then Φ(·, z) has far field pattern Φ∞(ˆ x, z) = 1 4πe−ikˆ

x·z

14

SLIDE 16

• 4. Linear Sampling Method (LSM)

Far field equation Let z ∈ R3. Consider Fgz(ˆ x) = Φ∞(ˆ 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 2 −3 −2 −1 1 2 3 10 20 30 40 50 60 k =4 −2 2 −3 −2 −1 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

SLIDE 17

LSM theorem

(SKP) 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 a gz ∈ 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

SLIDE 18

LSM motivation (Dirichlet)

• Assume u∞(ˆ

x, d) known for ˆ x, d ∈ Ω corresponding to ui(x, d) = eikx·d

• Let z ∈ D and g = gz ∈ L2(Ω) solve Fg = Φ∞(·, z):

∫

Ω u∞(ˆ

x, d)g(d)ds(d) = Φ∞(ˆ x, z)

• Rellich’s lemma:

∫

Ω us(x, d)g(d)ds(d) = Φ(x, z),

x ∈ R3 \ D

• Boundary condition us(x, d) = −eikx·d on ∂D implies:

−

∫

Ω eikx·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

SLIDE 19

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

SLIDE 20

Factorization: characterizes range of G (and therefore D by the previous theorem) in terms

• f 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 < ∞

SLIDE 21

(SKP)

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

SLIDE 22

• 5. Failure at eigenfrequencies

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 we devise an algorithm that works for

all k?

19

SLIDE 23

Modified far field operator

(SKP)

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

∞(ˆ

x, d) := 4π ik

∞

∑

n=0 n

∑

m=−n

amnY m

n (ˆ

x)Y m

n (d),

where Y m

n

= spherical harmonics

20

SLIDE 24

Modified far field operator (SKP) (F0 g)(ˆ x) :=

∫

Ω

(

u∞(ˆ x, d) − u0

∞(ˆ

x, d)

)

g(d)ds(d) Each term of the series of far field patterns 4π ik amnY m

n (d)Y m n (ˆ

x) corresponds to radiating Helmhotz solutions of the form us,0

mn(x) = 4πinamnY m n (d) h(1) n

(k|x|)Y m

n (ˆ

x)

21

SLIDE 25

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

SLIDE 26

5.2. Distinguishing obstacles at e-values

Claim: at interior eigenfrequencies, imaging ||gz|| indicates the zeros of the corresponding eigenfunctions (easy to see in the 2D/3D spher- ical case) Corollary: Given the far field data for k in an interval (containing e-freq.) then one can clas- sify 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

SLIDE 27

6.1. 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 5 10 15 −15 −10 −5 5 10 15 FM −15 −10 −5 5 10 15 −15 −10 −5 5 10 15

Cavity

FM −15 −10 −5 5 10 15 −15 −10 −5 5 10 15

24

SLIDE 28

References:

The factorization method for inverse problems (2008), Kirsch and Grinberg, Springer Qualitative methods in inverse scattering the-

• ry (2007), Cakoni and Colton , Springer

Stream of papers in Inverse problems, journal (upcoming transmission eigenvalue issue) Um problema inverso para detec¸ cao de objetos e a sua rela¸ cao com frequencias irregulares, Muniz, preprint

25

SLIDE 29

Near field inversions: 3D e 2D

26

SLIDE 30

3D inversions: synthetic data

Homogeneous background (free-space)

(Synthetic data without interaction between spheres!!) f = 10kHz, k− = k+ ≈ 2.1 · 10−4 144 meas./source points along the plane Γ = [−0.2, 0.2] × [−0.2, 0.2] × {0.05} Thanks to Ch. Schneider (Mainz) Reconstruction in perspective Zoomed reconstruction

27

SLIDE 31

What it seems we saw ...

Sampling methods

• No forward solver
• No a priori info on the scatterer
• No asymptotic approximation:
• Too much data!

– full model is considered

• Robust within various setting
• Exploitation of eigenfrequencies

28

SLIDE 32

Cross section at depth of 10 cm

−0.3 −0.2 −0.1 0.1 0.2 0.3 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

29

SLIDE 33

Multiple scatterers

−0.4 −0.2 0.2 0.4 −0.3 −0.2 −0.1 −0.4 −0.2 0.2 0.4 −0.3 −0.2 −0.1 −0.4 −0.2 0.2 0.4 −0.3 −0.2 −0.1 −0.4 −0.2 0.2 0.4 −0.3 −0.2 −0.1 −0.4 −0.2 0.2 0.4 −0.3 −0.2 −0.1 −0.4 −0.2 0.2 0.4 −0.3 −0.2 −0.1 −0.4 −0.2 0.2 0.4 −0.3 −0.2 −0.1 −0.4 −0.2 0.2 0.4 −0.3 −0.2 −0.1

30

SLIDE 34

Further examples Plastic only mine.

Linear sampling −0.4 −0.2 0.2 0.4 −0.3 −0.2 −0.1 Factorization −0.4 −0.2 0.2 0.4 −0.3 −0.2 −0.1

σout = σin = 10−1 (weakly conductive) ϵin

r = 3, ϵout r

= 3 (plastic/TNT) U-shape metal

Linear sampling −0.4 −0.2 0.2 0.4 −0.3 −0.2 −0.1 Factorization −0.4 −0.2 0.2 0.4 −0.3 −0.2 −0.1

σD = 106 (high) ϵr = 2.

31

SLIDE 35

• 7. Inhomogeneous medium scattering

(ff) Inhomogeneous medium (penetrable): ∆u + k2n(x)u = 0 in R3, where u = ui + us, ui(x, d) = eikx·d, |d| = 1 and us(x, d) is radiating such that as |x| → ∞ us(x, d) = eik|x| |x| u∞(ˆ x, d) + O( 1 |x|2) Inhomogeneity D = {x ∈ R3 : n(x) ̸= 1} = inhomogeneity EM: n(x) = ϵr + iσ/ωϵ0 Acoustics: n(x) = c2

0/c(x)2

32

SLIDE 36

Interior transmission problem

(ff) Let g ∈ L2(Ω) with Fg = 0:

∫

Ω u∞(ˆ

x, d)g(d)ds(d) = 0, ˆ x ∈ Ω Rellich’s Lemma

∫

Ω us(x, d)g(d)ds(d) = 0,

x ∈ R3 \ D Since us(x, d) = u(x, d) − eikx·d

∫

Ω u(x, d)g(d)ds(d)

• =:w(x)

=

∫

Ω eikx·dg(d)ds(d)

• =:v(x)=Herlgotz

, x ∈ R3 \ D Since ∆u + k2nu = 0 ∆w + k2n(x)w = 0, ∆v + k2v = 0 in D w = v in R3 \ D

Equivalent to the

Interior transmission problem (ITP) ∆w + k2n(x)w = 0, ∆v + k2v = 0 in D w = v, ∂w ∂ν = ∂v ∂ν

• n ∂D

33

SLIDE 37

Transmission eigenvalues

(ff) Definition: values of k for which there exist a nontrivial solution pair v, w of the interior transmission eigenvalue problem (ITP).

• Transmission eigenvalues do not exist if ℑm n > 0

(Green’s Theorem).

• Transmission eigenvalues do exist if ℑm n = 0 and

n = f(|x|) (spherically symmetric)

Open question: Do transmission eigen- values exist in the general case? Yes, Sylvester and Paiv¨

arinta and from then on:

Let m ̸= 0 in D and Ψ := w − v. Find Ψ ∈ H2

0(D) (equivalent ITP)

(∆ + k2n)m−1(∆ + k2)Ψ = 0 in D i.e. (L0 + λL1 + λ2L2)Ψ = 0 in D

34

SLIDE 38

Sampling × interior eigenvalues

(ff) Theor. F : L2(Ω) → L2(Ω) is injective with dense range iff transmission eigenvalues (for which v is Herglotz) do not exist. Sampling failure at transmission eigenval- ues. Analogous to Dirichlet or Neumann cases. Similar modification with F0

35

SLIDE 39

3D spherically stratified medium

(ff)

144 incident and measurement directions in S2 n(r) := 1 + 1 2 cos 5πr 2 , 0 ≤ r ≤ 1. Inhomogeneous medium reconstructions: Plane x3 = 0.

20 40 60 k =2.76 −2 −1 1 2 −2 −1 1 2 20 40 60 k =2.77 −2 −1 1 2 −2 −1 1 2 20 40 60 k =2.78 −2 −1 1 2 −2 −1 1 2 20 40 60 k =2.76 −2 −1 1 2 −2 −1 1 2 20 40 60 k =2.77 −2 −1 1 2 −2 −1 1 2 20 40 60 k =2.78 −2 −1 1 2 −2 −1 1 2

k = 2.76, 2.77 and 2.78 (all close to a transmission eigenvalue: 2.776)

Stopp.

36

SLIDE 40

Injectivity of F0, Dirichlet case

• Theor. F0 : L2(Ω) → L2(Ω) is injective with

dense range.

Outline of the injectivity proof: F0g = 0, g = ∑ βmnY m

n (ˆ

x) ∈ L2(Ω),

∫

Ω

u∞(ˆ x, d)g(d)ds(d) −

∫

Ω

u0

∞(ˆ

x, d)g(d)ds(d) = 0 where u0

∞(ˆ

x, d) := 4π

ik

∑∞

n=0

∑n

m=−n amnY m n (ˆ

x)Y m

n (d),

Rellich’s lemma implies

∫

Ω

us(x, d)g(d)ds(d) −

∫

Ω

us

0(x, d)g(d)ds(d) = 0 in R3 \ D

On ∂D we have −

∫

Ω

eikx·dg(d)ds(d)

• Herglotz

−

∫

Ω

us

0(x, d)g(d)ds(d) = 0 on ∂D

Let w(x) :=

∫

Ω

(eikx·d + us

0(x, d))g(d)ds(d)

= 4π

∑

inβmn

(

jn(kr) + amnh(1)

n (kr))

Let D ⊂ B(0, b), △w + k2w = 0 in B(0, b) \ D, w = 0 on ∂D Green’s identity to B(0, b) \ D & Wronskian relations imply:

∫

|x|=b

(w∂rw − w∂rw)ds =

∑

|βmn|2 ( 1 − |1 + 2amn|2) = 0 37