Inverse wave scattering problems: fast algorithms, resonance and applications

Transcription

1 Inverse wave scattering problems: fast algorithms, resonance and applications Wagner B. Muniz Department of Mathematics Federal University of Santa Catarina (UFSC) III Colóquio de Matemática da Região Sul 214

2 Inverse scattering (acoustics, EM) u s u i D u i (x) = known incident wave u s (x) = measured scattered wave incident u i + scattered u s = 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

3 Inverse scattering (acoustics, EM) u s u i D u i (x) = known incident wave u s (x) = measured scattered wave Direct problem: Given D (and its physical properties) describe the scattered field u s Inverse ill-posed problem : Determine the support (shape) of D from the knowledge of u s far away from the scatterer (far field region) 2

4 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

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: u s u i D Sampling: Collect the far field data u (or the near field data u s ) and solve an ill-posed linear integral equation for each sample point z 4

6 Inverse Scattering Methods Nonlinear optimization methods Kleinmann, Angell, Kress, Rundell, Hettlich, Dorn, Weng Chew, Hohage, 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/migration,...) 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, Factorization, Point-source, Ikehata s, MUSIC?...) Colton, Monk, Kirsch, Hanke-Bourgeois, Cakoni, Potthast, 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, ϵ r ) 5

7 2. Forward wave propagation 11 Wave equation (pressure p = p(x, t), velocity c) 2 t 2 p c2 p = Time-harmonic dependency: ω = frequency p(x, t) = Re { u(x)e iωt} Helmholtz (reduced wave) equation: ( i ω) 2 u c 2 u = u k 2 u = 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 c t)} = Re { e ikx d e iωt} Plane wave u i (x) = e ikx d satisfies u i k 2 u i = em R 3, where k = ω/c 6

8 Forward scattering Incident field (say plane wave or point source) u i k 2 u i = f in R 3, where k = ω/c Helmholtz equation for the total field u k 2 u = in R 3 \ D, Bu = on D, Total field u = u i + u s, u s perturbation due to D Boundary condition (impenetrable) Bu := ν u + iλu impedance (Neumann λ = ) = u Dirichlet/PEC Analogous to Maxwell with E k 2 E = F in R 3 \ D 7

9 Sommerfeld/Silver-Müller conditions Exterior boundary value problem for u s Uniqueness: u s travels away from the obstacle u s k 2 u s = in R 3 \ D, Bu s = f := Bu i on D, lim R r:= x =R r us iku s 2 ds(x) = (Sommerfeld radiation condition) Here x = x ˆx = rˆx, ˆx Ω Notation: Ω unit sphere Sommerfeld:... energy does not propagate from infinity into the domain... 8

10 Radiating solutions II Sommerfeld radiation condition on u s u s k 2 u s = in R 3 \ D, Bu s = f := Bu i on D, lim R r:= x =R r us iku s 2 ds(x) = Asymptotic behavior of radiating solutions Def. u s is radiating if it satisifies Helmholtz outside some ball and Sommerfeld radiation condition Theor. If u s is radiating then u s (x) = eik x x u (ˆx) + O ( ) 1 x

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 > ) If v 1 (ˆx) = v2 (ˆx) for infinitely many ˆx Ω then v s 1 (x) = vs 2 (x), x R3 \ D. That is, if v 1 (ˆx) = for ˆx Ω then v s 1 (x) =, x R3 \ D. Remark: R >> 1, x =R vs (x) 2 ds(x) Ω v (ˆx) 2 ds(ˆx) 1

12 3. Inverse Scattering Theory Inverse problem: ill-posed and nonlinear Given several incident plane waves with dir. d u i (x, d) = e ikx d, measure the corresponding far-field pattern u (ˆx, d), ˆx Ω and determine the support of D 35 Re 35 Im

13 Far field operator (data operator): F : L 2 (Ω) L 2 (Ω) (F g)(ˆx) := Ω u (ˆx, d)g(d)ds(d) Remark 1: F is compact (smooth kernel u ) Remark 2: F is injective and has dense range whenever k 2 interior eigenvalue Proof: F g = implies (Rellich) where Ω us (x, d)g(d)ds(d) =, x R 3 \ D B Ω ui (x, d)g(d)ds(d) =, x D that is, Bv g (x) =, x D Herglotz wave function: v g (x) := eikx d g(d)ds(d), kernel g L 2 (Ω) Ω so that v g satisfies the interior e-value problem v g k 2 v g = in D, Bv g = on D and v g =, g =, if k 2 eigenvalue 12

14 Far field operator (data operator): ( ) F : L 2 (Ω) L 2 (Ω) (F g)(ˆx) := Ω u (ˆx, d)g(d)ds(d) Obs.: F normal in the Dirichlet, Neumann and non-absorbing medium cases 13

15 Herglotz wave function Superposition with kernel g e ikx d g(d)ds(d) u s (x, d)g(d)ds(d) u (ˆx, d)g(d)ds(d) Ω Ω Ω v g (x) v s (x) (F g)(ˆx) By superposition the incident Herglotz function v g (x) induces the far field pattern (F g)(ˆx) The fundamental solution (R 3 ): Φ(x, z) := is radiating in R 3 \ {z}. eik x z 4π x z, x z, Fixing the source z R 3 as a parameter, then Φ(, z) has far field pattern ( ) Φ(x, z) := eik x 1 x Φ (ˆx, z) + O x 2, withφ (ˆx, z) = 1 4π e ikˆx z 14

16 4. Linear Sampling Method (LSM) Far field equation Let z R 3. Consider F g z (ˆx) = Φ (ˆx, z) It is solvable only in special cases, if z = z and D is a ball centered at z. In general a solution doesn t exist. Ex. 2D Neumann obstacle: (k = 3.4, k = 4) k =3.4 k = z inside D, g z remains bounded z outside D, g z becomes unbounded Nevertheless the regularized algorithm is numerically robust and the following approximation theorem holds 15

17 LSM theorem ( ) Theorem If k 2 Dirichlet eigenvalue for the Laplacian in D then (1) For any ϵ > and z D, there exists a g z L 2 (Ω) such that - F g z Φ (, z) L2 (Ω) < ϵ, and - lim z D g z L2 (Ω) =, lim z D v gz H 1 (D) =. (2) For any ϵ >, δ > and z R 3 \ D, there exists a g z L 2 (Ω) such that - F g z Φ (, z) L2 (Ω) < ϵ + δ and - lim δ g z L2 (Ω) =, lim δ v gz H 1 (D) = where v gz is the Herglotz function with kernel g z. 16

18 LSM motivation (Dirichlet) Assume u (ˆx, d) known for ˆx, d Ω corresponding to u i (x, d) = e ikx d Let z D and g = g z L 2 (Ω) solve F g = Φ (, z): u (ˆx, d)g(d)ds(d) = Φ (ˆx, z) Ω Rellich s lemma: Ω us (x, d)g(d)ds(d) = Φ(x, z), x R 3 \ D Boundary condition u s (x, d) = e ikx d on D implies: Ω eikx d g(d)ds(d) = Φ(x, z), x D, z D. If z D and z x D then g L 2 (Ω) since Φ(x, z) Same analogy: Neumann, impedance, inhomogeneous medium 17

19 Factorization method (Dirichlet) Generalized scattering problem: f H 1/2 ( D) v + k 2 v = in R 3 \ D, v = f on D, v radiating Data to far-field operator: G : H 1/2 ( D) L 2 (Ω), takes f into v f Gf := v Theorem z D iff Φ (, z) Range(G) Proof: Rellich + singularity of Φ(, z) at z. 18

20 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 k 2 Dirichlet e-value of in D. Let {σ j, ψ j, ϕ j } be the singular system of F. Then z D iff 1 (Φ (, z), ψ j ) 2 σ j <

21 ( ) Factorization method (Dirichlet) II Factorization of the far field operator: F = GS G where S is the adjoint of the single layer potential Obs. This corresponds to solving in L 2 (Ω) (F F ) 1/4 g = Φ (, z) i.e. Range(G) = Range(F F ) 1/4

22 5. And resonant frequencies? 2 Dirichlet eigenvalues (peanut) Lack of injectivity of F k =1.685 k =2.6 k =3.418 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

23 Modified far field operator ( ) Back to Jones, Ursell (196s), Kleinman & Roach and Colton & Monk (1988, 1993) Find a ball B R () of radius R >, B R D. Define a mn, n =, 1,..., m n, such that (1) 1+2a mn > 1 for all n =, 1,...,, m n (2) n n= m= n ( 2n ker ) 2n a mn <, R O D Define a series of far field patterns u 4π (ˆx, d) := ik n n= m= n a mn Y m n (ˆx)Y m n (d), where Y m n = spherical harmonics 2

24 Modified far field operator ( ) (F g)(ˆx) := Ω ( u (ˆx, d) u (ˆx, d)) g(d)ds(d) Each term of the series of far field patterns 4π ik a mny m n (d)y m n (ˆx) corresponds to radiating Helmhotz solutions of the form u s, mn (x) = 4πin a mn Y m n (d) h(1) n (k x )Yn m (ˆx) 21

25 Modified LSM valid for all k > Theor. F : L 2 (Ω) L 2 (Ω) is injective with dense range. Theor. (as before with F, without restriction on k) Jones/Ursell modification F : k =1.685 k =2.6 k = Before: k =1.685 k =2.6 k =

26 Object classification at e-frequencies Claim: at eigenfrequencies, imaging g z indicates the zeros of the corresponding eigenfunctions (easy to see in the 2D/3D spherical case) Corollary: Given the far field data for k [k, k 1 ] (containing e-freq.) then one can classify a scatterer as either a PEC (Dirichlet) or not. Dirichlet k = k =5 k = Neumann k =2.796 k =3 k =

27 6. Applications Landmine detection: near field inversions Real far-field 2D data inversions 24

28 Landmine detection Carl Baum:... we detect everything, we identify nothing! Metal detectors : high rate of false alarms (non landmine artifacts) air? sand high cost (due to false alarms) : USD 3 to buy, USD 2 1 to clear requires high level of detection accuracy (deminers safety) as opposed to military demining 1 million landmines world-wide 2 victims per month 25

29 Humanitarian Demining Project (HuMin/MD: Our goal: Decrease the number of false alarms through fast new imaging algorithms. 1. Local 3D imaging: Karlsruhe, Mainz, Cologne, Göttingen, & des Saarlandes 2. Signal analysis 3. 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

30 3D EM inversions: synthetic data Multi-static measurement on 12 x 12 grid (4 x 4 cm) Frequency 1 khz, k = k , PEC objects Reconstruction in perspective Zoomed reconstruction 27

31 2D inversions: synthetic data Two-layered background. Frequency 1 khz. Soil EM properties: σ = 1 3 S/m, ϵ r = 1 k.63(1 + i) k (δ = O(1m)) 3 meas./source points along Γ = [.4,.4] {.5}, Two penetrable obstacles σ D = 1 5 (high), ϵ D r = 8 U-shape metal Linear sampling Factorization σ D = 1 6 (high) ϵ r = 2. 28

32 Plastic only mine. Linear sampling Factorization σ out = σ in = 1 1 (weakly conductive) ϵ in r = 3, ϵ out r = 3 (plastic/tnt) Metal trigger. Linear sampling Factorization

33 Further multiple PEC scatterers

34 Experimental 2D far-field data Free-space parameters Frequency 1 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 15 FM 15 FM Cavity FM

35 Remark Superposition of the array data via u (ˆx, d)g(d)ds(d) Ω allows us to devise a criterion to determine whether a sampling point z belongs to the scatterer. This is done by testing the data against the background 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 compatible with the field due to a point source when z is inside D and not compatible when z is outside D (ranges...) References: The factorization method for inverse problems (28), Kirsch and Grinberg, Springer Qualitative methods in inverse scattering theory (27), Cakoni and Colton, Springer Inverse acoustic and EM scattering theory (213), 3rd ed., Colton and Kress, Springer Stream of papers in Inverse problems journal 31

36 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