Progress In Electromagnetics Research, Vol. 136, 607–622, 2013

STRUCTURE ANALYSIS OF SINGLE- AND MULTI-FREQUENCY SUBSPACE MIGRATIONS IN INVERSESCATTERING PROBLEMS

Young-Deuk Joh, Young Mi Kwon, Joo Young Huh, andWon-Kwang Park*

Department of Mathematics, Kookmin University, Seoul 136-702,Korea

Abstract—We carefully investigate the structure of single- and multi-frequency imaging functions, that are usually employed in inversescattering problems. Based on patterns of the singular vectors ofthe Multi-Static Response (MSR) matrix, we establish a relationshipbetween imaging functions and the Bessel function. This relationshipindicates certain properties of imaging functions and the reason behindenhancement in the imaging performance by multiple frequencies.Several numerical simulations with a large amount of noisy data areperformed in order to support our investigation.

1. INTRODUCTION

One of the main objectives of the inverse scattering problem isto identify the characteristics of unknown targets from measuredscattered field or far-field pattern. In research fields such as physics,medical science, and materials engineering, this is an interesting andimportant problem. Related works can be found in [2, 7, 9–11, 32] andreferences therein. In order to solve this problem, various algorithmsfor finding the locations and/or shapes of targets have been accordinglydeveloped.

In many research studies [1, 7, 9, 13, 17, 26], the shape reconstruc-tion method is based on Newton-type iterative algorithms. However,for a successful shape reconstruction using these algorithms, the iter-ative procedure must begin with a good initial guess that is close tothe unknown target because it highly depends on the initial guess; formore details, refer to [16, 26].

Received 3 December 2012, Accepted 21 January 2013, Scheduled 24 January 2013* Corresponding author: Won-Kwang Park (parkwk@kookmin.ac.kr).

608 Joh et al.

To find a good initial guess, alternative non-iterative recon-struction algorithms have been developed, such as the MUltipleSIgnal Classification (MUSIC)-type algorithm [5, 23, 25], linear sam-pling method [8, 12], topological derivative strategy [3, 6, 18, 21, 22],linear-δ, vector and multipolarized approaches [29, 30], and the multi-frequency based algorithm such as Kirchhoff and subspace migra-tions [2, 4, 15, 19, 20, 24]. Among them, although the multi-frequencybased subspace migration has exhibited potential as a non-iterativeimaging technique, a mathematical identification of its structure needsto be performed for its heuristical applications, which is the motivationbehind.

In this paper, by intensively analyzing the structure of single-and multi-frequency subspace migration, we discover some propertiesand confirm the reason behind the enhancement in the imagingperformance by applying multiple frequencies. In recent work [4], thisfact was verified by the Statistical Hypothesis Testing but our approachis to find a relationship between imaging functions and Bessel functionsof the first kind of the integer order.

This paper is organized as follows. In Section 2, we briefly reviewthe two-dimensional direct scattering problem, and an asymptoticexpansion formula for far-field patterns, and introduce the imagingfunction introduced in [19]. In Section 3, we analyze the single- andmulti-frequency based imaging functions and discuss their properties.In Section 4, we present several numerical experiments and discuss theeffectiveness, robustness, and limitation of imaging functions. Finally,a brief conclusion is given in Section 5.

2. REVIEW ON IMAGING FUNCTION

In this section, we survey the two-dimensional direct scatteringproblem and an imaging algorithm. A more detailed discussion canbe found in [4, 5, 19, 20, 24].

2.1. Direct Scattering Problem and Asymptotic ExpansionFormula

Let Σm be a homogeneous inclusion with a small diameter ρ in thetwo-dimensional space R2. Throughout this paper, we assume thatevery Σm is expressed as

Σm = rm + ρDm,

where rm and ρ denote the location and size of Σm, respectively. Here,Dm is a simple connected smooth domain containing the origin.

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Let ε0 and µ0 respectively denote the dielectric permittivity andmagnetic permeability of R2. Similarly, we let εm and µm be those ofΣm. For simplicity, let Σ be the collection of Σm, m = 1, 2, . . . , M,and we define the following piecewise constants:

ε(r) =

εm for r ∈ Σm

ε0 for r ∈ R2\Σ and µ(r) =

µm for r ∈ Σm

µ0 for r ∈ R2\Σ.

Throughout this paper, we assume that ε0 = µ0 = 1 and εm > ε0,µm > µ0 for m = 1, 2, . . . , M . At a given frequency ω, let utot(r, dl; ω)be the time-harmonic total field that satisfies the Helmholtz equation

∇ ·(

1µ(r)

∇utot(r,dl; ω))

+ ω2ε(r)utot(r,dl; ω) = 0 (1)

with transmission conditions at the boundaries of Σm.Let uinc(r, dl; ω) be the solution of (1) without Σ. In this paper,

we consider the following plane-wave illumination: for a vector dl ∈ C1,uinc(r,dl; ω) = exp(jωdl · r). Here, C1 denotes the unit circle in R2.

Generally, the total field utot can be divided into the incidentfield uinc and the unknown scattered field uscat, which satisfies theSommerfeld radiation condition

lim|r|→∞

√|r|

(∂uscat(r,dl;ω)

∂|r| − jk0uscat(r,dl; ω))

= 0,

uniformly in all directions r = r|r| . Since we assumed ε0 = µ0 = 1,

wavenumber k0 satisfies k0 = ω√

ε0µ0 = ω. As given in [5], uscat canbe written as the following asymptotic expansion formula

uscat(r,dl; ω) =ρ2M∑

m=1

(∇uinc(rm,dl; ω) · T(rm) · ∇Φ(rm, r, ω)

+ω2(ε−ε0)area(Dm)uinc(rm,dl;ω)Φ(rm,r,ω))+o(ρ2),(2)

where o(ρ2) is uniform in rm ∈ Σm and dl ∈ C1. Here area(Dm)denotes the area of Dm, T(rm) is a 2× 2 symmetric matrix:

T(rm) =2µ0

µm + µ0area(Dm)I2,

where In denotes the n × n identity matrix, and Φ(rm, r, ω) is thetwo-dimensional time harmonic Green function

Φ(rm, r, ω) = −µ0j

4H1

0 (ω|rm − r|),where H1

0 is the Hankel function of order zero and of the first kind.

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The far-field pattern is defined as function F (r,dl) that satisfies

uscat(r,dl; ω) =exp(jk0|r|)√

|r| F (r,dl) + o

(1√|r|

)(3)

as |r| → ∞ uniformly on r = r|r| .

2.2. Introduction to Subspace Migration

The imaging algorithm introduced in [19] used the structure of asingular vector of the Multi-Static Response (MSR) matrix M =(Fpq) = (F (rp,dq))N

p,q=1, whose elements F (rp, dq) is (3) withobservation number p and incident number q. Note that bycombining (2), (3), and the asymptotic behavior of the Hankel function,the far-field pattern F (rp,dq) can be represented as the asymptoticexpansion formula (see [5] for instance)

F (rp,dq) ≈ ρ2 ω2(1 + j)4√

ωπ

M∑

m=1

(ε− ε0√

ε0µ0area(Dm)− rp · T(rm) · dq

)

× exp(

jk0(dq − rp) · rm

). (4)

For the sake of simplicity, we eliminate the constant ω2(1+j)4√

ωπin (4).

Then, the incident and observation direction configurations are keptsame, i.e., for each rp = −dp, the pq-th element of the MSR matrix Mis given by

Fpq = F (rp,dq)∣∣∣∣rp=−dp

≈ ρ2M∑

m=1

[εm − ε0√

ε0µ0area(Dm)

+2µ0

µm + µ0area(Dm)dp · dq

]exp

(jk0(dp + dq) · rm

).

Based on the above representation of Fpq, we introduce a vectorD(r; ω) ∈ CN×3 as

D(r;ω) :=

e1 exp(jk0d1 · r)e2 exp(jk0d2 · r)

...eN exp(jk0dN · r)

, where ep = (1,dp)T . (5)

Then M can be decomposed as follows:

M =M∑

m=1

D(rm;ω)

ρ2 εm − ε0√

ε0µ0area(Dm) O1×2

O2×1 ρ2T(rm)

D(rm;ω)T ,

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where Op×q denotes the p × q zero matrix. This decomposition leadsus to introduce an imaging algorithm as follows. First, let us performthe Singular Value Decomposition (SVD) as follows:

M = USVT =M∑

m=1

σm(ω)Um(ω)Vm(ω)T , (6)

where Um and Vm are the left and right singular vectors, respectively,and a denotes the complex conjugate of a. Then, based on the structureof (5), we define a vector D(r; ω) ∈ CN×1:

D(r;ω) :=

c · (1,d1)T exp(jk0d1 · r)c · (1,d2)T exp(jk0d2 · r)

...c · (1,dN )T exp(jk0dN · r)

, c ∈ C3×1\0, (7)

and corresponding unit vector

W(r;ω) :=D(r;ω)|D(r;ω)| . (8)

With this, we can introduce a subspace migration as follows

W(r;ω) :=

∣∣∣∣∣M∑

m=1

(W(r; ω) ·Um(ω)

)(W(r; ω) · Vm(ω)

)∣∣∣∣∣ . (9)

Note that since the first M columns of the matrices U1, U2, . . . , UMand V1, V2, . . . , VM are orthonormal, we can observe that

W(r; ω) ·Um(ω) ≈ 1 and W(r; ω) · Vm(ω) ≈ 1 if r = rm

W(r; ω) ·Um(ω) ≈ 0 and W(r; ω) · Vm(ω) ≈ 0 if r 6= rm,

for m = 1, 2, . . . , M . Therefore,W(r; ω) will plots peaks of magnitudeof 1 at r = rm ∈ Σm, and of small magnitude at r /∈ Σm

(see [4, 19, 20, 24]). Complete algorithm is summarized in Algorithm 1.

3. STRUCTURE ANALYSIS OF IMAGING FUNCTIONS

3.1. Structure of Single-frequency Subspace Migration (9)

We now determine the structure of imaging function (9). Beforeproceeding, we assume that scatterers are well resolved by themeasurement array and based on the Rayleigh resolution limit fromfar-field data, any detail less than one-half of the wavelength cannotbe seen so that scatterers are well-separated to each other, refer to [2].

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Algorithm 1 Imaging algorithm via Subspace Migration (SM)1: procedure SM(ω)2: identify permittivity ε0 and permeability µ0 of R2

3: given ω, initialize W(r; ω)4: for q = 1 to N do5: for p = 1 to N do6: collect MSR matrix data F (rp,dq) ∈M . see (4)7: end for8: end for9: perform SVD of M = USVT

. see (6)10: choose U1,U2, . . . ,UM and V1,V2, . . . ,VM . see [25]11: for r ∈ Ω ⊂ R2 do . Ω is a search domain12: generate D(r; ω) and W(r; ω) . see (7) and (8)13: initialize I(r)14: for m = 1 to M do15: I(r) ← I(r) + (W(r; ω) ·Um(ω))(W(r; ω) ·Vm(ω))16: end for17: W(r;ω) = |I(r)|18: end for . see (9)19: plot W(r;ω) and find r = rm ∈ Σm . W(r; ω) ≈ 120: end procedure

For determining the structure, we recall some useful statements.Lemma 3.1 ([4]). A relation A ∼ B means that there exists a constantC such that A = CB. Then, for vectors Um and Vm in (6) andW(r; ω) in (8), the following relationship holds

W(rm; ω) ∼ Um(ω) and W(rm; ω) ∼ Vm(ω).

Lemma 3.2 ([14]). Let d, r ∈ R2, and ω > 0; then∫

C1

exp(jωd · r)dS(d) = 2πJ0(ω|r|),

where Jν(x) denotes the Bessel function of order ν of the first kind.Subsequently, we can explore the structure of (9) as follows

Lemma 3.3. If the total number of incident and observation directionsN is sufficiently large and satisfies N > M , then the imagingfunction (9) can be represented as follows:

W(r;ω) ∼M∑

m=1

J0(ω|rm − r|)2. (10)

Proof. By hypothesis, we assume that N is sufficiently large. Sincethe incident and observation direction configurations are same, we set

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(a) (b)

Figure 1. (a) 2-D plot of (10) for ω = 2π0.3 and (b) 2-D plot of (11) for

ω1 = 2π0.5 and ωS = 2π

0.3 when m = 1 and rm = 0.

4dp := |dp−dp−1| for p = 2, 3, . . . , N , and 4d1 := |d1−dN |. Then,applying Lemmas 3.1 and 3.2 yields

W(r;ω) =

∣∣∣∣∣M∑

m=1

(W(r; ω) ·Um(ω)

)(W(r;ω) · Vm(ω)

)∣∣∣∣∣

∼M∑

m=1

N∑

p=1

exp(jωdp · (rm − r))4dp

2π

2

≈ 14π2

M∑

m=1

(∫

C1

exp(jωd · (rm−r))dS(d))2

=M∑

m=1

J0(ω|rm−r|)2.

This completes the proof.Note that J0(x) has the maximum value 1 at x = 0. This is the

reason that the map of W(r; ω) plots magnitude 1 at r = rm ∈ Σm.Moreover, due to the oscillating property of J0(x), Theorem 3.3indicates why imaging function (9) plots unexpected replicas, as shownin Figure 1.

3.2. Reason Behind Enhancement in the ImagingPerformance by Applying Multiple Frequencies

According to Theorem 3.3, the oscillating pattern of the Bessel functionmust be reduced or eliminated in order to improve the imagingperformance. One way to do so is to apply the high-frequency ω = +∞in theory. Another way is to apply several frequencies to the imaging

614 Joh et al.

function (9) as follows:

W(r;S) :=1S

∣∣∣∣∣S∑

s=1

W(r; ωs)

∣∣∣∣∣

=1S

∣∣∣∣∣S∑

s=1

M∑

m=1

(W(r; ωs)·Um(ωs)

)(W(r; ωs) · Vm(ωs)

)∣∣∣∣∣ . (11)

Several researches in [2, 4, 19, 20] have confirmed on the basis ofStatistical Hypothesis Testing and numerical experiments that themulti-frequency imaging function (11) is an improved version of thesingle-frequency version (10). The reason for this is discussed asfollows.Theorem 3.4 If ωS and the total number of incident and observationdirections N is sufficiently large and satisfies N > M , then thestructure of the imaging function (11) is

W(r;S) ∼M∑

m=1

[ωS

ωS − ω1

(J0(ωS |rm − r|)2 + J1(ωS |rm − r|)2

)

− ω1

ωS − ω1

(J0(ω1|rm − r|)2 + J1(ω1|rm − r|)2

)]. (12)

Proof. According to (10), we can observe that

W(r; S) ≈ 1S

S∑

s=1

M∑

m=1

J0(ωs|rm − r|)2 ≈M∑

m=1

∫ ωS

ω1

J0(ω|rm − r|)2ωS − ω1

dω.

Using this, we apply an indefinite integral formula of the Besselfunction (see [27, page 35]):

∫J0(x)2dx = x

(J0(x)2 + J1(x)2

)+

∫J1(x)2dx

in addition to a change of variable ω|rm − r| = x. This yields∫ ωS

ω1

J0(ω|rm−r|)2dω

=1

|rm−r|∫ ωS |rm−r|

ω1|rm−r|J0(x)2dx

= ωS

(J0(ωS |rm−r|)2+J1(ωS |rm−r|)2

)

−ω1

(J0(ω1|rm − r|)2+J1(ω1|rm − r|)2

)+

∫ ωS

ω1

J1(ω|rm−r|)2dω.

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Now, we consider the upper bound of

Λ(|rm − r|, ω1, ωS) :=∫ ωS

ω1

J1(ω|rm − r|)2dω.

Note that since J1(0) = 0, let us assume that |rm − r| 6= 0 and0 < ωS |rm − r| ¿ √

2. Then applying asymptotic behavior

Jν(x) ≈ 1Γ(ν + 1)

(x

2

)ν

and boundedness property Jν(ω|rm − r|) ≤ 1√2

yields∫ ωS

ω1

J1(ω|rm − r|)2dω ≤ 1√2

∫ ωS

ω1

J1(ω|rm − r|)dω

=1√2

∫ ωS |rm−r|

ω1|rm−r|

J1(x)|rm − r|dx =

12√

2|rm − r|

∫ ωS |rm−r|

ω1|rm−r|xdx

=(ωS)2 − (ω1)2

4√

2|rm − r| < ωS

4√

2

(ωS |rm − r|

)¿ ωS

4= O(ωS).

Now, assume that ωS satisfies

ωS |rm − r| À√

2 i.e., |rm − r| À√

2ωS

> 0.

Then since ∫J1(x)dx = −J0(x),

we can obtain∫ ωS

ω1

J1(ω|rm − r|)dω

=∫ ωS |rm−r|

ω1|rm−r|

J1(x)|rm−r|dx

=1

|rm−r|(J0(ω1|rm −r|)−J0(ωS |rm−r|)

)≤ 2|rm − r| ¿

√2ωS .

Therefore, the term Λ(|rm − r|, ω1, ωS) can be disregarded because

ωS

ωS − ω1

(J0(ωS |rm − r|)2 + J1(ωS |rm − r|)2

)

− ω1

ωS − ω1

(J0(ω1|rm − r|)2 + J1(ω1|rm − r|)2

)= O(ωS)

and Λ(|rm − r|, ω1, ωS) ¿ O(ωS). Hence we can obtain (12). Thiscompletes the proof.

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Two-dimensional plot for (12) is shown in Figure 1. This showsthat (11) yields better images owing to less oscillation than (10) does sothat unexpected artifacts in the plot of W(r; S) are mitigated when Sis sufficiently large. This result indicates why a multi-frequency basedimaging function offers better results than a single-frequency basedone.

4. NUMERICAL EXPERIMENTS AND DISCUSSIONS

In this section, we describe the numerical experiments we conductedto validate our analysis. For this purpose, we choose a set of threedifferent small disks Σm. The common radii ρ of Σm are set to 0.1,and parameters ε0 and µ0 are chosen as 1. Locations rm of Σm areselected as r1 = (0.4, 0), r2 = (−0.6, 0.3), and r3 = (0.1, −0.5). Fora given wavelength λs, each frequency is selected as ωs = 2π

λs, for

s = 1, 2, . . . , S. Note that the test vector c in (7) is selected asc = (5, 1, 1)T and all the wavelengths λs are uniformly distributed inthe interval [λ1, λS ]. The observation directions dp are selected as

dp =(

cos2πp

N, sin

2πp

N

)for p = 1, 2, . . . , N,

and the incident directions dq ∈ C1 are selected analogously.In all the examples, scattered field data are computed within

the framework of the Foldy-Lax equation [31]. Then, a whiteGaussian noise with 10 dB signal-to-noise ratio (SNR) is added to theunperturbed data in order to exhibit the robustness of the proposedalgorithm via the MATLAB command awgn. In order to obtain thenumber of nonzero singular values M for each frequency ωs, a 0.01-threshold scheme is adopted (see [23, 25]). The search domain Ω ⊂ R2

is selected as a square Ω = [−1, 1]× [−1, 1].Figure 2 shows the map of W(r; 10) via the MSR matrix M for

N = 20 and S = 10 and different frequencies with λ1 = 0.5 andλS = 0.3. On the left-hand side of Figure 2, we set the same materialproperties εm ≡ 5 and µm ≡ 5, m = 1, 2, 3. As expected, locationsof Σm can be clearly identified. On the right-hand side of Figure 2,we set different material properties ε1 = µ1 = 5, ε2 = µ2 = 2,and ε3 = µ3 = 7. Notice that, if an inclusion has a much smallervalue of permittivity and/or permeability than the other, it does notsignificantly affect the MSR matrix and consequently, difference inamplitude appearing in the map of W(r; 10), refer to [19, 20, 24, 25].Hence, due to the small values of ε2 and µ2, the map of W(r; 10) plotsa small magnitude at r2 ∈ Σ2 but the locations of all Σm are wellidentified.

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Figure 2. Maps ofW(r; 10). (a) Same material property. (b) Differentmaterial property.

Figure 3 shows the influence of the number of applied frequenciesS. As we discussed in Section 3, increasing S yields a more accurateimage. Note that applying an infinite number of S would yield goodresults in theory, but in this experiment, S = 10 is sufficient forobtaining a good result.

Based on the recent work [19], the proposed algorithm can beapplied to imaging of extended, crack-like electromagnetic inclusion(s)Γ with a supporting curve γ and a small thickness h. However,note that even a sufficiently large number of N and S applied toobtain a good image of a complex-shaped thin inclusion using theproposed algorithm can occasionally yield poor results (see Figure 4).Furthermore, note that the elements of M are expressed as written by

F (dp,dq) ∼M∑

m=1

[ε− ε0√

ε0µ0+ 2

(1µ− 1

µ0

)dp · t(rm)dq · t(rm)

+2(

1µ0− µ

µ20

)dp ·n(rm)dq ·n(rm)

]exp

(jk0(dp+dq)·rm

).

Therefore, c in (7) must be a linear combination of a unit tangentialvector t(rm) and a normal vector n(rm) at rm ∈ γ. If we have a prioriinformation of t(rm) and n(rm), we can obtain a good result. However,because this is not the case, it is difficult to obtain a good result. Thisis further explained in detail in [25, Section 4.3.1].

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Figure 3. Maps of W(r; S) with (a) S = 1, (b) S = 3, (c) S = 7, and(d) S = 20.

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Figure 4. Maps of W(r;S) for complex-shaped thin inclusion.(a) N = 48 and S = 10. (b) N = 64 and S = 24.

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5. CONCLUSION

Using an integral representation formula and an indefinite integralof the Bessel function, we determined the structure of single andmultiple electromagnetic imaging functions. Because of the oscillationaspect of the Bessel function, we confirmed the reason behind theimproved imaging performance by successfully applying high andmultiple frequencies.

Based on recent works in [4, 6, 19, 20, 24], it has been shownthat subspace migration offers better results than the MUSIC andKirchhoff migration. Specially, subspace migration can be appliedto limited-view inverse scattering problems. However, in order todetermine the structure of subspace migration in the limited-viewproblem, the integration in Lemma 3.2 on the subset of a unit circlemust be evaluated; however, this evaluation is very difficult to perform.Therefore, identifying the imaging function structure in the limited-view problem will prove to be an interesting research topic. Moreover,in the imaging of crack-like inclusions, estimating unit tangential andnormal vectors on such an inclusion and yielding relatively good resultswill be an interesting work.

Finally, we consider the imaging function for penetrableelectromagnetic inclusions, but it will be applied to the perfectlyconducting inclusion(s) directly. Extension to the perfectly conductingtarget will be a forthcoming work. We further believe that the proposedstrategy can be extended to a three-dimensional problem. Basedon recent work [28], it is confirm that the multi-frequency MUSICalgorithm improves the single-frequency one. Analysis of structure ofMUSIC-type algorithm will be the forthcoming work.

ACKNOWLEDGMENT

W.-K. Park would like to thank Habib Ammari for introducing [2]and many valuable advices. The authors would like to acknowledgefour anonymous referees for their precious comments. This work wassupported by the research program of Kookmin University in Korea,the Basic Science Research Program through the National ResearchFoundation of Korea (NRF) funded by the Ministry of Education,Science and Technology (No. 2012-0003207), and the WCU (WorldClass University) program through the National Research Foundationof Korea (NRF) funded by the Ministry of Education, Science andTechnology R31-10049.

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