# Analysis of Weighted Multifrequency MUSIC-Type Algorithm for Imaging of Arc-Like, Perfectly Conducting Cracks

Authors: Young-Deuk Joh
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The main purpose of this paper is to investigate the structure of the weighted multifrequency multiple signal classification (MUSIC) type imaging function in order to improve the traditional MUSIC-type imaging. For this purpose, we devise a weighted multifrequency MUSIC-type imaging function and examine a relationship between weighted multifrequency MUSIC-type function and Bessel functions of integer order of the first kind. Some numerical results are demonstrated to support the survey.


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## Body

## 1. Introduction

Inverse problem, which deals with the reconstruction of cracks or thin inclusions in homogeneous material (or space) with physical features different from space, is of interest in a wide range of fields such as physics, engineering, and image medical science which are closely related to human life; refer to [1–9]. That is why inverse problem has been established as one interesting research field. Compared to the early studies on inverse problem in which much research had been done theoretically, in recent studies, more practical and applicable approaches have been undertaken and the reconstructive way appropriate to each specific study field started to be investigated thanks to the development of computational science using not only computers but also mathematical theory. As we can see through a series of papers [10, 11], the reconstruction algorithm, based on the iterative scheme such as Newton’s method, has been mainly studied. Generally, in regard to algorithms using Newton’s method, in the case of the initial shape quite different from the unknown target, the reconstruction of material leads to failure with the nonconvergence or yields faulty shapes even after the iterative methods are conducted. Hence, in such an iterative method, several noniterative algorithms have been proposed as a way to find the shape of initial value close to that of the unknown target as quickly as possible.The noniterative algorithms such as multiple signal classification (MUSIC), subspace migrations, topological derivative, and linear sampling method can contribute to yielding the appropriate image as an initial guess. Previous attempts to investigate MUSIC-type algorithm presented various experiments with the use of MUSIC-type algorithm. For instance, the use of MUSIC-type algorithm for eddy-current nondestructive evaluation of three-dimensional defects [12], and MUSIC-type algorithm designed for extended target, the boundary curves which have a five-leaf shape or big circle was presented [13]. In addition, MUSIC-type algorithm was introduced for locating small inclusions buried in a half space [2] and for detecting internal corrosion located in pipes [3]. Although the past phenomena about experimental results could not be theoretically explained because the mathematical structure about these algorithms was not verified, recent studies [14–19] managed to partially analyze the structure of some algorithms. On the basis of these studies, the present study examined the structure of algorithms to make improvements in imaging the defects. Therefore, this paper aims to improve traditional MUSIC-type imaging algorithm by weighting applied to each frequencies.This paper is organized as follows. In Section2, we discuss two-dimensional direct scattering problem in the presence of perfectly conducting crack and MUSIC-type algorithm. In Section 3, we introduce a weighted multifrequency MUSIC-type imaging algorithm and analyze its structure to confirm that it is an improved version of traditional MUSIC algorithm. In Section 4, we present several numerical experiments with noisy data. In Section 5, our conclusions are briefly presented.


## 2. Direct Scattering Problem and Single- and Multifrequency MUSIC-Type Algorithm

In this section, we simplify surveying the two-dimensional direct scattering problem for the existence of perfectly conducting cracks and the single- and multifrequency MUSIC algorithm. For more information, see [10, 20].


### 2.1. Direct Scattering Problem and MUSIC-Type Imaging Function

First, we consider the two-dimensional electromagnetic scattering by a perfectly conducting crack located in the homogeneous spaceℝ2. Throughout this paper, we assume that the crack Ω is a smooth, nonintersecting curve, and we represent Ω such that
(1)Ω={ξ(t):t∈[-1,1]},
where ξ:[-1,1]→ℝ2 is an injective piecewise smooth function. We consider only the transverse magnetic (TM) polarization case. Let us denote utotal to be the time-harmonic total field, which can be decomposed as
(2)utotal(x)=uincident(x)+uscatter(x),
where uincident(x) = exp(jωθ·x) is the given incident field with incident direction θ∈𝕊1 (unit circle) and uscatter(x) is the unknown scattered field that satisfies the Sommerfeld radiation condition
(3)lim|x|→∞|x|(∂uscatter(x)∂|x|-jωuscatter(x))=0
uniformly in all directions x^=x/|x|. Now, the total field utotal satisfies the two-dimensional Helmholtz equation
(4)Δutotal(x)+ω2utotal(x)=0inℝ2∖Ω,utotal(x)=0onΩ
with a given positive frequency ω. In the case that Ω is absent, incident field uincident can also be a solution of (4).The far-field pattern is defined as functionufar(x^,θ) that satisfies
(5)uscatter(x,ω)=exp(jk0|x|)|x|ufar(x^,θ)+o(1|x|)=exp(jω|x|)|x|ufar(x^,θ)+o(1|x|)
as |x|→∞ uniformly on x^. Then, based on [21], the far-field pattern ufar(x^;θ) of the scattered field uscatter(x) can be expressed by the following equation:
(6)ufar(x^;θ)=-exp(jπ/4)8πω∫Ω‍exp(-jωx^·y)φ(y;θ)dy,
where φ(y;θ) is an unknown density function (see [10]).Second, we present the traditional MUSIC-type algorithm for imaging of perfectly conducting cracks. For the sake of simplicity, we exclude the constant-exp(π/4)/8πk from formula (6). Based on [20, 22], we assume that the crack is divided into M different segments of size of the order of half the wavelength λ/2. Having in mind the Rayleigh resolution limit for far-field data, only one point at each segment is expected to contribute to the image space of the response matrix 𝕂 (i.e., see [20, 22, 23]). Each of these points, say ym, m=1,2,…,M, will be imaged via the MUSIC-type algorithm. With this assumption, we perform the following singular value decomposition (SVD) of the multistatic response (MSR) matrix 𝕂=[ufar(x^i;θl)]i,l=1N∈ℂN×N:
(7)𝕂=𝕌𝕊𝕍m*=∑m=1M‍τmUmVm*,
where superscript * is the mark of Hermitian, Um and Vm∈ℂN×1 are, respectively, the left- and right-singular vectors of 𝕂, and τm denotes singular values that satisfy
(8)τ1≥τ2≥⋯≥τm>0,τm=0,form≥M+1.
If so, {U1,U2,…,UM} are the basis for the signal and {UM+1,UM+2,…,UN} span the null space of 𝕂, respectively. Therefore, one can define the projection operator onto the null subspace, Pnoise:ℂN×1→ℂN×1. This projection is given explicitly by
(9)Pnoise:=𝕀N-∑m=1M‍UmUm*,
where 𝕀N denotes the N×N identity matrix. For any point z∈ℝ2, we define a test vector f(z,ω)∈ℂN×1 as
(10)f(z,ω)=[exp(jωθ1·z),exp(jωθ2·z),…,exp(jωθN·z)]T.
Based on this, we can design a MUSIC-type imaging function W:ℂN×1→ℝ such that
(11)𝔼(z)=∥Pnoise(f(z,ω))∥-1=1∥Pnoise(f(z,ω))∥.
Then, the map of 𝔼(z) will have peaks of large and small magnitudes at z∈Ω and z∈ℝ2∖Ω, respectively.


### 2.2. Multifrequency MUSIC-Type Imaging Function

We design multifrequency MUSIC-type imaging function and try to describe its structure. First, we introduce a multifrequency MUSIC-type algorithm𝔼MF:ℂN×1→ℝ defined by
(12)𝔼MF(z;S)=(1S∑s=1S‍∥Pnoise(f(z,ωs))∥2)-1/2.
Then, we can introduce the following lemma. A more detailed derivation can be found in [16].Lemma 1 (see [16]).
Assume thatkS and S are sufficiently large; then,
(13)𝔼MF(z;S)≈1N(1-∑m=1M‍Φ(∥z-ym∥;ω1,ωS))-1/2,
where function Φ(x;ω1,ωS) is defined as
(14)Φ(x;ω1,ωS):=1ωS-ω1[ωS(J0(ωSx)2+J1(ωSx)2)-ω1(J0(ω1x)2+J1(ω1x)2)]+∫ω1ωS‍J1(ωx)2dω.So we can recognize the mathematical structure of multifrequency MUSIC-type algorithm. However, the finite representation of∫‍J12(x)dx does not exist. Because of this term, although this can be negligible (see [15]), the map of 𝔼MF(z;S) should generate unexpected points of small magnitudes. In order to solve this problem, the last term of [14] should be eliminated. For this, we suggest a weighted multifrequency MUSIC-type imaging algorithm in the upcoming section.


## 2.1. Direct Scattering Problem and MUSIC-Type Imaging Function

First, we consider the two-dimensional electromagnetic scattering by a perfectly conducting crack located in the homogeneous spaceℝ2. Throughout this paper, we assume that the crack Ω is a smooth, nonintersecting curve, and we represent Ω such that
(1)Ω={ξ(t):t∈[-1,1]},
where ξ:[-1,1]→ℝ2 is an injective piecewise smooth function. We consider only the transverse magnetic (TM) polarization case. Let us denote utotal to be the time-harmonic total field, which can be decomposed as
(2)utotal(x)=uincident(x)+uscatter(x),
where uincident(x) = exp(jωθ·x) is the given incident field with incident direction θ∈𝕊1 (unit circle) and uscatter(x) is the unknown scattered field that satisfies the Sommerfeld radiation condition
(3)lim|x|→∞|x|(∂uscatter(x)∂|x|-jωuscatter(x))=0
uniformly in all directions x^=x/|x|. Now, the total field utotal satisfies the two-dimensional Helmholtz equation
(4)Δutotal(x)+ω2utotal(x)=0inℝ2∖Ω,utotal(x)=0onΩ
with a given positive frequency ω. In the case that Ω is absent, incident field uincident can also be a solution of (4).The far-field pattern is defined as functionufar(x^,θ) that satisfies
(5)uscatter(x,ω)=exp(jk0|x|)|x|ufar(x^,θ)+o(1|x|)=exp(jω|x|)|x|ufar(x^,θ)+o(1|x|)
as |x|→∞ uniformly on x^. Then, based on [21], the far-field pattern ufar(x^;θ) of the scattered field uscatter(x) can be expressed by the following equation:
(6)ufar(x^;θ)=-exp(jπ/4)8πω∫Ω‍exp(-jωx^·y)φ(y;θ)dy,
where φ(y;θ) is an unknown density function (see [10]).Second, we present the traditional MUSIC-type algorithm for imaging of perfectly conducting cracks. For the sake of simplicity, we exclude the constant-exp(π/4)/8πk from formula (6). Based on [20, 22], we assume that the crack is divided into M different segments of size of the order of half the wavelength λ/2. Having in mind the Rayleigh resolution limit for far-field data, only one point at each segment is expected to contribute to the image space of the response matrix 𝕂 (i.e., see [20, 22, 23]). Each of these points, say ym, m=1,2,…,M, will be imaged via the MUSIC-type algorithm. With this assumption, we perform the following singular value decomposition (SVD) of the multistatic response (MSR) matrix 𝕂=[ufar(x^i;θl)]i,l=1N∈ℂN×N:
(7)𝕂=𝕌𝕊𝕍m*=∑m=1M‍τmUmVm*,
where superscript * is the mark of Hermitian, Um and Vm∈ℂN×1 are, respectively, the left- and right-singular vectors of 𝕂, and τm denotes singular values that satisfy
(8)τ1≥τ2≥⋯≥τm>0,τm=0,form≥M+1.
If so, {U1,U2,…,UM} are the basis for the signal and {UM+1,UM+2,…,UN} span the null space of 𝕂, respectively. Therefore, one can define the projection operator onto the null subspace, Pnoise:ℂN×1→ℂN×1. This projection is given explicitly by
(9)Pnoise:=𝕀N-∑m=1M‍UmUm*,
where 𝕀N denotes the N×N identity matrix. For any point z∈ℝ2, we define a test vector f(z,ω)∈ℂN×1 as
(10)f(z,ω)=[exp(jωθ1·z),exp(jωθ2·z),…,exp(jωθN·z)]T.
Based on this, we can design a MUSIC-type imaging function W:ℂN×1→ℝ such that
(11)𝔼(z)=∥Pnoise(f(z,ω))∥-1=1∥Pnoise(f(z,ω))∥.
Then, the map of 𝔼(z) will have peaks of large and small magnitudes at z∈Ω and z∈ℝ2∖Ω, respectively.


## 2.2. Multifrequency MUSIC-Type Imaging Function

We design multifrequency MUSIC-type imaging function and try to describe its structure. First, we introduce a multifrequency MUSIC-type algorithm𝔼MF:ℂN×1→ℝ defined by
(12)𝔼MF(z;S)=(1S∑s=1S‍∥Pnoise(f(z,ωs))∥2)-1/2.
Then, we can introduce the following lemma. A more detailed derivation can be found in [16].Lemma 1 (see [16]).
Assume thatkS and S are sufficiently large; then,
(13)𝔼MF(z;S)≈1N(1-∑m=1M‍Φ(∥z-ym∥;ω1,ωS))-1/2,
where function Φ(x;ω1,ωS) is defined as
(14)Φ(x;ω1,ωS):=1ωS-ω1[ωS(J0(ωSx)2+J1(ωSx)2)-ω1(J0(ω1x)2+J1(ω1x)2)]+∫ω1ωS‍J1(ωx)2dω.So we can recognize the mathematical structure of multifrequency MUSIC-type algorithm. However, the finite representation of∫‍J12(x)dx does not exist. Because of this term, although this can be negligible (see [15]), the map of 𝔼MF(z;S) should generate unexpected points of small magnitudes. In order to solve this problem, the last term of [14] should be eliminated. For this, we suggest a weighted multifrequency MUSIC-type imaging algorithm in the upcoming section.


## 3. Weighted Multifrequency MUSIC-Type Algorithm and Its Structure

In order to propose the weighted multifrequency MUSIC-type imaging algorithm, we introduce the following lemma derived from [16].Lemma 2 ([16, page 218]).
For sufficiently largeN and ω, the following relationship holds:
(15)∥Pnoise(f(z,ω))∥≈N(1-∑m=1M‍J0(ω|z-ym|)2)1/2.With this, let us define an alternative projection operator weighted by applied frequencyPWN:ℂN×1→ℝ as
(16)PWN(f(z,ω))=Pnoise(ωf(z,ω)).
Then, the following result can be obtained.Theorem 3.
Assume thatN and ω are sufficiently large; then,
(17)∥PWN(f(z,ω))∥≈N(ω-∑m=1M‍ωJ0(ω|z-ym|)2)1/2.Proof.
The following equations are satisfied by the definition ofPWN(f(z,ω)) and by Lemma 2:
(18)∥PWN(f(z,ω))∥=∥Pnoise(ω(f(z,ω)))∥=ω∥Pnoise((f(z,ω)))∥≈ωN(1-∑m=1M‍J0(ω|z-ym|)2)1/2=N(ω-∑m=1M‍ωJ0(ω|z-ym|)2)1/2.Next, we introduce a weighted multifrequency MUSIC-type imaging function based on MUSIC-type imaging function𝔼WMF:ℂN×1→ℝ defined by
(19)𝔼WMF(z;S)=(1S∑s=1S‍∥PWN(f(z,ωs))∥2)-1/2.
Then, we can obtain the structure of 𝔼WMF(z;S).Theorem 4.
Assume thatS and ωS are sufficiently large; then,
(20)𝔼WMF(z;S)≈1N(ωS+ω12-∑m=1M‍Ψ(|z-ym|;ω1,ωS))-1/2,
where function Ψ(x;ω1,ωS) is defined as
(21)Ψ(x;ω1,ωS):=1ωS-ω1[ωS22(J0(ωSx)2+J1(ωSx)2)-ω122(J0(ω1x)2+J1(ω1x)2)].Proof.
By Theorem3, we can calculate the following:
(22)𝔼WMF(z;S)=(1S∑s=1S‍∥PWN(f(z,ωs))∥2)-1/2≈(1S∑s=1S‍(N(ω-∑m=1M‍ωJ0(ω|z-ym|)2)1/2)2)-1/2=(1S∑s=1S‍N(ω-∑m=1M‍ωJ0(ω|z-ym|)2))-1/2=1N(∑s=1S‍(ω-∑m=1M‍ωJ0(ω|z-ym|)2)1S)-1/2.
Then, since S is sufficiently large, we can observe that
(23)∑s=1S‍(ω-∑m=1M‍ωJ0(ω|z-ym|)2)1S≈1ωS-ω1∫ω1ωS‍(ω-∑m=1M‍ωJ0(ω|z-ym|)2)dω,
and applying an indefinite integral of the Bessel function (see [24, page 106])
(24)∫‍xJ0(x)2dx=x22(J0(x)2+J1(x)2)
yields
(25)1ωS-ω1∫ω1ωS‍ωJ0(ω|z-ym|)2dω=1ωS-ω1[ωS22(J0(ωS|z-ym|)2+J1(ωS|z-ym|)2)-ω122({ωS22}J0(ω1|z-ym|)2+J1(ω1|z-ym|)2{ωS22})ωS22]=Ψ(ω|z-ym|;ω1,ωS).
Hence, we can obtain(26)1ωS-ω1∫ω1ωS‍(ω-∑m=1M‍ωJ0(ω|z-ym|)2)dω=ωS+ω12-∑m=1M‍Ψ(ω|z-ym|;ω1,ωS).
Therefore,(27)𝔼WMF(z;S)≈1N(ωS+ω12-∑m=1M‍Ψ({|z-ym|}|z-ym|;ω1,ωS{|z-ym|}))-1/2.
This completes the proof.Looking at the results of Theorem4, in contrast to the 𝔼MF(z;S), the 𝔼WMF(z;S) does not have the ∫‍J12(x)dx term. Therefore, we expect that the imaging results of the 𝔼WMF(z;S) will be better than 𝔼MF(z;S). In the next section, numerical experiments will be presented to support this.


## 4. Numerical Experiments

In this section, some numerical examples are displayed in order to support our analysis in the previous section. Applied frequencies are of the formωs=2π/λs, where λs, s=1,2,…,S(=10) is the given wavelength. The observation directions θn∈𝕊1 are taken as
(28)θn=[cos2πnN,sin2πnN]T.For illustrating arc-like cracks, threeΩl are chosen:
(29)Ω1={[s,12cossπ2+15sinsπ2-110cos3sπ2]T:s∈[-1,1][s,12cossπ2+15sinsπ2-110cos3sπ2]T},Ω2={[2sins2,sins]T:s∈[π4,7π4]},Ω3=Ω3(1)∪Ω3(2),
where
(30)Ω3(1)={[s-15,-s22+35]T:s∈[-12,12]},Ω3(2)={[s+15,s3+s2-35]T:s∈[-12,12]}.It is worth emphasizing that all the far-field dataufar of (6) are generated by the method introduced in [25, Chapter 3, 4]. After generating the data, a 20 dB white Gaussian random noise is added to the unperturbed data. In order to obtain the number of nonzero singular values M for each frequency, a 0.1-threshold scheme (choosing first m singular values τm such that τm/τ1≥0.1) is adopted. A more detailed discussion of thresholding can be found in [20, 22].Figures1 and 2 show the imaging results via multifrequency MUSIC and weighted multifrequency MUSIC algorithms for single crack Ω1 and Ω2, respectively. As we already mentioned, since the term ∫‍J12(x)dx can be disregarded, it is very hard to compare the improvements via visual inspection of the reconstructions. However, based on Figure 3, we can examine that the proposed weighted multifrequency MUSIC algorithm successfully reduces these artifacts, so we can conclude that this is an improved version.Shape reconstruction ofΩ1 via MUSIC algorithm.
(a)
Map of𝔼MF(z;10)
(b)
Map of𝔼WMF(z;10)Same as Figure1 except the crack is Ω2.
(a)
Map of𝔼MF(z;10)
(b)
Map of𝔼WMF(z;10)Blue- and red-colored lines are𝔼WMF(z;10) and 𝔼MF(z;10), respectively, at z=[-0.8,y]T ((a) and (b)) and z=[x,0]T ((c) and (d)).
(a)
Map of𝔼MF(z;10)
(b)
Graph of oscillating pattern
(c)
Map of𝔼MF(z;10)
(d)
Graph of oscillating patternFigure4 shows the imaging results via multifrequency MUSIC and weighted multifrequency MUSIC algorithms for multiple cracks Ω3. Similar to the imaging of single crack, we can observe that weighted multifrequency MUSIC algorithm improves the traditional one, although it is hard to compare the improvements via visual inspection.Same as Figure1 except the crack is Ω3.
(a)
Map of𝔼MF(z;10)
(b)
Map of𝔼WMF(z;10)Figure5 shows the noise contribution in terms of SNR. In order to observe the effect of noise, 30 dB and 10 dB white Gaussian random noises are added to the unperturbed data. Based on these results, we can easily observe that both traditional and proposed MUSIC algorithms offer very good result when 30 dB noise is added. However, when 10 dB noise is added, the traditional MUSIC algorithm yields a poor result while the proposed algorithm yields an acceptable result.Influence of noise. 30 dB ((a) and (b)) and 10 dB ((c) and (d)) white Gaussian random is added.
(a)
Map of𝔼(z)
(b)
Map of𝔼WMF(z;10)
(c)
Map of𝔼(z)
(d)
Map of𝔼WMF(z;10)Now, we consider the imaging of oscillating crack. For this, we consider the following cracks (see Figure6):
(31)Ω4={[s,12s2+110sin(4π(s+1))]T:s∈[-1,1]}Ω5={[s,12s2+120sin(20π(s+1))-1100cos(15πs)]T:s∈[-1,1][s,12cossπ2+15sinsπ2-110cos3sπ2]T}.Shape of oscillating cracksΩ4 and Ω5.
(a)
CrackΩ4
(b)
CrackΩ5Figure7 shows the maps of 𝔼(z) and 𝔼WMF(z;10) for the crack Ω4. In this result, we can observe that both traditional and proposed MUSIC algorithms produce acceptable result, but the proposed algorithm successfully eliminates replicas.Shape reconstruction ofΩ4 via 𝔼(z) (a) and 𝔼WMF(z;10) (b).
(a)
Map of𝔼(z)
(b)
Map of𝔼WMF(z;10)Figure8 shows the maps of 𝔼(z) and 𝔼WMF(z;10) for highly oscillating crack Ω5. Opposite to Figure 7, both traditional and proposed MUSIC algorithms yield poor result. This example shows the limitation of proposed algorithm.Same as Figure7 except the crack is Ω5.
(a)
Map of𝔼(z)
(b)
Map of𝔼WMF(z;10)


## 5. Conclusion

Based on the structure of multifrequency MUSIC-type imaging function, we introduced a weighted multifrequency MUSIC-type imaging function. Through a careful analysis, a relationship between imaging function and Bessel function of the first kind of integer order is established, and we have confirmed that the proposed imaging algorithm is an improved version of the traditional one.Although, the proposed algorithm produces very good results and improves the traditional MUSIC algorithm, it still needs some upgrade, for example, imaging of highly oscillating cracks. Development of this should be an interesting and remarkable research project. In this paper, we considered the MUSIC algorithm in full-view inverse scattering problem. Based on the result in [26], the MUSIC algorithm cannot be applied to limited-view problems but the reason is still unknown. Identifying the structure of the MUSIC algorithm in the limited-view inverse scattering problems will be the forthcoming work.



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*Source: 461251-2013-11-06.xml*