DETECTION AND IMAGING IN A STRONGLY CLUTTERED ...

DETECTION AND IMAGING IN A STRONGLY CLUTTERED ENVIRONMENT SONGMING HOU

∗,

KNUT SOLNA

† , AND

HONGKAI ZHAO

‡

Abstract. In this paper, we study method for identification of targets in a cluttered environment using the inter-element response, the response matrix, of an active array. We examine the properties of the response matrix and its singular value decomposition (SVD). In particular we analyze the patterns of singular values for scatterers of different sizes and use it for detection and characterization of scatterers. We use the corresponding singular vectors to image the scatterer and present a novel approach using the SVD of the response matrix for imaging the shape of extended targets. We form the difference matrix between response matrices measured at two different times and use its SVD to detect significant changes in a cluttered medium. We also demonstrate that by measuring the response matrix at consecutive times we can track the motion of a target in a cluttered environment. Numerical experiments are presented to illustrate and validate our approach.

Keywords: Helmholtz equation, singular value decomposition, time reversal, response matrix, difference field. 1. Introduction. Detection and imaging targets using waves in a highly cluttered environment can be a challenging task. Multiple scattering due to medium heterogeneities makes the problem complicated. For examples of situations with significant multiple scattering consider for instance target detection under foilage using electromagnetic waves, underground mine detection, non destructive testing in composite materials, medical imaging using ultrasound, etc. In many applications imaging the whole medium, an inverse problem, is impractical. In a typical inverse problem approach, an inverse or pesudo-inverse of the forward operator has to be approximated and solved. The inverse problem is typically nonlinear even if the forward problem is linear. Moreover, the inverse problem is usually very ill-posed and regularization has to be introduced. Here we apply techniques based on time reversal methods to detect significant pattern changes in a cluttered environment, such as the appearance and disappearance of targets or motion of targets, without knowing the background medium. The time reversal phenomenon and related techniques have been the subject of extensive studies in both the physics and the mathematics community recently. The main property of time reversal is its source auto-focusing capability. The time reversal process and its focusing property can also be viewed as a spatial and tempero matched filter [1, 7, 20] from the signal processing point of view. The time reversal techniques can be exploited to image active sources and scatterers more efficiently and robustly than when imaging the whole medium by a standard inverse problem approach. To probe the medium we use an active array of transducers that can both send and record waves. We form the response matrix of the active array, i.e., measure the inter-element response by firing one transducer and recording the responses at another transducer. The response matrix and its properties have been under extensive study. For examples, the product of the response matrix and its adjoint corresponds to the time reversal operator. The operator and its singular value decomposition (SVD) have been studied in [19, 18, 16, 4, 1, 6, 21, 14, 15]. Important information can be extracted from this operator. In particular the pattern of the singular value distribution can reveal the number of scatterers and the size of the scatterers, moreover, the singular vectors contain information about locations and geometry of the targets. For point scatterers, which are small compared to the wavelength, the rank of the response matrix corresponds to the number of point scatterers and the eigenspace of the response matrix is spanned by the illumination vectors, which are the wavefields at the array corresponding to a point source at one of the scatterers [19, 18, 16, 6, 17, 8]. If the point scatterers are well separated then there is a one to one correspondence between the singular vectors with nonzero singular values and the illumination vectors. These relations have been explored to focus a wave field on selected scatterer using iterated time reversal, called D.O.R.T [18, 16, 14, 15, 10, 13]. The iterated time reversal procedure corresponds to the power method for finding the singular vectors for the time reversal operator. ∗ Dept

of Math, UCI, Irvine, CA, 92697, [email protected] of Math, UCI, Irvine, CA, 92697, [email protected] ‡ Dept of Math, UCI, Irvine, CA, 92697, [email protected] The research is partially supported by ONR grant N00014-02-1-0090, DARPA grant N00014-02-1-0603, NSF grant 0307011 and the Sloan Foundation † Dept

1

Another interesting imaging algorithm, called MUSIC (MUltiple SIgnal Classification), can find the locations of point scatterers using the SVD of the response matrix [6, 11, 17, 8]. For extended scatterers that have finite sizes the above statements for point scatterers are not relevant. Even for a single extended targets there could be a continuum spectrum of singular values for the response matrix. For extended scatterers the eigenspace of the response matrix becomes more complicated. For example, it was shown in [4] that compressibility contrast and density contrast can generate different wave fields and hence multiple eigenstates even for a small spherical scatterer. The study was extended to general extended scatterers in [1] and also to electric magnetic waves in [2, 3]. In [20], the number of significant singular values for a finite aperture array is analyzed. In [21] the leading singular values and corresponding singular vectors of the response matrix are further characterized in terms of the location and the dimensions of the extended scatterers in a particular regime. In this paper we show that if the scattering target has a size that is large compared to the resolution of the active array, we can choose an appropriate signal space from the SVD of the response matrix. The signal space can be used in a version of the MUSIC algorithm to image the shape of the scatterers. This algorithm is different from the one proposed in [9], in which a shape optimization is used to match all measurements in the response matrix. The method can be parallelized easily since the evaluation of the imaging function at different grids are independent. In physical time reversal experiments the recorded signals are time reversed and sent back into the same medium. The medium will perform a matched filter and automatically focuses the wavefield back to the source. In all imaging procedures, based on post-processing of measurements, approximate Green’s function for the medium has to be used to imitate the matched filtering or time reversal process. For a cluttered heterogeneous medium the Green’s function is very difficult to estimate. We propose to use the differential field between two times to take out the unknown Green’s function for the background medium to some extent. Changes in the medium become active sources in the differential field. In particular, we apply SVD to the difference of two response matrices taken at two different times to detect and image the significant changes in the cluttered medium. The pattern of the singular values of the difference matrix corresponds to the differential field with the same active array at different times and can reveal the significant changes in the medium. We show that the SVD of the difference matrix is very similar to the case in homogeneous medium. We then apply the known results in homogeneous case to detect and image significant changes in the medium. In this paper we first briefly review the basic properties of the response matrix in Section 2. We set up the difference field formulation and the corresponding difference matrix approach in Section 3. We propose the MUSIC algorithm for shape identification, give derivations and resolution analysis in Section 4. We show extensive numerical experiments to illustrate our analysis and algorithms in Section 5. Finally, conclusions and future research are presented in Section 6. 2. The Response Matrix of an Active Array. We use an active array of transducers that can both send and receive signals in order to probe the medium. We model the propagation phenomenon in terms of the scalar wave equation corresponding to for instance acoustic waves. Figure (2.1) is a schematic illustration of the setup. Here the array of transducers is aligned on one side with limited aperture in the figure, which is usually the case for remote sensing. We can also surround the region of interest with transducers which gives a larger aperture. In the paper we will test several different setups of array locations and illustrate the differences. In the medium we put in tiny scatterers in the region to simulate the cluttered environment. The background medium could be either homogeneous or weakly heterogeneous and random. Define the interelement response Pij (t) to be the received signal at j − th transducer corresponding to an impulse sent out from i − th transducer. For an array consisting of N transducers, the matrix P (t) = [p ij (t)]N ×N is called the response matrix. Since the medium is static we have Pij (t) = Pji (t) due to spatial reciprocity. The medium and the array responses are linear and for a source signal ~e(t) = [e1 (t), e2 (t), . . . , eN (t)]T , where ei (t) is the output signal at i − th transducer and T means transpose, the reflected signal at the array is, ~r(t) = [r1 (t), r2 (t), . . . , rN (t)]T = P (t) ∗ ~e(t) , with ∗ denoting convolution in time. We have therefore ~rˆ(ω) = Pˆ (ω)~eˆ(ω), where ω is the frequency and Pˆ (ω) is the Fourier transform of P (t). In this paper, we focus on a frequency domain formulation with time harmonic waves. We briefly review the basic structure of the response matrix Pˆ (ω) 2

Calculation domain, perfectly matched layer(PML) is used. PML

array of transducers scatter

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Fig. 2.1. Setup for Target Detection and Imaging in a Strongly Cluttered Environment

for a fixed frequency and we omit ˆ notation below. Denote the Green’s function of the medium at frequency ω by G(ξ, x). Due to the spatial reciprocity, G(x, ξ) = G(ξ, x). Here we also suppress the dependence of the Green’s function on the frequency. Assume that there are M point scatterers located at x 1 , x2 , . . . , xM in the medium with reflectivity τ1 , τ2 , . . . , τM , if we neglect the multiple scattering among the scatters, then for a signal ~e(ω) = [e1 (ω), e2 (ω), . . . , eN (ω)]T sent out from the active array, the reflected signal at j − th transducer is rj (ω) =

M X N X

G(ξ j , xk )τk G(ξ i , xk )ei (ω),

k=1 i=1

where ξ 1 , ξ 2 , . . . , ξ N are the locations of the transducers. If we define the illumination vectors, ~g k , k = 1, 2, . . . , M , to be ~gk = [G(ξ 1 , xk ), G(ξ 2 , xk ), . . . , G(ξ N , xk )]T , i.e., the wave field at the array of transducers corresponding to a point source at the k − th scatter, we have P (ω) =

M X

τk~gk~gkT

and ~r(ω) = P (ω)~e(ω).

(2.1)

k=1

Due to the spatial reciprocity P (ω) is symmetric. The time reversal step is equivalent to phase conjugation in the frequency domain, and R(ω) = P (ω)P (ω) = P ∗ (ω)P (ω), which corresponds to one cycle of time reversal operation, is called the time reversal matrix (operator), with ∗ denoting the adjoint. It is shown that the time reversal operator is an optimal spatial matched filter in [20]. The matrix R(ω) is Hermitian and from (2.1) we have R(ω) =

M X

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τk0 ~gk0 ~gkT0 =

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Λk,k0 ~g k~gkT0 ,

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where T

Λk,k0 = τ k τk0 < ~gk , ~gk0 >= τ k τk0 ~g k ~gk0 . 3

(2.2)

All medium properties are embedded in the Green function in the above formulations. From representations (2.1) and (2.2), we can easily see that both the response matrix P (ω) and the time reversal matrix R(ω) are of rank M , i.e., the number of scatters, and that the range is the span of the illumination vectors ~g k , k = 1, 2, . . . , M . Define the point spread function Γ(x0 , x) =

N X

G(ξ i , x0 )G(ξ i , x).

(2.3)

i=1

Then Γ(x0 , x) is exactly the wave field at point x after phase conjugating the signal received at the active array for a point source at x0 and sending it back into the medium. That the scatterers are well resolved by the active array means Γ(xk , xk0 ) = ~gkT ~gk0 ≈ 0 if k 6= k 0 i.e., the wave field corresponding to the time reversal of a point source at one scatterer is almost zero at all other scatterers. Then ~gk (~gk ) is the left (right) singular vectors for P (ω) with singular values τk k~gk k2 since P (w)~gk = τk k~gk k2~gk ,

P ∗ (w)~gk = τk k~gk k2~gk .

(2.4)

The structure of the response matrix (2.1) can be used to image locations of point scatterers. If the Green’s function is known approximately, then the singular vectors corresponding to non-zero singular values can be time reversed to focus on the scatterers based on the one to one correspondence between the illumination vectors and the singular vectors (2.4). However, the success and resolution of this time reversal based imaging procedure depends on the resolution of the point spread function for the active array, i.e., if the point scatterers are well resolved by the active array. Another more powerful imaging procedure based on the structure of the response matrix (2.1) for point scatterers is called MUSIC (MUltiple SIgnal Classification) [6]. In MUSIC, one of the crucial steps is the definition of the signal space V S and the noise space V N for the response matrix in terms of the SVD. Denote ~g 0 (x) to be the illumination vector corresponding to a point source at x, then the imaging function is defined as I(x) =

1 , kPV N ~g 0 (x)k2

where PV N is the projection operator.

(2.5)

For point scatterers, ideally the signal space is spanned by the singular vectors corresponding to those non-zero singular values with the noise space being the complement. In MUSIC the projection to the noise space is used instead of using the one to one correspondence (2.4). Recently it is shown in [8] that also in the case with multiple scattering between point scatterers the response matrix has the structure P (ω) =

M X

τk~gk0~gkT ,

k=1

where ~gk0 is the illumination vector in the background medium and ~gk is the illumination vector in the medium that includes the interactions among all point scatterers. Hence, all singular vectors corresponding to non-zero singular values are still linear combinations of the illumination vectors in the background medium, i.e., ~g k0 . Thus, the MUSIC algorithm can be used also in this case. The main constraint for MUSIC is the requirement of knowledge of the background medium. If the scatterers have finite sizes (compared to the wave length) the above analysis is not valid anymore. The response matrix has a more complicated structures. Even for a single extended scatterer, there will be many nonzero singular values. We can classify a scatterer into three regimes in terms of the size r of the support of the point spread function, i.e., the resolution of time reversal, of an active array defined in (2.3). If the size of the scatterer s is much smaller than the resolution r of the active array then the scatterer can be regarded as a point scatterer. The response matrix for point scatterers contains only their location information. If s is not much smaller than or comparable to r then the response matrix contains both the location and qualitative shape information about the scatterer as studied in for instance [21]. For a single scatterer there are R a groups of leading singular values and singular vectors. The first singular value is related to the size of the target Ω , and the corresponding singular vector 4

to the illumination vector located at the center of the target. The next group of leading singular values includes two in two dimensions R R (threeR in three dimensions) and are related to the second moment integration of the target in each direction, Ω x2 , RΩ y 2 , ( Ω z 2 ), and the corresponding singular vectors to differentiation of the illumination vectors at the array. Here, Ω denotes the integration on the target and direction x, y, (z), are the three orthogonal directions according to the symmetry of the extended target. If s is much larger than r, then the rank of the response matrix depends on the ratio s/r [20]. Moreover, the response matrix contains detailed shape (geometry) information of the scatterer as well. In the next section we show that if we choose the proper signal space according to the singular value distribution we can use MUSIC to image the shape of the extended scatterer. Our objective is moreover to image the extended target when it is located in a heterogeneous background medium, that is, a strongly cluttered environment. In our setup the heterogeneous background medium has two components, first a set of tiny scatterers modeled with perfect reflection that simulate strongly cluttered environment, second, weakly inhomogeneous random medium fluctuations. In our model we assume the target is perfectly reflecting. These assumptions are just for implementation and discussion simplicity. Figure (2.1) shows the setup of the problem. Although the array of transducers is aligned with limited aperture in the figure, we will use several different setups of array locations in the paper with in some cases the transducers surrounding the target. 3. The Difference Matrix. In this and the next sections we use direct numerical simulations to study the response matrix and its SVD. We first examine the pattern of the singular value distribution for scatterers of various sizes in homogeneous medium, in particular we demonstrate that using SVD and MUSIC we can image the shape of a large extended scatterer. We then formulate the differential field in terms of the changes in the medium and show the structure of the associated difference matrix, next we illustrate how to use the difference matrix and its SVD to detect emerging targets in a cluttered medium. As mentioned, the basic numerical setup is illustrated in Figure 2.1. To simulate a cluttered environment a set of small scatterers are randomly positioned in the computational domain and on top of this we may also put in a weak random medium. The number of transducers and the array position vary in different tests. 3.1. The SVD of Response Matrix in Homogeneous Media. Here we use direct numerical simulations to demonstrate the basic properties of the response matrix and its SVD in homogeneous medium. For point scatterers we show that: • The number of leading singular values of the response matrix corresponds to the number of scatterers if the number of transducers is larger than the number of scatterers. • If the scatterers are well separated, the singular vectors corresponding to those leading singular values have a one to one correspondence to the illumination vector located at each scatterer, moreover, by carrying out time reversal of these singular vectors one can generate a wavefield that focuses on each of the scatterers. For extended but small scatterers, with sizes that are small or comparable to the resolution of the active array, our numerical simulations demonstrate that the spectrum of the singular values of the response matrix is separated into groups. The pattern is very similar to the analysis in [21] qualitatively. • The number of singular values in the first group of leading singular values corresponds to the number of scatterers and the corresponding singular vectors are related to the illumination vectors located at those scatterers and can be time reversed to determine the location of the scatterers if they are well separated. • The second group of leading singular values are related to the second moment integration on the scatterers in each direction. Each scatterer contributes two (three) singular values in two (three) dimensions. For large extended scatterers, the sizes of which are larger than the resolution of the active array, there are no good theory for the SVD of the response matrix thus far. Here we show numerically that • There is a certain number of significant singular values with a smooth distribution. The number of significant singular values is roughly proportional to the ratio between size of the scatterer and the resolution of the active array. • The signal space spanned by the singular vectors corresponding to the significant singular values contains the geometry information of the scatterer. As a consequence MUSIC algorithm can be used to image the shape of the scatterer. Figures 3.1, 3.2 and 3.3 show typical spectra of the singular values in log scale for scatterers of different sizes as classified above. The wavelength is 48h and the target sizes are 1h,5h,35h respectively, where h is the grid size, and 5

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Fig. 3.1. The spectrum(log scale) of the response matrix in the case with point target. The leading singular vector is roughly the illumination vector associated with the target location. 0

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Fig. 3.2. The spectrum(log scale) of the response matrix in the case with a small target. The leading groups of singular values correspond to moment information about the target shape.

the background medium is homogeneous. We use 40 transducers and they are located on all 4 sides and they are about 200h away from the target. We first consider the two cases where the target is small compared to the time reversal resolution. We see that with point and small scatterers there are clear separations in the spectrum as shown in Figures 3.1 and 3.2. The singular vector corresponding to the leading singular value is closely related to the illumination vector at the scatterer and can be time reversed to focus on the scatterer. The Example below demonstrates this in the case of a homogeneous medium. Figure 3.4(left) shows the focusing of the time reversed signal using the leading singular vector which takes place exactly on the target. In these two cases we can easily define the signal space in the MUSIC algorithm to be spanned by the leading singular vectors due to the clear spectral separation. Figure 3.4(right) show the result and also illustrates that imaging using MUSIC has better focusing resolution. The main reasons are: (1) MUSIC does not need the one to one correspondence between the illumination vector and the singular vectors of the response matrix since it uses separation of the signal space and noise space, (2) the imaging function associated with MUSIC increases the signal to noise ratio, or in other words is a better matched filter. The better resolution is also observed in experiments in [17] We remark that the above procedure can be repeated in the case with several small targets. For large scatterers the SVD structure of the response matrix is more complicated. The shape information is contained in the response matrix. From the SVD plot in Figure 3.3 we see a spectrum with many ‘large’ singular values and a smoother variation. As will be explained and demonstrated in section 4, if we choose an appropriate collection of singular vectors corresponding to these large singular values to form the signal space V S and define the orthogonal complement to V S as the noise space V N , the illumination vector corresponding to a point on the target 6

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Fig. 3.3. The spectrum(log scale) of the response matrix in the case with an extended target. The spectrum is ‘continuous’ and contains detailed information about the target shape.

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Fig. 3.4. Time reversal(left) and MUSIC(right) with one point source in a homogeneous medium

will in general be in or close to the signal space. By projection to the noise space using MUSIC algorithm, we can image the shape of the target(s). However, one important point is that each singular vector corresponding to one of the large singular values does not have a clear physical interpretation. It does not correspond to the illumination vector of a point on the target. In another word it does not focus on a point on the target when it is time reversed and sent back. 3.2. Using the Difference Matrix. In a heterogeneous medium, the main difficulty for imaging or time reversal on computers is that the Green’s function for the medium is unknown. The response matrix contains both the information of the cluttered medium and the targets. The properties of the SVD of the response matrix in a homogeneous medium no longer holds. For example, time reversal of the leading singular vectors in a homogeneous background may not focus on the locations of the targets. It might focus on some scatterers in the medium that we are not interested in. Here we propose to use the difference matrix of two response matrices to detect and image significant changes in a cluttered background. Consider a cluttered environment, e.g., there are many randomly positioned tiny scatterers. We first measure the response matrix for this medium without a target. Then we measure the response matrix for this medium with targets. We take the difference between these two response matrices and analyze properties of the difference matrix by computing the SVD of the difference matrix. Although superposition does not hold for these two measurement due to multiple scattering between the background and the emerging targets, we shall see that the pattern of the SVD of the difference matrix is much clearer and is very similar to the pattern of the SVD of the response matrix for targets in a homogeneous medium. Moreover, if we apply time reversal in a homogeneous medium, to each of the leading singular vectors, the back propagated field will focus on each target. The main motivation for using the difference field is the approximate cancelation of the scattered field due to the cluttered environment as we 7

show next. Suppose the wave field in the cluttered medium, u1 (x), satisfies the Helmholtz equation: 4u1 (x) + k 2 (1 + n(x))u1 (x) = f (x),

(3.1)

where k is the wave number in the homogeneous medium, n(x) is the variation of the refraction index in the cluttered medium, and f (x) is the source. After some localized change in medium, for instance, the appearance of new targets, the wave field, u2 (x), in the medium with the same source satisfies 4u2 (x) + k 2 (1 + n(x) + δn(x))u2 (x) = f (x),

(3.2)

where δn(x) denotes localized changes in the medium. Now the difference field u ˜(x) = u 2 (x) − u1 (x) satisfies 4˜ u(x) + k 2 (1 + n(x))˜ u(x) = −k 2 δn(x)u2 (x).

(3.3)

Here we point out two key aspects of the above formulations. 1. The localized changes become active sources in equation (3.3) for the differential field. 2. The equation for the differential field, (3.3), can also be written as 4˜ u(x) + k 2 u ˜(x) = −k 2 δn(x)u2 (x) − k 2 n(x)˜ u(x)

(3.4)

If the wave energy reaching the changes in the medium is significant (by designing an appropriate wave form such as beam forming) then the wave field generated by the changes δn(x)u 2 (x) is stronger than the wave field corresponding to multiple scattering of the cluttered medium u ˜(x) = u 2 (x) − u1 (x), the differential field is changed from the homogeneous formulation 4˜ u(x) + k 2 u ˜(x) = −k 2 δn(x)u2 (x)

(3.5)

by a high order term n(x)˜ u(x), while the wave field for equation (3.2) contains information of both the cluttered medium n(x) and the targets δn(x). Hence directly using the response matrix and not forming the difference matrix will not reveal target information in a cluttered environment clearly. Below we present numerical experiments that demonstrate how we can use the difference matrix formulation for robust imaging even if the background is dynamically changing. We remark that if there is a scale separation between the size of the background scatterers and the targets, we can choose a wavelength that is large compared to the scatterers and small or comparable to the targets, which will enhance the cancelation (or superposition) for the differential field. Difference in material properties between the cluttered background and the targets can also be exploited by choosing appropriate tuned wave length or wave forms that interacts with the material contrast. 4. Imaging the geometry of extended targets using SVD of the response matrix. Recall that the imaging function of the MUSIC algorithm to locate point scatterers using singular vectors of the response matrix P : I(x) =

1 , kPV N ~g 0 (x)k2

(4.1)

where x is a search point and PV N is projection to the noise space V N which is the orthogonal complement of the signal space V S spanned by the significant singular vectors. This function peaks at locations of scatterers. From the matched filter point of view, imaging using either time reversal or MUSIC will result in focusings at places where the wave field or signal comes from. In our case if we can model the scattered wavefield, we can locate where strong scatterings occur. Based upon the above motivation we can imagine the geometry of extended scatterers using MUSIC. Again we select a set of significant singular vectors, ~vi , i = 1, 2, . . . , r, to span our signal space VS and then use MUSIC. We have the following claims: 1. For sound-soft targets, i.e., targets with Dirichlet boundary condition u = 0, the imaging function peaks on the boundary of the targets 2. For targets with smooth variation of contrast, the imaging function peaks in the interior of the targets. 3. For targets with a constant contrast, the imaging function peaks on or near the boundary of the targets 8

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Fig. 4.1. A sound-soft target of circular shape with radius about 3 wavelength

We justify our claims by studying the scattered wavefield and verify the analysis using numerical results below. For sound-soft targets, Huygen’s principle gives Z ∂u i u(x) = u (x) − G0 (x, y)dS(y) (4.2) ∂D ∂ν where u is the total field and ui is the incident field, G0 is the homogeneous Green’s function, D is the support of the target(s). Since we model the incoming wave ui (x) as a point source the response matrix for the scattered field is Z ∂G(xi , y) Pij = − G0 (xj , y)dS(y) (4.3) ∂ν ∂D where xi is the source (ith transducer) and xj is the receiver (jth transducer) location. Here, G is the inhomogeneous Green’s function in the presence of the scatterers. This formula says that our measured field, the response matrix, comes from scattering at the boundary of the targets. In particular, it can be viewed as a weighted integral of the single layer potential of the homogeneous Green’s function. The matrix form is T  Z ∂~g (y) dS(y) (4.4) ~g0 (y) P =− ∂ν ∂D where ~g0 (y) and ~g (y) are the illumination vectors in the homogeneous and inhomogeneous medium respectively. The h iT boundary parts that are well illuminated by the array, i.e., where ∂~g∂ν(y) is significant, can be viewed as sources for the scattered field and should be in the signal space of the response matrix. Let σ i be the singular values with corresponding singular vectors ~vi of the response matrix P , , i = 1, 2, . . . , n. Let  be an appropriate threshold such that the signal space V S = span{~vi |σi > } and the noise space V N is the orthogonal complement of VS . We apply the standard MUSIC algorithm (2.5), the imaging function will peak on those well illuminated boundary parts. The dimension of the signal space or the choice of  will be discussed in more details in our resolution study below. Figure (4.1) shows the result of the imaging function above for a circular sound-soft target. A circular array with 80 transducers with equal spacing are used and the wavelength is λ = 16h. The radius of the circular target is 45h. The distance from the transducers to the target center is about 200h. The medium is homogeneous medium. The first 24 singular vectors are used in the imaging function. The SVD pattern is similar to Figure (3.3). Due to the symmetry of our setup the whole boundary is illuminated uniformly by the array and hence the imaging function peaks on the boundary of the target fairly uniformly, as expected. For target with a smooth variation of contrast, Lippmann-Schwinger equation gives Z u(x) = ui (x) + G0 (x, y)σ(y)u(y)dy. (4.5) D

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Fig. 4.2. Circular shape with radius about 3 wavelength and smooth transition of contrast

Here D is the target region(s) and σ is the contrast which is a smooth function with compact support. The interelement response is Z σ(y)G0 (xi , y)G(xj , y)dy (4.6) Pij = D

and the matrix form is P =

Z

σ(y)~g0 (y)~g T (y)dy,

(4.7)

D

The difference between this case and the previous case is that the integral is in the whole region D. Well illuminated parts and/or high contrast parts, i.e., where σ(y)G0 (xi , y) is large, can be viewed as the sources for the scattered wavefield. Hence the imaging function from the MUSIC will peak on those parts in region D. Figure (4.2) shows the result of the imaging function above for a circular target with smooth variation of contrast. The profile of the contrast is 0.5(3 + cos(2π Rr )), where r is the distance to the center of the target, R is the radius of the target. This function has value 1 when r = R. Consequently, the transition from the background homogeneous medium to the target is smooth. Other parameters are the same as the example in the previous case. The first 37 singular vectors are used in the imaging function. We observe big value inside the target, as expected. For target with a constant contrast, we have the following equation from potential theory: Z ∂G0 (x, y) u(x) = ui (x) + ψ(y) + G0 (x, y)φ(y)dS(y) (4.8) ∂ν(y) ∂D where D is the target region(s), φ and ψ are density functions for single and double layer potentials. So the interelement response is Z ∂G0 (xj , y) ψ(xi , y) + G0 (xj , y)φ(xi , y)dS(y) (4.9) Pij = ∂ν(y) ∂D and the matrix form is P =

Z

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∂~g0 (y) ~ T ~ T (y)dS(y) ψ (y) + ~g0 (y)φ ∂ν(y)

(4.10)

Again for the scattered field the sources, which are a combination of monopoles and dipoles in this case, are located on the boundary. Figure (4.3) shows the result of the imaging function for a circular target with a constant contrast that equals 3. The setup is the same as previous examples. The first 20 singular vectors are used in the imaging function. 10

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Fig. 4.3. Circular shape with radius about 3 wavelength and a constant contrast

Next, we carry out a resolution analysis to characterize the signal space and give a guideline for choosing the number of significant singular vectors for imaging. We consider only sound-soft target(s) in this study. Let N be the number of significant singular vectors(these significant singular vectors span the signal space). In the case of uniform illumination of the boundary, we have the relation D α = CN , where D is the perimeter of the target(s) boundary, α is the diameter of the resolution for the array and C is a dimensionless factor of proportionality. In more general situations the above relation will be more complicated. For example, the material properties of the target, such as contrast, may affect the relation. Moreover, the geometry of boundary, such as concavity, and the configuration of the array, such as limited aperture, will affect the illumination strength at different parts of the boundary. Define the following parameters: 1. λ: wavelength 2. L: average distance from the transducers to the target(s) 3. a: aperture of the transducer array 4. δ: spacing between two adjacent transducers 5. D: perimeter of the target(s) boundary that is well illuminated. When the distance between the target and the array is large compared with the wavelength, the resolution of the Da array is proportional to λL a so we have the following relation [20]: λL = CN , where C is a constant. (Since all parts of the boundary are well illuminated by our circular array, we can apply this relation for transversal resolution.) Let P = U ΣV H be the singular value decomposition of the response matrix. Assume σ 1 > σ2 > . . . > σn are the singular values for the response matrix and the signal space V S is spanned by the first N singular vectors. Define kP k2 = σNσ1+1 , where Psignal = UN ΣN VNH , UN , VN denote the first N columns of U and the signal-to-noise ratio: kPsignal noise k2 V , respectively, ΣN is corresponding submatrix of Σ, Pnoise = P − Psignal . We claim that σNσ1+1 depend only weakly on the parameters above and the shape of the target(s). We design the following numerical tests to verify our claim. In our experiments we use a circular array with evenly spaced transducers in a homogeneous medium. We change the parameters listed above as well as geometry and size of the targets. In the case of non-convex geometry, such as the two well separated circles and seven leave geometry, they are weakly concave. Let N0 be the dimension of the signal space when the parameters (λ0 , L0 , a0 , δ0 , D0 ) are used. Pick a threshold  such that σσN1 <  ≤ σNσ1+1 . This experiment is set as the reference case. For any other parameter sets, we first use the relation Da λL = CN to predict the value N and then use the same  to find the actual N . The following table gives the results for the homogeneous medium.

In practice calibration and noise level may be used to determine this . In our numerical experiments for extended targets, we follow this guideline to choose the number of significant singular vectors and the imaging function always gives reasonably good result, as shown in Section 5. Remark1: The above analysis only claims that the imaging function will peak on the well illuminated boundary 11

shape circle circle circle circle circle circle ellipse rectangle triangle 7 leaves 2 circles

λ λ0 λ0 3 2 λ0 λ0 λ0 λ0 λ0 λ0 λ0 λ0 λ0

L L0 L0 L0 L0 L0 2 3 L0 L0 L0 L0 L0 ≈ L0

a a0 a0 a0 a0 5 8 a0 2 3 a0 a0 a0 a0 a0 a0

δ δ0 δ0 δ0 2δ0 δ0 2 3 δ0 δ0 δ0 δ0 δ0 δ0

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Table 4.1 Numerical justification of the invariance of the signal-to-noise ratio. The signal-to-noise ratio is close to a constant since the last two columns of the table almost agree.

parts. However it does not preclude the possibility of illumination of other points in the domain. Those points are usually inside the boundary and their corresponding illumination vectors may be close to some linear combination of those significant singular vectors due to some symmetry of the the setup. Physically this can by explained by resonance or interference patterns. We do find such situations in our numerical test, see Figure (5.7). Remark2: The reason that we use the MUSIC algorithm instead of time reversal procedure to image the geometry of extended scatterers is because of the better resolution and better signal to noise ratio of MUSIC. This allow us to get a sharp resolution of the geometry. If we use time reversal of the significant singular vectors we will get smeared target boundary. Remark3: Our approach is different from linear sampling method [5, 12]. The linear sampling method is based on the far field scattering operator and uses an inverse approximation. Here we use a direct approach which is essentially a matched filter. No inverse operation is used.

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5. Numerical Experiments. We present numerical experiments that demonstrate imaging of extended targets and the use of difference matrix and the robustness of this procedure in a strongly cluttered environment. In all the examples below the calculation domain is 499h-by-499h, where h is the grid size, and we use a Helmholtz solver with PML for the forward problem in order to generate the response matrix and difference matrix. The search box for imaging is chosen to be (101:400,101:400). For all the imaging figures we use the coordinate system that puts the origin at (101,101). • We consider the case with the random background medium with strong scatterers. Due to the strong inhomogeneities in the medium, the pattern of the response matrix in homogeneous medium will be destroyed, it will be difficult to tell how many targets there are if we use the response matrix only. We use the difference matrix to cancel out the effect from the scatterers. Consider a random medium with one percent standard deviation, correlation length 10h and 16 tiny scatterers of size h surrounding two targets of size 9h. We use the wave length λ = 30h and 10 transducers on the left and 10 more at the top of the calculation domain. Figure (5.1) shows the distribution of the singular values for response matrix(left) and the difference matrix(right). Due to the signature from the scatterers we do not see a clear pattern if we use the response matrix. However, if we use the difference matrix that cancels out the effect from the scatterers to some extent, the SVD pattern shows two dominant singular values corresponding to the two targets. Figure (5.2) shows the result of focusing at the target locations using time reversal. The first target is centered at (110,180) and the numerical global maximum(excluding the neighborhood of the transducers) occurs at (109,178). The second target is centered at (200,90) and the numerical global maximum (excluding the neighborhood of the transducers) occurs at (198,88). We could also use the imaging function associated with the music algorithm. Figure (5.3) shows the result. Again MUSIC has better signal-to-noise ratio than time reversal. The difference matrix idea can also be naturally applied to tracking moving targets by taking measurements at different times. Assume that there is one moving target and the background is static, we first form the response matrix at certain time, then form another at a later time and take the difference. This is equivalent to taking the difference between the response matrix with and without two targets, except that now one of the targets is a pseudo image. Following the same procedure as above, we can find the locations of the target at two different times and in general track the moves of the target. • Consider now tracking a moving target when the parameters are the same as the previous example. Figure (5.4) shows the distribution of the singular values. We observe two dominant singular values. When taking the difference, one moving target is equivalent to two emerging targets and the pattern is exactly as we would expect. Figure (5.5) show the result of time reversal focusing. The target center is moved from (110,180) to (200,90). The numerical global maximum (excluding the neighborhood of the transducers) occurs at (110,178), (197,87) for the two singular vectors. Figure (5.6) shows the corresponding result using MUSIC imaging function. 13

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• Finally, we demonstrate how the MUSIC algorithm can be used to image the shape of extended targets whose size is comparable or larger than the resolution of the active array. We use 80 transducers with equal spacing that surround the target(s). The radius of the transducer array is 200h and the wave length is 16h. Figure (5.7) shows the MUSIC imaging function for a sound-soft target with the shape of 5 leaves: r = 40(1 + 0.2 cos(5θ))h in the homogeneous medium. We observe large values on the boundary of the target. There are also some spots inside the targets with large value, as predicted above in Remark1. Figure (5.8) shows the imaging function for 3 targets of different shapes: a circle, an ellipse and a 3-leaf shape in the homogeneous medium. As long as the targets are well separated and the multiple scattering is not strong, we observe good results. Next we have a group of tests to show the significance of using the difference matrix. Consider a 5-leaf target with 16 random tiny scatterers of size approximately 0.2λ in a random background medium with two percent standard deviation and with correlation length 10h. Figure (5.9) shows the imaging function by using the response matrix. Due to the signature of the random variation and the 16 scatterers the result is poor. Figure (5.10) shows the imaging function by using the difference matrix in the following way: first make a measurement with the background random medium only. Then make a measurement with the presence of the 16 scatterers and the target. By using the difference matrix to some extent we cancel out the effect from the background randomness. The result is clearly much better than the previous one. Figure (5.11) shows the imaging function by using the difference matrix in the following way: first make a measurement with the 16 tiny scatterers and the background random medium. Then make a measurement when the target is introduced. Again, by using the difference matrix to some extent we cancel out the effect 15

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from the background randomness as well as the 16 scatterers. We no longer see the 16 scatterers and the target shape is clear. Figure (5.12) shows the imaging function for tracking a moving target with the shape of three leaves in the random medium with one percent standard deviation and correlation length 10h. By using the difference matrix, we detect both the old and the new location and shape of the target. 6. Conclusions. Detection of point scatterers in a homogeneous medium is well understood and in the time harmonic case the MUSIC algorithm provides an important tool for this task. Here we have generalized the MUSIC framework to image the shape of extended targets using SVD of the response matrix. We use the difference matrix to image medium changes and to track moving targets in a cluttered background. In future work we will show how the approach can be improved further by using multiple frequencies and travel time information to estimate the essential random modulation of the background Green’s function.

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