A novel direct imaging method for passive inverse obstacle scattering problem

Assume that $D\subset\mathbb{R}^{2}$ is a bounded simply connected sound-soft obstacle with a $C^{2}$-boundary $\partial{D}$. Given the time-harmonic point source located at $z\in\mathbb{R}^{2}\setminus\overline{D}$, the incident field $u^{i}$ can be described as

$\displaystyle u^{i}(x,z):=\phi(x,z)=\frac{{\mathrm{i}}}{{4}}H_{0}^{(1)}(k|x-z|),\ \ x\in\mathbb{R}^{2}\setminus\{\overline{D}\cup\{z\}\},$

(2.1)

where $\phi(x,z)$ denotes the outgoing Green function of Helmholtz equation in $\mathbb{R}^{2}$, $H_{0}^{(1)}$ is the Hankel function of the first kind of order zero, $k\in\mathbb{R}_{+}$ is the wave number and $\mathrm{i}:=\sqrt{-1}$ is the imaginary unit. The presence of the sound-soft obstacle $D$ will interrupt the propagation of the incident wave $u^{i}$, leading to the exterior scattered field $u^{s}$. The direct scattering problem is to find $u^{s}=u-u^{i}$ that satisfies the following Helmholtz system:

$\displaystyle\bigtriangleup u^{s}(x,z)+k^{2}u^{s}(x,z)=0\ \ \mbox{in}\ \mathbb{R}^{2}\backslash\overline{D},$

(2.2)

$\displaystyle u^{s}(x,z)=-u^{i}(x,z)\ \ \mbox{on}\ \partial{D},$

(2.3)

$\displaystyle\lim_{r=|x|\rightarrow\infty}\sqrt{r}\Big{(}\frac{\partial{u^{s}(x,z)}}{\partial{r}}-\mathrm{i}ku^{s}(x,z)\Big{)}=0,$

(2.4)

where $u$ is the total field and the Sommerfeld radiation condition (2.4) describes the outgoing nature of the scattered field $u^{s}$ and holds uniformly in all directions. The well-posedness of the forward scattering problem (2.2)-(2.4) is well known (cf. [16, 35]).

We are now going to introduce the considered passive inverse obstacle scattering problem. Let $\partial{B}$, which is a circle of the radius $r_{B}$ and contains $D$ in its interior, be the measurement curve. Measurement points $x_{j}(j=1,\cdots,J)$ are located on $\partial{B}$. Let $z_{l}(l=1,\cdots,L)$ be the positions of incident point sources $\phi(x,z_{l})$ that are randomly distributed on $\Sigma$, where $\Sigma$ denotes the incident curve that encloses $\partial{B}$. Moreover, the random position $z_{l}$ is far away from the obstacle $D$. Then, the corresponding inverse problem is to determine the location and shape of the sound-soft obstacle $D$ from the passively collected total field $u(x_{j},z_{l})$. We refer to Fig. 1(a) for the specific illustration of this passive inverse scattering problem. In order to more intuitively describe the difference of passive imaging and active imaging, we also present illustration of active imaging in Fig. 1(b). In the active setup, measurement points $x_{j}$ and source positions $x_{m}$ are both controlled and located on $\partial{B}$. Clearly, the main difference of these two problems is whether excitation sources are controlled. Unfortunately, we can not place active excitation sources particularly in the practical scenario that one needs to face the limitation of safety or environment, and meanwhile passive imaging viewpoint can be naturally adopted.

In this paper, we want to design a direct sampling method for solving this problem. The direct sampling method has emerged as one of the most popular reconstruction techniques for inverse scattering problems due to its efficient computing capabilities. The fundamental idea is to develop a distinct imaging functional that can be used to qualitatively visualize the scatterer by applying its values at various probing points. However, in our considered passive imaging setting, directly using the scattered field $u^{s}(x_{j},z_{l})$ to design the traditional direct sampling method may not be a good choice because of the randomness of the position $z_{l}$. In order to overcome this challenge, this paper proposes a novel direct imaging scheme whose specific details are presented in the following section.

To describe our method, we use mathematical notations similar to works [17, 39]. Denote by $\Omega$ a sampling domain that contains the obstacle $D$. Before we continue, we recall $\phi(x,y)=\frac{{\mathrm{i}}}{{4}}H_{0}^{(1)}(k|x-y|)$ that is the fundamental solution of the Helmholtz equation

$\displaystyle\bigtriangleup\phi(x,y)+k^{2}\phi(x,y)=-\delta_{y}(x)\ \ \mbox{in}\ \mathbb{R}^{2},$

(3.1)

and satisfies the Sommerfeld radiation condition

$\displaystyle\lim_{r=|x|\rightarrow\infty}\sqrt{r}(\frac{\partial{\phi}}{\partial{r}}-\mathrm{i}k\phi)=0,$

(3.2)

where $\delta_{y}(\cdot)$ is the Dirac source located at $y$. In order to alleviate the difficulty caused by random sources at positions $z_{l}$, we compute the cross-correlation matrix with its entries $C_{jm}$ defined at $x_{j}$ and $x_{m}$ by

$\displaystyle C_{jm}=\frac{2\mathrm{i}k|\Sigma|}{L}\sum_{l=1}^{L}\overline{u\left(x_{j},z_{l}\right)}u\left(x_{m},z_{l}\right)-\left[\phi\left(x_{j},x_{m}\right)-\overline{\phi\left(x_{j},x_{m}\right)}\right],\quad 1\leq j,m\leq J,$

(3.3)

where $|\Sigma|$ denotes the area of the surface $\Sigma$ on which incident sources are randomly distributed, and $\phi\left(x_{j},x_{m}\right)$ is the measurement collected at $x_{j}$ of the point source located at $x_{m}$.

In what follows, we shall connect the cross-correlation (3.3) to the active scattering model. To this end, let $x,y$ be far from $\Sigma$. Then, the Helmholtz-Kirchhoff identity for $\phi(x,y)$ is described as [17, 39]

$\displaystyle\phi(x,y)-\overline{\phi(x,y)}=2\mathrm{i}k\int_{\Sigma}\overline{\phi(x,z)}\phi(y,z)\mathrm{d}s(z).$

(3.4)

We remark that (3.4) holds for the case that $x$ and $y$ are far from $\Sigma$, namely for $z\in\Sigma$, $|x-z|$ and $|y-z|$ are both required to be large enough. Viewing $\phi(x,y)$ as the incident source located at $y$, the corresponding total field measured at $x$ is given by $u(x,y)=\phi(x,y)+u^{s}(x,y)$. The Helmholtz-Kirchhoff identity also holds for $u(x,y)$ [17, 39], that is,

$\displaystyle u(x,y)-\overline{u(x,y)}=2\mathrm{i}k\int_{\Sigma}\overline{u(x,z)}u(y,z)\mathrm{d}s(z).$

(3.5)

The formulas (3.4) and (3.5) immediately give rise to

$\displaystyle u^{s}(x,y)-\overline{u^{s}(x,y)}=2\mathrm{i}k\int_{\Sigma}\overline{u(x,z)}u(y,z)\mathrm{d}s(z)-[\phi(x,y)-\overline{\phi(x,y)}].$

(3.6)

By replacing $x$ and $y$ by $x_{j}$ and $x_{m}$ respectively, we obtain

$\displaystyle u^{s}(x_{j},x_{m})-\overline{u^{s}(x_{j},x_{m})}=2\mathrm{i}k\int_{\Sigma}\overline{u(x_{j},z)}u(x_{m},z)\mathrm{d}s(z)-[\phi(x_{j},x_{m})-\overline{\phi(x_{j},x_{m})}].$

(3.7)

Note that $u^{s}(x_{j},x_{m})$ can be regarded as the active scattered field measured at $x_{j}$ due to the illumination by an active source located at $x_{m}$. For the sake of the following use, we define

$\displaystyle N^{s}_{jm}=u^{s}(x_{j},x_{m})-\overline{u^{s}(x_{j},x_{m})},\quad 1\leq j,m\leq J.$

(3.8)

Therefore, the relationship between the cross-correlation (3.3) and the imaginary scattered field (3.8) is

$\displaystyle C_{jm}\approx N^{s}_{jm}.$

(3.9)

In fact, (3.3) is the discrete approximation form of (3.8), and the approximation accuracy between these two parts relies on the total number $L$ and the random position $z_{l}$. See Fig. 1(b) for the illustration of the introduced active scattering problem.

Before we proceed, let us explain why traditional direct sampling methods will fail in passive imaging. Indeed, traditional direct sampling methods are constructed by using the scattered field $u^{s}(x_{j},z_{l})$, but in passive imaging setup only total field $u(x_{j},z_{l})$ is accessible and the scattered field can not be attainable due to the randomness of $z_{l}$. Even though one can have $u^{s}(x_{j},z_{l})$, the influence of randomness of $z_{l}$ still exists. To show this, we write the indicator functional of classical reverse time migration method by using $u^{s}(x_{j},z_{l})$, which is in the form

$\displaystyle I_{RTM}(\tau)=-k^{2}\mathrm{Im}\left\{\frac{2\pi r_{B}|\Sigma|}{J}\sum\limits_{l=1}^{L}\sum\limits_{j=1}^{J}\phi(\tau,z_{l})\phi(\tau,x_{j})\overline{u^{s}(x_{j},z_{l})}\right\},$

(3.10)

where $\tau\in\Omega$ is the sampling point. Clearly, in the imaging process this indicator (3.10) shall inevitably fail since positions $z_{l}$ are unknown. The above discussed two issues of traditional direct sampling approaches can be overcome by employing (3.3) instead of $u^{s}(x_{j},z_{l})$. Thus in what follows, we shall introduce the specific design of our imaging functional. Because of the approximation relationship (3.9), we can naturally benefit from the existing active imaging methods. Comparable to the reverse time migration method, our imaging algorithm contains two phases. The first phase is the back-propagation stage in which the complex conjugated data $\overline{C_{jm}}$ is back-propagated into the domain. Specifically, for $m=1,2,\cdots,J$, we compute the solution $v_{b}$ to the following problem:

$\displaystyle\bigtriangleup v_{b}(x,x_{m})+k^{2}v_{b}(x,x_{m})=\frac{2\pi r_{B}}{J}\sum\limits_{j=1}^{J}\overline{C_{jm}}\delta_{x_{j}}(x)\ \ \mbox{in}\ \mathbb{R}^{2},$

(3.11)

$\displaystyle\lim_{r=|x|\rightarrow\infty}\sqrt{r}(\frac{\partial{v_{b}}}{\partial{r}}-\mathrm{i}kv_{b})=0.$

(3.12)

By (3.1), (3.2) and the linearity, the solution $v_{b}$ to (3.11) and (3.12) can be represented by

$\displaystyle v_{b}(x,x_{m})=-\frac{2\pi r_{B}}{J}\sum\limits_{j=1}^{J}\overline{C_{jm}}\phi(x,x_{j}).$

(3.13)

Then for $\tau\in\Omega$, the imaginary part of the cross-correlation of the fundamental solution $\phi(\tau,x_{m})$ and the back-propagated field $v_{b}(\tau,x_{m})$ is employed in the second phase, which has the form

	$\displaystyle I(\tau)$	$\displaystyle=k^{2}\mathrm{Im}\left\{\frac{2\pi r_{B}}{J}\sum\limits_{m=1}^{J}\phi(\tau,x_{m})v^{b}(\tau,x_{m})\right\}$		(3.14)
		$\displaystyle=-k^{2}\mathrm{Im}\left\{\frac{(2\pi)^{2}r_{B}^{2}}{J^{2}}\sum\limits_{m=1}^{J}\sum\limits_{j=1}^{J}\phi(\tau,x_{m})\phi(\tau,x_{j})\overline{C_{jm}}\right\}.$

The formula (3.14) is the main imaging functional of the proposed DCM. (3.3) and (3.14) are the so-called two cross-correlations of DCM.

In this subsection, we first investigate the resolution analysis of the proposed DCM, which aims to show different properties of indicator functional (3.14) when probing point $\tau$ approaches the boundary $\partial D$ and goes away from $\partial D$. To this end, in the following we estimate the bound of (3.14) when $\tau$ changes. Then we further study the stability analysis that indicates our DCM is robust to the added noise. Considering the relationship (3.9), we define

$\displaystyle I_{N^{s}}(\tau)=-k^{2}\mathrm{Im}\left\{\frac{(2\pi)^{2}r_{B}^{2}}{J^{2}}\sum\limits_{m=1}^{J}\sum\limits_{j=1}^{J}\phi(\tau,x_{m})\phi(\tau,x_{j})\overline{N^{s}_{jm}}\right\}.$

(3.15)

If the sampling point $\tau$ is contained inside the curve $\partial{B}$, then $\phi(\tau,x_{j})$ and $\phi(\tau,x_{m})$ are both smooth functions. Furthermore, the imaginary scattered field $N^{s}_{jm}$ is also smooth. As a result, $I_{N^{s}}$ can be regarded as the discrete approximation of the following continuous indicator function:

	$\displaystyle\hat{I}_{N^{s}}(\tau)$	$\displaystyle=-k^{2}\mathrm{Im}\int_{\partial{B}}\int_{\partial{B}}\phi(\tau,x_{m})\phi(\tau,x_{j})\overline{N^{s}_{jm}}\mathrm{d}s(x_{m})\mathrm{d}s(x_{j})$		(3.16)
		$\displaystyle=-k^{2}\mathrm{Im}\int_{\partial{B}}\int_{\partial{B}}\phi(\tau,x_{m})\phi(\tau,x_{j})\overline{u^{s}(x_{j},x_{m})}\mathrm{d}s(x_{m})\mathrm{d}s(x_{j})$
		$\displaystyle+k^{2}\mathrm{Im}\int_{\partial{B}}\int_{\partial{B}}\phi(\tau,x_{m})\phi(\tau,x_{j})u^{s}(x_{j},x_{m})\mathrm{d}s(x_{m})\mathrm{d}s(x_{j}).$

Clearly, the first term at the right hand side of (3.16) is the standard indicator of reverse time migration method that is presented in [9]. Its analysis is also thoroughly studied in [9]. Thus, for the following resolution analysis, we mainly focus on estimating the second term. To this end, we first present some lemmas for the following use.

Lemma 3.1 depicts the well-posedness of the forward scattering problem. We refer to [16, 35] for more relevant discussions. Define the Dirichlet-to-Neumann mapping $T:H^{1/2}(\partial D)\rightarrow H^{-1/2}(\partial D)$ for any $\widetilde{\omega}\in H^{1/2}(\partial D)$ by $T(\widetilde{\omega})=\frac{\partial\omega}{\partial\nu}$. It follows from Lemma 3.1 that $T$ is a bounded linear operator. Denote the norm by $\|T\|$. Since the obstacle $D$ and the sampling point $\tau$ are both contained inside $\partial B$, we define $h_{1}=\mathrm{dist}(\partial{B},\partial{D})$ and $h_{2}=\mathrm{dist}(\tau,\partial{B})$. The following lemma on the estimation of the second term at the right hand side of (3.16) holds.

Now, we are ready to present our main results.