Factorization method with one plane wave: from model-driven and data-driven perspectives

Before stating the one-wave version of the factorization method, we first present a corollary of theorems 2.2 and 2.3. Denote by ${\Omega}\subset {\mathbb{R}}^{2}$ a convex Lipschitz domain which may represent a sound-soft or impedance scatterer. From numerical point of view, Ω will play the role of test domains for imaging the unknown scatterer D. Here we use the notation Ω in order to distinguish from our target scatterer D. The far-field operator corresponding to Ω is therefore given by

$$\begin{equation}({F}_{{\Omega}}g)(\hat{x})={\int }_{\mathbb{S}}{u}_{{\Omega}}^{\infty }(\hat{x},d)g(d)\mathrm{d}s(d),\quad {F}_{{\Omega}}:{L}^{2}(\mathbb{S})\to {L}^{2}(\mathbb{S}),\end{equation} \tag{ 3.1 }$$

where ${u}_{{\Omega}}^{\infty }(\hat{x},d)$ is the far-field pattern corresponding to the plane wave e^ikx⋅d incident onto Ω. The eigenvalues and eigenfunctions of F_Ω will be denoted by $({\lambda }_{{\Omega}}^{(j)},{\varphi }_{{\Omega}}^{(j)})$.

Corollary 3.1. Let ${v}^{\infty }\in {L}^{2}(\mathbb{S})$ and assume that k² is not an eigenvalue of −Δ over Ω with respect to the boundary condition under consideration. Then

$$\begin{equation}I({\Omega})=\sum\limits _{j\in \mathbb{Z}}\frac{{\left\vert {\left\langle {v}^{\infty },{\varphi }_{{\Omega}}^{(j)}\right\rangle }_{\mathbb{S}}\right\vert }^{2}}{\left\vert {\lambda }_{{\Omega}}^{(j)}\right\vert }< +\infty \end{equation} \tag{ 3.2 }$$

if and only if v^∞ is the far-field pattern of some radiating solution v^s , where v^s satisfies the Helmholtz equation

$$\begin{equation}{\triangle}{v}^{s}+{k}^{2}{v}^{s}=0\quad \text{in}\quad {\mathbb{R}}^{2}{\backslash}\bar{{\Omega}},\end{equation} \tag{ 3.3 }$$

with the boundary data v^s |_∂Ω ∈ H^1/2(∂Ω).

Remark 3.2. If ${v}^{\infty }(\hat{x})={\mathrm{e}}^{-\mathrm{i}k\hat{x}\cdot z}$, then by theorems 2.2 and 2.3 it holds that I(Ω) < ∞ if and only if z ∈ Ω. This implies that the scattered field v^s (x) = Φ(x, y) is a well-defined analytic function in ${\mathbb{R}}^{2}{\backslash}\bar{{\Omega}}$ and v^s |_∂Ω ∈ H^1/2(∂Ω) if and only if I(Ω) < ∞. Hence, the results in corollary 3.1 follow directly from theorems 2.2 and 2.3 in this special case.

Proof. In Ω is sound-soft, by (2.2) we have ${F}_{{\Omega}}=-{G}_{{\Omega}}{S}_{{\Omega}}^{\ast }{G}_{{\Omega}}^{\ast }$, where ${G}_{{\Omega}}:{H}^{1/2}(\partial {\Omega})\to {L}^{2}(\mathbb{S})$ is the data-to-pattern operator corresponding to Ω. Obviously, I(Ω) < +∞ if and only if ${v}^{\infty }\in \mathrm{R}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}({({F}_{{\Omega}}^{\ast }{F}_{{\Omega}})}^{1/4})$. Since $\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}({({F}_{{\Omega}}^{\ast }{F}_{{\Omega}})}^{1/4})=\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}({G}_{{\Omega}})$, we get v^∞ ∈ Range(G_Ω) if and only if I(Ω) < +∞. Recalling the definition of G_Ω, it follows that v^s satisfies the Helmholtz equation (3.3) and the Sommerfeld radiation condition (1.4) with the boundary data v^s |_∂Ω ∈ H^1/2(∂Ω). The impedance case can be proved in an analogous manner by applying the factorization ${F}_{{\Omega}}=-{G}_{{\Omega},\mathrm{i}\mathrm{m}\mathrm{p}}{T}_{{\Omega},\mathrm{i}\mathrm{m}\mathrm{p}}^{\ast }{G}_{{\Omega},\mathrm{i}\mathrm{m}\mathrm{p}}^{\ast }$ (see (2.7)) and the range identity Range(F_Ω,# ) = Range(G_Ω).

Corollary 3.1 relies essentially on the factorization form (see e.g., (2.2), (2.7)) of the far-field operator and is applicable to both penetrable and impenetrable scatterers Ω. However, establishing the abstract framework of the factorization method turns out to be nontrivial in some cases, for instance, time-harmonic acoustic scattering from mixed obstacles and electromagnetic scattering from perfectly conducting obstacles. In this subsection we show that corollary 3.1 can be justified independently of the factorization form, if the test domain Ω is chosen to be a disk of acoustically Dirichlet or impedance type. This is mainly due to the explicit form of far-field patterns for Dirichlet and impedance disks.

Let B_h (z) be the disk centered at $z\in {\mathbb{R}}^{2}$ with radius h > 0. The boundary of B_h (z) is denoted by Γ_z,h := {x: |x − z| = h}. It is supposed that B_h (z) is either a sound-soft disk in which k² is not the Dirichlet eigenvalue of −Δ, or an impedance disk with the constant impedance coefficient $\eta \in \mathbb{C}$ such that Im(η) > 0. Denote by ${u}_{z,h}^{\infty }(\hat{x},d)={u}^{\infty }(\hat{x};{B}_{h}(z),d,k)$ the far-field pattern incited by the plane wave e^ikx⋅d incident onto B_h (z) and by F_z,h the associated far-field operator, that is,

$$\begin{equation}({F}_{z,h}g)(\hat{x})={\int }_{\mathbb{S}}{u}_{z,h}^{\infty }(\hat{x},d)g(d)\mathrm{d}s(d)\quad \text{for}\enspace \hat{x}\in \mathbb{S},\quad g\in {L}^{2}(\mathbb{S}).\end{equation} \tag{ 3.4 }$$

Using the translation formula

$$\begin{equation}{u}^{\infty }(\hat{x};{B}_{h}(z),d,k)={\mathrm{e}}^{\mathrm{i}kz\cdot (d-\hat{x})}{u}^{\infty }(\hat{x};{B}_{h}(O),d,k),\end{equation} \tag{ 3.5 }$$

together with the spectral system for B_h (O), we can get the spectral system $({\lambda }_{z,h}^{(n)},{\varphi }_{z,h}^{(n)})$ of F_z,h under the Dirichlet or impedance boundary condition:

• If B_h (z) is a sound-soft disk, then
  $$\begin{equation}{\lambda }_{z,h}^{(n)}=-2\pi C\frac{{J}_{n}(kh)}{{H}_{n}^{(1)}(kh)},\quad {\varphi }_{z,h}^{(n)}(\hat{x})={\text{e}}^{\text{i}n\hat{\theta }-\mathrm{i}kz\cdot (\mathrm{cos}\enspace \hat{\theta },\mathrm{sin}\enspace \hat{\theta })}.\end{equation} \tag{ 3.6 }$$
  In particular, ${\lambda }_{z,h}^{(n)}\ne 0$ if k² is not the Dirichlet eigenvalue of −Δ in B_h (z).
• If B_h (z) is an impedance disk with the impedance constant $\eta \in \mathbb{C}$, then (see e.g., (2.11))
  $$\begin{equation}{\lambda }_{z,h}^{(n)}=-2\pi C\frac{k{J}_{n}^{\prime }(kh)+\eta {J}_{n}(kh)}{k{H}_{n}^{(1)\prime }(kh)+\eta {H}_{n}^{(1)}(kh)},\quad {\varphi }_{z,h}^{(n)}(\hat{x})={\text{e}}^{\text{i}n\hat{\theta }-\mathrm{i}kz\cdot (\mathrm{cos}\enspace \hat{\theta },\mathrm{sin}\enspace \hat{\theta })}.\end{equation} \tag{ 3.7 }$$
  In particular, we have ${\lambda }_{z,h}^{(n)}\ne 0$.

Note that the above eigenvalues are independent of z and the eigenfunctions are independent of h. Taking Ω = B_h (z), we can rewrite corollary 3.1 as

Corollary 3.3. Let ${v}^{\infty }\in {L}^{2}(\mathbb{S})$. Then

$$\begin{equation}I(z,h){:=}\sum\limits _{j\in \mathbb{Z}}\frac{{\left\vert {\left\langle {v}^{\infty },{\varphi }_{z,h}^{(j)}\right\rangle }_{\mathbb{S}}\right\vert }^{2}}{\left\vert {\lambda }_{z,h}^{(j)}\right\vert }< +\infty \end{equation} \tag{ 3.8 }$$

if and only if v^∞ is the far-field pattern of the radiating solution v^s , where v^s satisfies Helmholtz equation Δv^s + k² v^s = 0 in |x − z| > h with the boundary data $f{:=}{v}^{s}{\vert }_{{{\Gamma}}_{z,h}}\in {H}^{1/2}({{\Gamma}}_{z,h})$.

Proof. Without loss of generality, we assume that the center z coincides with the origin, i.e., B_h (z) = B_h (O). By the Jacobi–Anger expansion (see e.g., [9]), v^s can be expanded into the series

$$\begin{equation}{v}^{s}(x)=\sum\limits _{n\in \mathbb{Z}}{A}_{n}{H}_{n}^{(1)}(k\vert x\vert ){\text{e}}^{\text{i}n\hat{\theta }},\quad \vert x\vert > R,\quad x=(\vert x\vert ,\hat{\theta }),\end{equation} \tag{ 3.9 }$$

for some sufficiently large R > 0, with the far-field pattern given by (see [9, (3.82)])

$$\begin{equation}{v}^{\infty }(\hat{x})=\sum\limits _{n\in \mathbb{Z}}{A}_{n}{C}_{n}{\mathrm{e}}^{\mathrm{i}n\hat{\theta }},\quad {C}_{n}{:=}\sqrt{\frac{2}{k\pi }}{\text{e}}^{-\text{i}\left(\frac{n\pi }{2}+\frac{\pi }{4}\right)}.\end{equation} \tag{ 3.10 }$$

By the asymptotic behavior of Bessel functions (see [9]), the function I(z, h) with z = O can be written as

$$\begin{equation}\begin{aligned}\hfill I(z,h)& =\sum\limits _{j\in \mathbb{Z}}\frac{{\left\vert {\left\langle \sum {A}_{n}{C}_{n}{\mathrm{e}}^{\mathrm{i}n\hat{\theta }},{\varphi }_{z,h}^{(j)}\right\rangle }_{\mathbb{S}}\right\vert }^{2}}{\left\vert {\lambda }_{z,h}^{(j)}\right\vert }=\sum\limits _{j\in \mathbb{Z}}{\left\vert {\lambda }_{z,h}^{(j)}\right\vert }^{-1}\vert 2\pi {A}_{j}{C}_{j}{\vert }^{2}\hfill \\ \hfill & =\sum\limits _{j\in \mathbb{Z}}\sqrt{\frac{8}{\pi k}}\vert {A}_{j}{\vert }^{2}\frac{{2}^{2j}j!(j-1)!}{{(kh)}^{2j}}\left(1+O\left(\frac{1}{j}\right)\right).\hfill \end{aligned}\end{equation} \tag{ 3.11 }$$

On the other hand, by (3.9) and the definition of the norm ${\Vert}\cdot {{\Vert}}_{{H}^{1/2}({{\Gamma}}_{z,h})}$, it is easy to see when z = O that

$$\begin{equation}\begin{aligned}\hfill {\Vert}{v}^{s}{{\Vert}}_{{H}^{1/2}({{\Gamma}}_{z,h})}^{2}& =\sum\limits _{j\in \mathbb{Z}}{(1+{j}^{2})}^{1/2}\vert {A}_{j}{H}_{j}^{(1)}(kh){\vert }^{2}\hfill \\ \hfill & =\sum\limits _{j\in \mathbb{Z}}\vert {A}_{j}{\vert }^{2}\frac{{2}^{2j}j!(j-1)!}{{\pi }^{2}{(kh)}^{2j}}\left(1+O\left(\frac{1}{j}\right)\right).\hfill \end{aligned}\end{equation} \tag{ 3.12 }$$

Obviously, the series (3.11) and (3.12) have the same convergence. On the other hand, by [9, theorem 2.15], the boundedness of ${\Vert}{v}^{s}{{\Vert}}_{{L}^{2}({{\Gamma}}_{z,h})}$ means that v^s is a radiating solution in |x − z| > h with the far-field pattern v^∞. This proves that I(z, h) < ∞ if and only if v^s is a radiating solution in |x − z| > h with the far-field pattern v^∞ and with the H^1/2-boundary data on Γ_z,h . □

In our applications of corollaries 3.1 and 3.3, we will take v^∞ to be the measurement data ${u}_{D}^{\infty }(\hat{x};{d}_{0}){:=}{u}^{\infty }(\hat{x};{d}_{0},D)$ that corresponds to our target scatterer D and the incident plane wave ${e}^{ikx\cdot {d}_{0}}$ for some fixed ${d}_{0}\in \mathbb{S}$. We shall omit the dependance on d₀ if it is always clear from the context. Our purpose is to extract the geometrical information on D from the domain-defined indicator function I(Ω) or I(z, h). By corollary 3.1, I(Ω) < ∞ if the scattered field ${u}_{D}^{s}(x)={u}^{s}(x;{d}_{0},D)$ can be extended to the domain ${\mathbb{R}}^{2}{\backslash}\bar{{\Omega}}$ as a solution to the Helmholtz equation. This implies that ${u}_{D}^{s}$ admits an analytical extension across the boundary ∂D when the inclusion relation D ⊂ Ω does not hold. Therefore, the one-wave factorization method requires us to exclude the possibility of analytical extension, which however is possible only if ∂D is not everywhere analytic. If ∂D is analytic, it follows from the Cauchy–Kovalevski theorem (see e.g. [27, chapter 3.3]) that ${u}_{D}^{s}$ can be locally extended into the inside of D across the boundary ∂D. In the special case that D = B_h (O), by the Schwartz reflection principle the scattered field ${u}_{D}^{s}$ can be globally continued into ${\mathbb{R}}^{2}{\backslash}\left\{O\right\}$. Below we shall discuss the absence of the analytic extension of ${u}_{D}^{s}$ in corner domains, which is an import ingredient in establishing the one-wave factorization method. The results in the subsequent section have been used in the literature but without proofs. We intend to present their proofs for the readers' convenience.

We first consider time-harmonic acoustic scattering from a convex polygon of sound-soft, sound-hard or impedance type. In the impedance case, the impedance function is supposed to be a constant.

Lemma 3.4. Assume that D is either a sound-soft, sound-hard or impedance obstacle occupying a convex polygon. Then the scattered field ${u}_{D}^{s}(x,{d}_{0})$ for a fixed ${d}_{0}\in \mathbb{S}$ cannot be analytically extended from ${\mathbb{R}}^{2}{\backslash}\bar{D}$ into D across any corner of D.

As one can imagine, the proof of lemma 3.4 is closely related to uniqueness in determining a convex polygonal obstacle with a single incoming wave (see e.g., [7], [9, theorem 5.5] and [20]). In fact, the result of lemma 3.4 implies that a convex polygonal obstacle of sound-soft, sound-hard or impedance type can be uniquely determined by one far-field pattern. There are several approaches to prove lemma 3.4. Below we present a unified method valid for any kind boundary condition under consideration. The original version of this approach was presented in [14] for proving uniqueness in inverse conductivity problems.

Proof. Assume on the contrary that ${u}_{D}^{s}$ can be analytically continued across a corner of ∂D. By coordinate translation and rotation, we may suppose that this corner coincides with the origin, so that ${u}_{D}^{s}$ and also the total field ${u}_{D}={u}_{D}^{s}+{u}^{i}$ satisfy the Helmholtz equation in B (O) for some > 0. Since u_D is real analytic in $({\mathbb{R}}^{2}{\backslash}\bar{D})\cup {B}_{{\epsilon}}(O)$ and D is a convex polygon, u_D satisfies the Helmholtz equation in a neighborhood of an infinite sector ${\Sigma}\subset {\mathbb{R}}^{2}{\backslash}\bar{D}$ which extends the finite sector B (O) ∩ D to ${\mathbb{R}}^{2}{\backslash}\bar{D}$. On the other hand, u_D fulfills the boundary condition on the two half lines ∂Σ starting from the corner point O. By the Schwartz reflection principle of the Helmholtz equation, u_D can be extended onto ${\mathbb{R}}^{2}$ (see also [14] for discussions on the conductivity equation). We remark that u_D ≡ 0 when the angle of Σ is irrational and that the impedance case follows from the arguments in [20]. This implies that the scattered field ${u}_{D}^{s}$ is an entire radiating solution. Hence, ${u}_{D}^{s}$ must vanish identically in ${\mathbb{R}}^{2}$ and u_D = uⁱ must fulfill the boundary condition on ∂D. However, this is impossible for a plane wave incidence under either of the Dirichlet, Neumann or Robin condition. □

Next we consider the source radiating problem where D is a polygonal source term. The absence of analytical extension in this case implies that a polygonal source term cannot be a non-radiating source (that is, the resulting far-field pattern vanishes identically). This can be proved based on the idea of constructing CGO solutions [3, 4, 19] or analyzing corner singularities for solutions of the inhomogeneous Laplace equation [12, 18]. Below we consider the case of a special analytic source function, which will significantly simplify the arguments employed in [11, 12] for inverse medium scattering problems and those in [3, 18] for inverse source problems. It is worthy mentioning that the results of the following lemma also appear in [32] but with insufficient regularity assumption on the source term. Here we intend to present an elementary and simple proof, which together with lemma 3.4 can be considered as supplementary arguments to [32] for planar corners.

Lemma 3.5. Assume that D is a convex polygon and let χ_D be the characteristic function for D. If $u\in {H}_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2}({\mathbb{R}}^{2})$ is a radiating solution to

$$\begin{equation}{\Delta}u(x)+{k}^{2}u(x)={\chi }_{D}(x)f(x)\quad \text{in}\enspace {\mathbb{R}}^{2},\end{equation} \tag{ 3.13 }$$

where f(x) is real-analytic and non-vanishing near the corner point O of D and the lowest order Taylor expansion of f at O is harmonic. Then u cannot be analytically extended from ${\mathbb{R}}^{2}{\backslash}\bar{D}$ to D across the corner O.

Proof. Assume that u can be analytically extended from ${\mathbb{R}}^{2}{\backslash}\bar{D}$ to B_δ (O) for some δ > 0 (see figure 2), as a solution of the Helmholtz equation. Without loss of generality, the corner point O is supposed to coincide with the origin. Set ${v}^{\pm }=u{\vert }_{{D}^{\pm }}$ where ${D}^{+}{:=}{B}_{\delta }(O)\cap ({\mathbb{R}}^{2}{\backslash}\bar{D})$ and D⁻ := B_δ (O) ∩ D. This implies that Δv⁺ + k² v⁺ = 0 in B_δ (O) and thus the Cauchy data v⁻ = v⁺, ∂_ν v⁻ = ∂_ν v⁺ on Γ := ∂D ∩ B_δ (O) are analytic. Since f is also analytic in B_δ (O), by the Cauchy–Kovalevskaya theorem (see e.g. [27, chapter 3.3]) the function v⁻ can also be analytically continued into B_δ (O) as a solution of Δv⁻ + k² v⁻ = f in B_δ (O). Setting w = v⁻ − v⁺ in B_δ (O), we have

$$\begin{equation}{\Delta}w+{k}^{2}w=f\quad \text{in}\enspace {B}_{\delta }(O),\quad w={\partial }_{\nu }w=0\quad \text{on}\enspace {\Gamma}.\end{equation} \tag{ 3.14 }$$

Using [10, lemma 2.2], the analytic functions w and f can be expanded in polar coordinates into the series

$$\begin{equation}\begin{aligned}\hfill & w(x)=\sum\limits _{n+2m\geqslant 0}{r}^{n+2m}({a}_{n,m}\enspace \mathrm{cos}\enspace n\theta +{b}_{n,m}\enspace \mathrm{sin}\enspace n\theta ),\hfill \\ \hfill & f(x)={r}^{N}({\tilde{a}}_{N}\enspace \mathrm{cos}\enspace N\theta +{\tilde{b}}_{N}\enspace \mathrm{sin}\enspace N\theta )+\sum\limits _{n+2m\geqslant N+1}{r}^{n+2m}({\tilde{a}}_{n,m}\enspace \mathrm{cos}\enspace n\theta +{\tilde{b}}_{n,m}\enspace \mathrm{sin}\enspace n\theta ),\hfill \end{aligned}\end{equation} \tag{ 3.15 }$$

in B_δ (O), where (r, θ) denote the polar coordinates of $x\in {\mathbb{R}}^{2}$ and $N\in {\mathbb{N}}_{0}$. Since f does not vanish identically, we may suppose that $\vert {\tilde{a}}_{N}\vert +\vert {\tilde{b}}_{N}\vert > 0$. Recalling the Laplace operator in polar coordinates, ${\Delta}=\frac{{\partial }^{2}}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial }^{2}}{\partial {\theta }^{2}}$, we have

$$\begin{align*}\hfill f(x)& ={\Delta}w(x)+{k}^{2}w(x)\hfill \\ \hfill & =\sum\limits _{n+2m\geqslant 0}[4(n+m+1)(m+1){r}^{n+2m}({a}_{n,m+1}\enspace \mathrm{cos}\enspace n\theta +{b}_{n,m+1}\enspace \mathrm{sin}\enspace n\theta )]\hfill \\ \hfill & \quad +\sum\limits _{n+2m\geqslant 0}{k}^{2}{r}^{n+2m}({a}_{n,m}\enspace \mathrm{cos}\enspace n\theta +{b}_{n,m}\enspace \mathrm{sin}\enspace n\theta ).\hfill \end{align*}$$

Inserting the expansion of f(x) in (3.15) and comparing the coefficients of r^l , $l\in {\mathbb{N}}_{0}$, we can get the recurrence relations for a_n,m ,

$$\begin{equation}\begin{aligned}\hfill 4(N+1){a}_{N,\enspace 1}+{k}^{2}{a}_{N,\enspace 0}& ={\tilde{a}}_{N},\hfill \\ \hfill 4(n+m+1)(m+1){a}_{n,m+1}+{k}^{2}{a}_{n,m}& =0,\quad n+2m< N\enspace \hfill \\ \hfill 4(n+m+1)(m+1){a}_{n,m+1}+{k}^{2}{a}_{n,m}& =0,\quad n+2m=N,m > 0.\hfill \end{aligned}\end{equation} \tag{ 3.16 }$$

The same relations hold for b_n,m . Now, we suppose without loss of generality that Γ = {(r, ±θ₀): |r| < δ} for some θ₀ ∈ (0, π/2). From the boundary conditions w = ∂_θ w = 0 on Γ, we have

$$\begin{equation}\begin{cases}\sum\limits _{n,m\in {\mathbb{N}}_{0},n+2m=l}{a}_{n,m}\enspace \mathrm{cos}\enspace n{\theta }_{0}=0,\quad \hfill \\ \sum\limits _{n,m\in {\mathbb{N}}_{0},n+2m=l}n{a}_{n,m}\enspace \mathrm{sin}\enspace n{\theta }_{0}=0,\quad \hfill \\ \sum\limits _{n,m\in {\mathbb{N}}_{0},n+2m=l}n{b}_{n,m}\enspace \mathrm{cos}\enspace n{\theta }_{0}=0,\quad \hfill \\ \sum\limits _{n,m\in {\mathbb{N}}_{0},n+2m=l}{b}_{n,m}\enspace \mathrm{sin}\enspace n{\theta }_{0}=0,\quad \hfill \end{cases}\end{equation} \tag{ 3.17 }$$

for any $l\in {\mathbb{N}}_{0}$. From the second formula in (3.16) and the first two formulas in (3.17), we can easily obtain a_n,m = 0 if n + 2m < N.

Now we prove that a_n,m = 0 if l = n + 2m = N. In fact, for m ⩾ 1, setting m' = m − 1 ⩾ 0 we derive from the second formula in (3.16) that

$$\begin{equation*}{a}_{n,m}={a}_{n,{m}^{\prime }+1}=-\frac{{k}^{2}}{4(n+m)m}{a}_{n,{m}^{\prime }}=0,\end{equation*}$$

since n + 2m' < N. The above relations together with the first two formulas in (3.17) with l = N lead to

$$\begin{equation*}{a}_{N,0}\enspace \mathrm{cos}\enspace N{\theta }_{0}=0,\qquad N{a}_{N,0}\enspace \mathrm{sin}\enspace N{\theta }_{0}=0,\end{equation*}$$

which imply that a_N,0 = 0.

When l = n + 2m = N + 2, we observe that n + 2m' = N, where m' = m − 1. Hence one can get a_n,m = 0 if m > 1 and l = N + 2, by using the third formula in (3.16) and the fact that a_n,m' = 0 for all n + 2m' = N. Then it follows from the first two formulas in (3.17) with l = N + 2 that

$$\begin{equation}\begin{cases}{a}_{N+2,0}\enspace \mathrm{cos}(N+2){\theta }_{0}+{a}_{N,1}\enspace \mathrm{cos}\enspace N{\theta }_{0}=0,\quad \hfill \\ (N+2){a}_{N+2,0}\enspace \mathrm{sin}(N+2){\theta }_{0}+N{a}_{N,1}\enspace \mathrm{sin}N{\theta }_{0}=0.\quad \hfill \end{cases}\end{equation} \tag{ 3.18 }$$

Since $0< {\theta }_{0}< \frac{\pi }{2}$, we have (see [11])

$$\begin{equation}\left\vert \begin{aligned}\hfill & \mathrm{cos}(N+2){\theta }_{0}\quad \hfill & \hfill \mathrm{cos}\enspace N{\theta }_{0}\\ \hfill & (N+2)\mathrm{sin}(N+2){\theta }_{0}\quad \hfill & \hfill N\enspace \mathrm{sin}\enspace N{\theta }_{0}\end{aligned}\right\vert =(N+1)\mathrm{sin}\enspace 2\enspace {\theta }_{0}-\mathrm{sin}(N+1){\theta }_{0}\ne 0.\end{equation} \tag{ 3.19 }$$

Then we have a_N+2,0 = a_N,1 = 0. By the first relation in (3.16) we get ${\tilde{a}}_{N}=0$. Analogously one can prove ${\tilde{b}}_{N}=0$. This implies that f ≡ 0, which is a contradiction. □

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Remark 3.6. 

• (a)  
  If f satisfies the elliptic equation

  $$\begin{equation*}{\Delta}f(x)+A(x)\cdot \nabla f(x)+b(x)\enspace f(x)=0,\end{equation*}$$

  where A(x) = (a₁(x), a₂(x)) and b(x) are both real-analytic, then the lowest Taylor expansion of f at any point must be harmonic (see [18]). Therefore, the class of source functions f specified in lemma 3.5 covers at least harmonic functions, including constant functions.
• (b)  
  The analyticity of f in lemma 3.5 can be weakened to be Hölder continuous near O with the asymptotic behavior (see [18, 32])

  $$\begin{equation}f(x)={r}^{N}({A}_{n}\enspace \mathrm{cos}\enspace N\theta +{B}_{N}\enspace \mathrm{sin}\enspace N\theta )+o({r}^{N}),\quad \vert x\vert \to 0,\end{equation} \tag{ 3.20 }$$

  for some $N\in {\mathbb{N}}_{0}$ and ${A}_{N},{B}_{N}\in \mathbb{C}$ with |A_N | + |B_N | > 0. Moreover, the corner O ∈ ∂D can be weakened to be a weakly singular point of arbitrary order such that ∂D is not of C^∞-smooth at O; see [34].

To state the one-wave factorization method, we shall restrict our discussions to convex polygonal impenetrable scatterers of sound-soft, sound-hard or impedance type and to convex polygonal source terms where the source function satisfies the condition of lemma 3.5. In the former case, ${u}_{D}^{\infty }$ represents the far-field pattern of the scattered field caused by some plane wave incident onto D; in the latter case, ${u}_{D}^{\infty }$ denotes the far-field pattern of the radiating solution to (3.13). Recall from subsection 3.1 that Ω is a convex sound-soft or impedance scatterer such that k² is not the eigenvalue of −Δ in Ω. Denote by $({\lambda }_{{\Omega}}^{(j)},{\varphi }_{{\Omega}}^{(j)})$ the eigenvalues and eigenfunctions of the far-field operator F_Ω. Below we characterize the inclusion relationship between our target scatterer D and the test domain Ω through the interaction of the measurement data ${u}_{D}^{\infty }$ and the spectra of F_Ω.

Theorem 3.7. Define

$$\begin{equation*}W({\Omega}){:=}\sum\limits _{j\in \mathbb{Z}}\frac{{\left\vert {\left\langle {u}_{D}^{\infty },{\varphi }_{{\Omega}}^{(j)}\right\rangle }_{\mathbb{S}}\right\vert }^{2}}{\left\vert {\lambda }_{{\Omega}}^{(j)}\right\vert }.\end{equation*}$$

Then W(Ω) < ∞ if and only if D ⊆ Ω.

Proof. ⟹: By corollary 3.1, W(Ω) < +∞ implies that u is analytic in ${\mathbb{R}}^{2}{\backslash}\bar{{\Omega}}$. If D ⊈ Ω and D is a convex polygon, three cases might happen (see figure 3): (i) Ω ⊂ D; (ii) Ω ∩ D = ∅; (iii) Ω ∩ D ≠ ∅ and ${\Omega}\cap ({\mathbb{R}}^{2}{\backslash}\bar{D})\ne \varnothing$. In either of these cases, we observe that u can be analytically continued from ${\mathbb{R}}^{2}{\backslash}\bar{D}$ to D across a corner of ∂D, which however is impossible by lemmas 3.4 and 3.5. This proves the relationship D ⊆ Ω.

⟸: We only consider the case where D is an impenetrable scatterer. The source problem can be proved analogously. Assume D ⊆ Ω. Then the scattered field u^s satisfies the Helmholtz equation Δu^s + k² u^s = 0 in ${\mathbb{R}}^{2}{\backslash}\bar{{\Omega}}$ with the boundary data u^s |_∂Ω ∈ H^1/2(∂Ω). Then we can get W(Ω) < +∞ by applying corollary 3.1. □

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Taking the test domain Ω as the disk B_h (z), we can immediately get

Corollary 3.8. Let $({\lambda }_{z,h}^{(j)},{\varphi }_{z,h}^{(j)})$ be an eigensystem of the far-field operator F_z,h . Define

$$\begin{equation}W(z,h){:=}\sum\limits _{j\in \mathbb{Z}}\frac{{\left\vert {\left\langle {u}_{D}^{\infty },{\varphi }_{z,h}^{(j)}\right\rangle }_{\mathbb{S}}\right\vert }^{2}}{\left\vert {\lambda }_{z,h}^{(j)}\right\vert }.\end{equation} \tag{ 3.21 }$$

Then we have W(z, h) < ∞ if h ⩾ max_y∈∂D |y − z| and W(z, h) = ∞ if h < max_y∈∂D |y − z|.

Remark 3.9. Theorem 3.7 and corollary 3.8 explain how do the a priori data ${u}_{{\Omega}}^{\infty }(\hat{x},d)$ for all $\hat{x},\mathrm{d}\in \mathbb{S}$ encode the information of the unknown target D. Evidently, their proofs rely essentially on mathematical properties of the scattering model. On the other hand, in the terminology of learning theory and data sciences, the test domains Ω can be regarded as samples and the a priori data ${u}_{{\Omega}}^{\infty }(\hat{x},d)$ as the associated sample data, which are usually calculated off-line. In this sense, the one-wave factorization method is a both model-driven and data-driven approach.

Corollary 3.8 says that the maximum distance between a sampling point $z\in {\mathbb{R}}^{2}$ and our target D is coded in the function h ↦ W(z, h). Hence, changing the sampling points z on a large circle |z| = R such that D ⊆ B_R (O) and computing max_y∈∂D |z − y| for each z would give an approximation of D as follows:

$$\begin{equation}D\approx \bigcap\limits _{\vert z\vert =R,h\in (0,2R)}^{W(h,z)< \infty }{B}_{h}(z).\end{equation} \tag{ 3.22 }$$

Remark 3.10. In (3.22), it is supposed that R > 0 is a fixed large number. A precise characterization of the convex polygon D is given by taking the limit R → ∞, that is,

$$\begin{equation*}D=\underset{R\to +\infty }{\mathrm{lim}}\bigcap\limits _{\vert z\vert =R,h\in (0,2R)}^{W(h,z)< \infty }{B}_{h}(z).\end{equation*}$$

In fact, this can be easily proved using corollary 3.8.

We remark that a proper regularization scheme should be employed in computing the truncated indicator (3.21), because the far-field operator F_z,h is compact and the eigenvalues ${\lambda }_{z,h}^{(j)}$ decay almost exponentially as j → ∞; see figure 4.

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If Ω = B_h (z) is a sound-soft test disk, the ${({F}^{\ast }F)}^{1/4}$-method yields the relation $W(z,h)={\Vert}{g}_{z,h}{{\Vert}}_{{L}^{2}(\mathbb{S})}^{2}$, where $g={g}_{z,h}\in {L}^{2}(\mathbb{S})$ solves the operator equation

$$\begin{equation}{({F}_{z,h}^{\ast }{F}_{z,h})}^{1/4}g={u}_{D}^{\infty }.\end{equation} \tag{ 3.23 }$$

The solution of the equation (3.23) is

$$\begin{equation*}g=\sum\limits _{j}\frac{{\left\langle {u}_{D}^{\infty },{\varphi }_{j}\right\rangle }_{\mathbb{S}}}{\sqrt{{\lambda }_{z,h}^{(j)}}}{\varphi }_{z,h}^{(j)}\quad \text{if}\quad {u}_{D}^{\infty }\in \mathrm{R}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}({({F}_{z,h}^{\ast }{F}_{z,h})}^{1/4}).\end{equation*}$$

Using the Tikhonov regularization we aim to solve the equation

$$\begin{equation}\alpha I+{({F}_{z,h}^{\ast }{F}_{z,h})}^{1/2}{g}_{\alpha }={({F}_{z,h}^{\ast }{F}_{z,h})}^{1/4}{u}_{D}^{\infty }\end{equation} \tag{ 3.24 }$$

with the solution given by

$$\begin{equation}{g}_{\alpha }=\sum\limits _{j}\frac{\sqrt{{\lambda }_{z,h}^{(j)}}}{\vert \alpha +{\lambda }_{z,h}^{(j)}\vert }{\left\langle {u}_{D}^{\infty },{\varphi }_{z,h}^{(j)}\right\rangle }_{\mathbb{S}}{\varphi }_{z,h}^{(j)},\end{equation} \tag{ 3.25 }$$

where α > 0 is the regularization parameter. This implies that

$$\begin{equation*}{\Vert}{g}_{\alpha }{{\Vert}}_{{L}^{2}(\mathbb{S})}^{2}=\sum\limits _{j}\frac{\vert {\lambda }_{z,h}^{(j)}\vert }{{(\vert {\lambda }_{z,h}^{(j)}+\alpha \vert )}^{2}}{\left\vert \left\langle {u}_{D}^{\infty },{\varphi }_{z,h}^{(j)}\right\rangle \right\vert }^{2}.\end{equation*}$$

Hence, numerically one should use the modified indicator

$$\begin{equation}\tilde{W}(z,h)={\left[\sum\limits _{j\leqslant N}\frac{\vert {\lambda }_{z,h}^{(j)}\vert {\left\vert {\left\langle {u}_{D}^{\infty },{\varphi }_{z,h}^{(j)}\right\rangle }_{\mathbb{S}}\right\vert }^{2}}{{\left\vert {\lambda }_{z,h}^{(j)}+\alpha \right\vert }^{2}}\right]}^{-1}=1/{\Vert}{g}_{\alpha }{{\Vert}}_{{L}^{2}(\mathbb{S})}^{2}.\end{equation} \tag{ 3.26 }$$

Our imaging scheme I is described as follows (see figure 5):

• Suppose that D ⊆ B_R (O) for some R > 0 and collect the measurement data ${u}_{D}^{\infty }(\hat{x})$ for all $\hat{x}\in \mathbb{S}$;
• Choose sampling points z_n ∈ Γ_R := {x: |x| = R} for n = 1, ..., N_z ;
• Choose h_m ∈ (0, 2R) to get different spectral systems $({\lambda }_{{z}_{n},{h}_{m}}^{(j)},{\varphi }_{{z}_{n},{h}_{m}}^{(j)})$ (see (3.6) or (3.7));
• For each z_n ∈ Γ_R , calculate the maximum distance between z_n and D by ${h}_{{z}_{n}}{:=}\mathrm{inf}\left\{{h}_{m}\in (0,2R):\tilde{W}({z}_{n},{h}_{m})\geqslant \delta \right\}$ where δ > 0 is a threshold.
• Take $D\approx {\bigcap }_{1\leqslant n\leqslant {N}_{z}}{B}_{{h}_{{z}_{n}}}({z}_{n})$.

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In the above numerical scheme, we take the test domain Ω as sound-soft or impedance disks B_h (z), because the spectral systems $({\lambda }_{z,h}^{(j)},{\varphi }_{z,h}^{(j)})$ are given explicitly. For a general test domain, the spectral systems $({\lambda }_{{\Omega}}^{(j)},{\varphi }_{{\Omega}}^{(j)})$ should be calculated off-line, so that they are available before inversion. The sample variables (disks) in the above one-wave factorization method consist of the centers z ∈ ∂B_R and radii h ∈ (0, 2R). Obviously, the number of these variables is comparable with that of the original factorization method with infinitely many plane waves.

Assume that the obstacle (resp. source support) D has been shrank to a point located at z* (for instance, if the wavelength λ = 2π/k is much bigger than the diameter of D). In this case, the scattered (resp. radiated) wave field can be asymptotically written as c*Φ(x, z*), where ${c}^{\ast }\in \mathbb{C}$ depends on the incoming wave, the scattering strength of D as well as the location point z*. For simplicity we suppose that c* = 1. Then the far-field pattern ${u}^{\infty }={u}_{D}^{\infty }$ takes the simple form

$$\begin{equation}{u}^{\infty }(\hat{x})={\mathrm{e}}^{-\mathrm{i}k{z}^{\ast }\cdot \hat{x}}.\end{equation} \tag{ 5.1 }$$

To perform numerical examples, we set the wave number k = 6 and suppose that z* ∈ B_R with R = 4. The number of sampling centers z_n lying on |x| = 4 is taken to be N_z = 8, and the parameter for truncating the infinite series (3.21) is chosen as N = 60. The threshold specified in our imaging scheme is set as δ = 4 × 10⁻⁴. In these settings we obtain figure 6 where z* = [2, 2], [−1, 2], [2, −1], [−1, −1] can be accurately located with a single far-field pattern without polluted noise. In figure 6, the dotted curve represents the circle Γ_R where the centers of the test disks are located. The solid circles are boundaries of the test disks ${B}_{{h}_{{z}_{n}}}({z}_{n})$, where ${h}_{{z}_{n}}$ denotes the distance between z_n and z*. In figure 7, we plot the function $h\to \tilde{W}(z,h)$ with z = [4, 0] for locating z* = [−2, 0]. Obviously, we have |z − z*| = 6. It is seen from figure 7 that the values of the function $h\to \tilde{W}(z,h)$ for h ∈ (0, 6) are much smaller than those for h ∈ (6, 8). With our threshold the distance between z and z* is calculated as 5.98.

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Although the first scheme can be used previously to locate a point-like obstacle/source, it is not straightforward for imaging an extended obstacle/source. Below we describe a more direct imaging scheme II.

• Suppose that B_R ⊃ D for some R > 0 and collect the measurement data ${u}_{D}^{\infty }(\hat{x})$ for all $\hat{x}\in \mathbb{S}$. Let Q ⊃ D be our search/computational region for imaging D;
• Choose sampling centers z_n ∈ Γ_R := {x: |x| = R} for n = 1, ..., N_z and choose sampling radii h_m ∈ (0, 2R) to get different spectral systems $({\lambda }_{{z}_{n},{h}_{m}}^{(j)},{\varphi }_{{z}_{n},{h}_{m}}^{(j)})$ (see (3.6) or (3.7));
• For each z_n ∈ Γ_R , define the function ${\mathcal{I}}_{n}(x)=\tilde{W}({z}_{n},\vert x-{z}_{n}\vert )$ for x ∈ Q (see (3.26));
• The imaging function for recovering D is defined as $\mathcal{I}(x)={\sum }_{n=1}^{{N}_{z}}{\mathcal{I}}_{n}(x)$, x ∈ Q;

Remark 5.1. By definition, the values of the indicator function $\mathcal{I}$ in the interior of D are smaller than those in the exterior. Hence, the function $\mathcal{I}$ indeed indicates the position and shape of an unknown extended scatterer. This was rigorously justified at least for convex scatterers of polygonal type; see the blue parts in figures 8–10 for imaging multiple point-like scatterers and extended polygonal sources.

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Note that, by corollary 3.8, ${\mathcal{I}}_{n}(x) > 0$ if |x − z_n | > max_y∈D |z_n − y| and ${\mathcal{I}}_{n}(x)=0$ if otherwise. Hence, the values of $\mathcal{I}(x)$ for $x\in Q{\backslash}\bar{D}$ should be larger than those for x ∈ D. As an example, we apply this new scheme to image $D={\bigcup }_{j=1,2,\dots ,M}\left\{{z}_{j}^{\ast }\right\}$ which consists of multiple point-like scatterers. For simplicity we neglect the multiple scattering between them and write the far-field pattern as

$$\begin{equation}{u}^{\infty }({z}^{\ast })={\mathrm{e}}^{-\mathrm{i}k{z}_{1}^{\ast }\cdot \hat{x}}+{\mathrm{e}}^{-\mathrm{i}k{z}_{2}^{\ast }\cdot \hat{x}}+\cdots +{\mathrm{e}}^{-\mathrm{i}k{z}_{M}^{\ast }\cdot \hat{x}}.\end{equation} \tag{ 5.2 }$$

The results for reconstructing one point ${z}_{1}^{\ast }=[1,1]\enspace (M=1)$, two points ${z}_{1}^{\ast }=[-2,4]$, ${z}_{2}^{\ast }=[2,-3]\enspace (M=2)$ and three points ${z}_{1}^{\ast }=[3,3]$, ${z}_{2}^{\ast }=[-2,2]$, ${z}_{3}^{\ast }=[0,-4]\enspace (M=3)$ are shown in figure 8. As one can imagine, with a single far-field pattern our approach can only recover the convex hull of these points. In the case of two points, the line segment connecting ${z}_{1}^{\ast }$ and ${z}_{2}^{\ast }$ is shown in the middle of figure 8. The triangle formed by ${z}_{1}^{\ast }$, ${z}_{2}^{\ast }$ and ${z}_{3}^{\ast }$ is shown in the right figure.

Remark 5.2. It is important to remark that there exist other approaches for locating a finite number of point scatterers using several incident waves, for example the MUSIC algorithm (which can be regarded as the discrete analogue of the classical factorization method [30]) and the expansion method [1]. Our concern here is to show the capability of the one-wave factorization method for capturing singularities of the scattered/radiated wave field. In 2D, the singularity is of logarithmic type at the point-like scatterers.

Suppose that the source support D is a triangle with the three corners located at (−2, −2), (−2, 2) and (2, –2). The source function is supposed to be a constant. For simplicity we assume that χ_D f(x) ≡ 1 on $\bar{D}$, so that the far-field pattern takes the explicit form

$$\begin{equation}{u}^{\infty }(\hat{x})=\frac{i}{4}{\int }_{D}{\text{e}}^{-\text{i}k\hat{x}\cdot z}\enspace \mathrm{d}z=\frac{i}{4}{\int }_{-2}^{2}{\text{e}}^{-\text{i}k{\hat{x}}_{1}{z}_{1}}{\int }_{-2}^{-{z}_{1}}{\text{e}}^{-\text{i}k{\hat{x}}_{2}{z}_{2}}\enspace \mathrm{d}{z}_{2}\enspace \mathrm{d}{z}_{1},\end{equation} \tag{ 5.3 }$$

where $\hat{x}=({\hat{x}}_{1},{\hat{x}}_{2})\in \mathbb{S}$ and $z=({z}_{1},{z}_{2})\in {\mathbb{R}}^{2}$. We want to test the sensitivity of our approach to the incident wavenumber k and to the circle Γ_R = {x: |x| = R} = ∂B_R . In this subsection, the number of sampling centers z_n lying on |x| = R is set to be N_z = 64 and the truncation parameter to be N = 80.

Firstly, we fixed R = 4 and change the excited frequencies. The far-field data corresponding to B_h (z) are supposed to be excited at the same frequency as for D. In figure 9, the recovery of D from a single far-field pattern at different frequencies are illustrated. We observe that a regularization parameter depending on k must be properly selected to get a satisfactory image of D. In our numerical tests, α > 0 is chosen by the method of trial and error. Consequently, we take α = 1e − 22, 1e − 13 and 1e − 8 corresponding to the wavenumbers k = 1.5, 6, 12, respectively. From figure 9, one can conclude that a better image can be achieved at higher frequencies.

Secondly, we fix k = 6 and recovery D by using sampling disks with the centers equally distributed on Γ_R with R = 4, 8, 12. The regularization parameter are chosen as α = 1e − 13, 1e − 13 and 1e − 20. Note that a smaller R gives more a priori information on D. It is seen from figure 10 that a larger R yields a worse image of D.