Uniqueness for the phaseless inverse scattering problem of the generalized 3-D Helmholtz equation: removing the simple-metric and high-smoothness hypotheses, and the non-overdetermined data question

For the generalized Helmholtz equation $\Delta u+k^{2}c(x)u=-\delta(x-y)$ in $\mathbb{R}^{3}$ with the Sommerfeld radiation condition, Klibanov (2016; Inverse Problems & Imaging 2017) proved that the spatially distributed dielectric constant $c(x)$ is uniquely determined by phaseless point-source data, i.e. by the modulus $|u(x,y,k)|$ of the complex-valued wave field measured outside the scatterer for sources on a closed surface and wavenumbers in an interval, without any phase information. The theorem rests on two auxiliary hypotheses — the conformal travel-time metric $d\tau=\sqrt{c(x)}\,|dx|$ is simple (a single geodesic joins every pair of points; a sufficient condition is that the Hessian of $\ln c$ be positive semidefinite), and $c$ is fifteen times continuously differentiable — and it uses overdetermined data (more data variables than unknowns). The author explicitly noted that no uniqueness theorem exists for any non-overdetermined formulation of this point-source problem even when the full complex field is known. Across the Klibanov-Romanov line, the Romanov-Yamamoto interference line, numerical single-illumination methods, and adjacent obstacle-phaseless literature, no theorem removing the simple-metric assumption, no reduction of the $C^{15}$ regularity, and no non-overdetermined uniqueness result is known. All three questions remain open; this record states them as one problem about the true geometric, regularity, and data requirements for unique recovery of a dielectric constant from point-source scattering data.

Background

In optical and X-ray imaging of micron- and nano-scale objects (fabricated nanostructures, biological cells), the phase of the electromagnetic field cannot be measured at the relevant wavelengths; detectors record only intensities. A widely used scalar model for a single incident polarization component is the generalized Helmholtz equation

$$\Delta u+k^{2}c(x)u=-\delta(x-y),\quad x\in\mathbb{R}^{3},$$

where $k>0$ is the wavenumber, $y$ is the source position, and $c(x)$ is the spatially distributed dielectric constant (relative permittivity), normalized so that $c=1$ in vacuum: $c=1+\beta$ with $\beta\ge 0$ supported in a bounded domain $\Omega$ that is enclosed by a measurement surface $S=\partial G$. The solution $u$ satisfies the Sommerfeld radiation condition $\partial u/\partial r-iku=o(1/r)$ as $r\to\infty$, which selects outgoing waves; $u_{0}(x,y,k)=e^{ik|x-y|}/(4\pi|x-y|)$ is the incident spherical wave and $u_{sc}=u-u_{0}$ the scattered wave. The phaseless inverse scattering problem (PISP) asks whether $c$ is determined by the modulus of the field — e.g. by $f(x,y,k)=|u(x,y,k)|$ for source positions $y\in S$, receiver positions $x$ in a small ball $B_{\omega}(y)$ around each source with $\omega<\mathrm{dist}(S,\partial\Omega)$, and all wavenumbers $k$ in an interval $(a,b)\subset(0,\infty)$ — or by $|u_{sc}(x,y,k)|^{2}$ for $x,y\in S$.

Uniqueness for phaseless inverse scattering was first established in one dimension, then in 3-D for the Schrödinger equation $\Delta v+k^{2}v-q(x)v=-\delta(x-y)$ (Klibanov, Applied Mathematics Letters, 2014). The Schrödinger case is structurally simpler because the unknown coefficient $q$ is not multiplied by $k^{2}$. For the dielectric problem the factor $k^{2}c(x)$ makes the high-frequency behavior of the point-source field governed by the travel-time geometry of the conformal Riemannian metric $d\tau=\sqrt{c(x)}\,|dx|$: the travel time $\tau(x,y)$ between two points solves the eikonal equation $|\nabla_{x}\tau|^{2}=c(x)$ with $\tau=O(|x-y|)$ as $x\to y$, and equals the length $\int_{\Gamma}\sqrt{c}\,d\sigma$ of the geodesic $\Gamma(x,y)$ of this metric joining the points.

Klibanov's uniqueness proof (arXiv:1607.03978, published in Inverse Problems and Imaging, 2017) proceeds in four steps: (i) for each fixed pair $(x,y)$ the function $u(x,y,k)$ is analytic in $k$ in a complex half-plane, and its modulus on a real interval determines the function itself for all real $k$ (phase retrieval via the zero structure of analytic functions); (ii) unique continuation for elliptic equations extends the recovered field to all $x\notin\overline{\Omega}$, $y\in S$; (iii) passing to the time-domain Green function of the associated hyperbolic equation $c(x)v_{tt}=\Delta v+\delta(x-y,t)$, whose leading singularity sits at $t=\tau(x,y)$, extracts the travel times $\tau(x,y)$ for all $x,y\in S$; (iv) $c$ is recovered from these boundary travel times via the inverse kinematic problem (IKP) — the boundary rigidity problem for the metric — whose 3-D uniqueness is classical under the simplicity assumption. A refined theorem of Klibanov and Romanov (Inverse Problems 33, 095007, 2017) reaches the same conclusion from the more practical data $|u_{sc}|^{2}$ on $S\times S$, measured entirely outside the inhomogeneity.

Two auxiliary hypotheses enter this architecture. First, the metric $d\tau=\sqrt{c}\,|dx|$ is assumed simple: every pair of points of $\mathbb{R}^{3}$ is connected by exactly one geodesic. By the Hadamard-Cartan theorem this holds for simply connected complete manifolds of nonpositive sectional curvature, and a sufficient condition on the coefficient (Romanov) is that the matrix Hessian $D^{2}(\ln c)$ be positive semidefinite on $\overline{\Omega}$; this is condition (4) of the source paper. The assumption is used both in the construction of the hyperbolic fundamental solution and in the final IKP step, where uniqueness of recovering a metric from boundary travel times is open beyond simple metrics (with multiple geodesics, caustics, or trapped rays) — related injectivity results for the geodesic ray transform on Cartan-Hadamard manifolds (simply connected, complete, nonpositive curvature, with decay) are recent partial progress. Second, $c\in C^{15}(\mathbb{R}^{3})$: the high smoothness is required by the Vainberg-type construction of the hyperbolic fundamental solution (a method, going back to Vainberg, of building the time-domain Green function as a superposition over wavenumbers) and its geometric-optics asymptotics $u\approx A(x,y)e^{ik\tau(x,y)}+O(1/k)$ as $k\to\infty$, on which the travel-time extraction rests.

Finally, there is a data-dimension gap. The unknown $c$ depends on three variables, while the data $f(x,y,k)$ above depend on up to six; the problem is overdetermined. The author noted that, even with the full complex field (phase included), no uniqueness theorem is known for any non-overdetermined formulation of this 3-D point-source (Dirac delta right-hand side) inverse scattering problem — e.g. for data depending on exactly three variables such as the field $u(x,y_{0},k)$ on $S$ over a frequency interval at a single fixed source position $y_{0}$. The only known non-overdetermined phaseless uniqueness theorems for Helmholtz-type equations replace the point source by a right-hand side that does not vanish on $\overline{\Omega}$, an idealization introduced to make the proof work. Thus the uniqueness theory for the physically natural delta-source dielectric problem currently stands on a restrictive geometric assumption, an unrealistically high smoothness requirement, and more data than the unknown warrants.

Problem Statement

Let $\Omega\subset G\subset\mathbb{R}^{3}$ be bounded domains with $S=\partial G$ and $\mathrm{dist}(S,\partial\Omega)>0$, and let $c(x)=1+\beta(x)$ be real-valued with $\beta\ge 0$ and $\mathrm{supp}\,\beta\subset\overline{\Omega}$. Let $u(x,y,k)$ solve $\Delta u+k^{2}c(x)u=-\delta(x-y)$ in $\mathbb{R}^{3}$ with the radiation condition $\partial u/\partial r-iku=o(1/r)$. The phaseless inverse scattering problem is to determine $c$ in $\Omega$ from the modulus data $f(x,y,k)=|u(x,y,k)|$ for $y\in S$, $x\in B_{\omega}(y)$ (the ball of radius $\omega$ around the source, with $0<\omega<\mathrm{dist}(S,\partial\Omega)$), and $k$ in an interval $(a,b)\subset(0,\infty)$ (or, in the refined formulation of Klibanov-Romanov 2017, from $|u_{sc}(x,y,k)|^{2}$ for $x,y\in S$, $k\in(a,b)$). The known uniqueness theorem requires that the conformal metric $d\tau=\sqrt{c(x)}\,|dx|$ be simple (every pair of points of $\mathbb{R}^{3}$ is joined by exactly one geodesic of this metric) and that $c\in C^{15}(\mathbb{R}^{3})$, and it uses overdetermined data (more free data variables than the three variables of the unknown $c$). The problem is to settle whether uniqueness holds in each of the following three situations:

1. Geometry: is the phaseless problem uniquely solvable when the metric $d\tau=\sqrt{c}\,|dx|$ is not simple — in particular when the sufficient condition $D^{2}(\ln c)\succeq 0$ on $\overline{\Omega}$ (the Hessian matrix of $\ln c$ positive semidefinite) fails, in which case simplicity is not guaranteed and some point pairs may be joined by several geodesics, or the metric may have conjugate points or trapped rays? The target class here is the whole family of coefficients whose metric is not simple, i.e. for which some pair of points is not joined by exactly one geodesic; membership is fixed by this defining property alone, and failure of the sufficient condition $D^{2}(\ln c)\succeq 0$ only makes a coefficient a candidate for the class, not a member of it. The several-geodesics, conjugate-point, and trapped-ray situations are illustrations of the class, not the whole of it.
2. Regularity: does uniqueness persist when $c$ is substantially less smooth than $C^{15}$ — concretely, for $c\in C^{2}(\mathbb{R}^{3})$ or any weaker regularity class (for example $c\in C^{1,\alpha}$) — with the same data? The target class here is that of coefficients no smoother than $C^{2}$, and it suffices to settle the question for any single such class.
3. Data dimension: does uniqueness hold for the non-overdetermined data set consisting of the full complex field $u(x,y_{0},k)$ for $x\in S$ and $k\in(a,b)$ at a single fixed source position $y_{0}$ (three data variables, with the phase known)?

These three questions jointly determine the true geometric, regularity, and data requirements for unique recovery of a dielectric constant from point-source scattering data: every known uniqueness theorem for the $\delta$-source Helmholtz problem currently needs the simple-metric hypothesis, fifteen derivatives, and more data variables than the three unknown variables warrant, and it is not known which of these requirements are essential and which are artifacts of the proof.

References

1. Michael V. Klibanov, 'Uniqueness of a phaseless inverse scattering problem for the generalized 3-D Helmholtz equation', arXiv:1607.03978 [math-ph] (2016), DOI 10.48550/arXiv.1607.03978, https://arxiv.org/abs/1607.03978
2. Michael V. Klibanov, 'A phaseless inverse scattering problem for the 3-D Helmholtz equation', Inverse Problems and Imaging 11(2):263-276 (2017), DOI 10.3934/ipi.2017013, https://doi.org/10.3934/ipi.2017013 (published version of arXiv:1607.03978, which was retitled from 'Uniqueness of a phaseless inverse scattering problem for the generalized 3-D Helmholtz equation' at publication)
3. Michael V. Klibanov and Vladimir G. Romanov, 'Uniqueness of a 3-D coefficient inverse scattering problem without the phase information', Inverse Problems 33(9):095007 (2017), DOI 10.1088/1361-6420/aa7a18, arXiv:1703.01031, https://arxiv.org/abs/1703.01031
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