Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A holographic global uniqueness in passive imaging

Roman Novikov

TL;DR

This work proves holographic global uniqueness for passive imaging in the Helmholtz setting: a radiation solution $\psi$ of $- abla^2\psi=\kappa^2\psi$ in an exterior region is uniquely determined by the imaginary part of $\psi$ measured on a ray interval or a plane patch. The approach blends the Atkinson–Wilcox expansion with a two-point approximation, Green-type integral formulas, and analytic continuation to propagate local boundary information into global reconstructions. Consequently, $\psi$ (and, via the Gelfand–Krein–Levitan framework, the potential $v$) can be uniquely recovered from phaseless data such as $\Im\psi$ or $\Im R_v^+$ on suitably chosen surfaces. The results extend to other measurement surfaces and establish holographic-type guarantees for inverse scattering problems in passive imaging, with potential applications to helioseismology and related fields.

Abstract

We consider a radiation solution $ψ$ for the Helmholtz equation in an exterior region in $\mathbb R^3$. We show that the restriction of $ψ$ to any ray $L$ in the exterior region is uniquely determined by its imaginary part $\Imψ$ on an interval of this ray. As a corollary, the restriction of $ψ$ to any plane $X$ in the exterior region is uniquely determined by $\Imψ$ on an open domain in this plane. These results have holographic prototypes in the recent work Novikov (2024, Proc. Steklov Inst. Math. 325, 218-223). In particular, these and known results imply a holographic type global uniqueness in passive imaging and for the Gelfand-Krein-Levitan inverse problem (from boundary values of the spectral measure in the whole space) in the monochromatic case. Some other surfaces for measurements instead of the planes $X$ are also considered.

A holographic global uniqueness in passive imaging

TL;DR

This work proves holographic global uniqueness for passive imaging in the Helmholtz setting: a radiation solution of in an exterior region is uniquely determined by the imaginary part of measured on a ray interval or a plane patch. The approach blends the Atkinson–Wilcox expansion with a two-point approximation, Green-type integral formulas, and analytic continuation to propagate local boundary information into global reconstructions. Consequently, (and, via the Gelfand–Krein–Levitan framework, the potential ) can be uniquely recovered from phaseless data such as or on suitably chosen surfaces. The results extend to other measurement surfaces and establish holographic-type guarantees for inverse scattering problems in passive imaging, with potential applications to helioseismology and related fields.

Abstract

We consider a radiation solution for the Helmholtz equation in an exterior region in . We show that the restriction of to any ray in the exterior region is uniquely determined by its imaginary part on an interval of this ray. As a corollary, the restriction of to any plane in the exterior region is uniquely determined by on an open domain in this plane. These results have holographic prototypes in the recent work Novikov (2024, Proc. Steklov Inst. Math. 325, 218-223). In particular, these and known results imply a holographic type global uniqueness in passive imaging and for the Gelfand-Krein-Levitan inverse problem (from boundary values of the spectral measure in the whole space) in the monochromatic case. Some other surfaces for measurements instead of the planes are also considered.
Paper Structure (13 sections, 9 theorems, 46 equations)

This paper contains 13 sections, 9 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. The Atkinson-Wilcox expansion
  5. A two-point approximation for $\psi$
  6. A Green type formula
  7. Some facts of direct scattering
  8. Proof of Theorem 1
  9. Case $L\subseteq L_{0, \theta}$
  10. General case
  11. Proof of Theorem 3
  12. Proof of Theorem 4
  13. Proof of Theorem 5

Key Result

Theorem 1

Let $\psi$ be a radiation solution of equation (eq:1.1) as in (eq:1.2). Let $L$ be a ray as in (eq:1.3) such that $L \subset {\cal U}$, where ${\cal U}$ is the region in (eq:1.1). Then $\psi$ on $L$ is uniquely determined by $\mathop{\mathrm{Im}} \psi$ on $\Lambda$, where $\Lambda$ is an arbitrary n

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Remark 1
  • Theorem 3
  • Remark 2
  • Remark 3
  • Example 1
  • Theorem 4
  • Theorem 5
  • ...and 8 more