[Paper Review] Uniqueness results on phaseless inverse scattering with a reference ball

Zhang, Deyue, Guo, Yukun|arXiv (Cornell University)|Feb 10, 2018

Numerical methods in inverse problems45 citations

TL;DR

This paper establishes uniqueness in phaseless inverse scattering for the Helmholtz equation by introducing a reference ball and superposing a fixed plane wave with point sources as incident waves. The method overcomes translation invariance, enabling unique recovery of obstacle shape, boundary condition, or medium refractive index from modulus-only far-field data at a single frequency.

ABSTRACT

This paper is devoted to the uniqueness in inverse acoustic scattering problems for the Helmholtz equation with phaseless far-field data. Some novel techniques are developed to overcome the difficulty of translation invariance induced by a single incident plane wave. In this paper, based on adding a reference ball as an extra artificial impenetrable obstacle (resp. penetrable homogeneous medium) to the inverse obstacle (resp. medium) scattering system and then using superpositions of a fixed plane wave and some point sources as the incident waves, we rigorously prove that the location and shape of the obstacle as well as its boundary condition or the refractive index can be uniquely determined by the modulus of far-field patterns. The reference ball technique in conjunction with the superposition of incident waves brings in several salient benefits. First, the framework of our theoretical analysis can be applied to both the inverse obstacle and medium scattering problems. Second, for inverse obstacle scattering, the underlying boundary condition could be of a general type and be uniquely determined. Finally, only a single frequency is needed.

Motivation & Objective

• Address the fundamental challenge of translation invariance in phaseless inverse scattering, where only intensity (magnitude) of far-field data is available.
• Overcome the non-uniqueness issue in recovering scatterer location and shape from phaseless far-field patterns.
• Develop a theoretical framework that uniquely determines both the geometry and boundary condition (or refractive index) of a scatterer using only modulus of far-field data.
• Establish a unified approach applicable to both inverse obstacle and medium scattering problems.
• Enable uniqueness with a single frequency, avoiding the need for multi-frequency or multi-directional data.

Proposed method

• Introduce a reference sound-soft ball as an artificial obstacle to break translation invariance in phaseless scattering.
• Use superposition of a fixed plane wave and multiple point sources as incident fields to generate rich far-field data.
• Apply the reciprocity relation and analyticity of far-field patterns to derive uniqueness via contradiction arguments.
• Utilize Green's second theorem and the Helmholtz equation to derive integral identities on the boundary of the reference ball.
• Employ complex exponential phase relations and the non-vanishing property of far-field patterns to eliminate spurious solutions.
• Prove that identical modulus of far-field patterns implies identical total fields and thus identical scatterers and boundary conditions.

Experimental results

Research questions

• RQ1 How can translation invariance in phaseless inverse scattering be overcome to uniquely determine the location and shape of a scatterer?
• RQ2 Can the boundary condition of an impenetrable obstacle be uniquely identified from phaseless far-field data alone?
• RQ3 Is it possible to uniquely reconstruct the refractive index of a penetrable inhomogeneous medium using only the modulus of far-field patterns?
• RQ4 Does the combination of a reference ball and superposed incident waves enable uniqueness in both obstacle and medium scattering problems?
• RQ5 Can uniqueness be achieved with only a single frequency and phaseless far-field data?

Key findings

• The reference ball technique, combined with superposition of a fixed plane wave and point sources, successfully breaks translation invariance in phaseless inverse scattering.
• Uniqueness of the obstacle shape and its boundary condition is proven for general mixed boundary conditions, including Dirichlet, Neumann, and impedance types.
• The refractive index of a penetrable medium is uniquely determined by the modulus of far-field patterns when using the proposed incident wave configuration.
• Only a single frequency is required to achieve uniqueness, which is a significant improvement over previous methods requiring multi-frequency data.
• Analyticity of the far-field pattern and the reciprocity relation are key tools in ruling out spurious solutions and proving uniqueness.
• The method is theoretically robust and applicable to both inverse obstacle and medium scattering problems with phaseless data.