A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers

Jun Lai,,School Of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027, China , ,,Institute For Mathematics And Its Applications, Minneapolis, Mn 55455, Usa ,,Department Of Mathematics, Purdue University, West Lafayette, In 47907, Usa ,Wei Li,Ming Li,Peijun Li

doi:10.3934/ipi.2018027

Jun Lai, ,School Of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027, China + Show 6 more

Open Access

https://doi.org/10.3934/ipi.2018027

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Journal: Inverse Problems and Imaging	Publication Date: Jan 1, 2018
Citations: 1	License type: cc-by

Abstract

Consider the scattering of a time-harmonic plane wave by heterogeneous media consisting of linear or nonlinear point scatterers and extended obstacles. A generalized Foldy–Lax formulation is developed to take fully into account of the multiple scattering by the complex media. A new imaging function is proposed and an FFT-based direct imaging method is developed for the inverse obstacle scattering problem, which is to reconstruct the shape of the extended obstacles. The novel idea is to utilize the nonlinear point scatterers to excite high harmonic generation so that enhanced imaging resolution can be achieved. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Highlights

In scattering theory, one of the basic problems is the scattering of a time-harmonic plane wave by an impenetrable medium, which is called the obstacle scattering problem [20]
Number of point scatterers number of points to discretize the boundary of extended scatterer(s) number of incident and observation directions number of sampling points along the x- and y-direction time to invert the scattering matrix time to solve the linear system for one incidence time to evaluate the far-field patterns time to apply the non-uniform fast Fourier transform (NUFFT) to evaluate the imaging function
Where Dij represents the nonlinear interaction between φ(1) and φ(2) at the point scatterers, Hij is the nonlinear interaction from the point scatterers to the extended obstacle, Mκj denotes the linear interaction from the extended obstacle to the point scatterers at the wavenumber κj, and Kκj is the linear interaction for the extended obstacle

Summary

Introduction

One of the basic problems is the scattering of a time-harmonic plane wave by an impenetrable medium, which is called the obstacle scattering problem [20]. We introduce the Foldy–Lax formulation for the scattering of a plane incident wave by a group of linear, quadratically nonlinear, or cubically nonlinear point scatterers. The method can be applied directly to the scattered field generated by the first order frequency wave in the nonlinear case. If we fix the location of the nonlinear point scatterers, the imaging method will fail for testing the higher order frequency waves. Npoint Nboundary Ndirection Nsampling Tinvert Tsolver Tffp TNUFFT number of point scatterers number of points to discretize the boundary of extended scatterer(s) number of incident and observation directions number of sampling points along the x- and y-direction time to invert (factorize) the scattering matrix time to solve the linear system for one incidence time to evaluate the far-field patterns time to apply the NUFFT to evaluate the imaging function. The generalized Foldy– Lax for nonlinear point scatterers can be written as (69)

D11 D12 Mκ1

Conclusion