Imaging with Power Controlled Source Pairs

Bardsley, Patrick; Vasquez, Fernando Guevara

doi:10.1137/15m1023191

Cited by 4 publications

(11 citation statements)

References 36 publications

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“…Thus, we have computationally generated the data for exactly the same microspheres as we have used in experiments. We have observed that computational results for the forward problem have a very good similarity with experimental data, see Figures 4,5,8. In addition, inversion results for both experimental data sets are very similar with the those of computationally simulated data, see Figures 6,7,9 and Tables 1,2. Finally, the reconstruction error of the refractive index is between 5.2% and 5.4% in all cases, which is small.…”

Section: Discussionsupporting

confidence: 58%

“…We now refer to other approaches to phaseless inverse scattering problems. In [7,8] a phaseless CIP for Helmholtz equation was solved numerically using Kirchhoff migration and Born approximation. While coefficients of partial differential equations are subjects of interests in all above cited works, there is also a significant interest in the reconstruction of surfaces of scatterers from the phaseless data.…”

Section: Introductionmentioning

confidence: 99%

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A Numerical Method to Solve a Phaseless Coefficient Inverse Problem from a Single Measurement of Experimental Data

Klibanov¹,

Koshev²,

Nguyen³

et al. 2018

SIAM J. Imaging Sci.

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We propose in this paper a globally numerical method to solve a phaseless coefficient inverse problem: how to reconstruct the spatially distributed refractive index of scatterers from the intensity (modulus square) of the full complex valued wave field at an array of light detectors located on a measurement board. The propagation of the wave field is governed by the 3D Helmholtz equation. Our method consists of two stages. On the first stage, we use asymptotic analysis to obtain an upper estimate for the modulus of the scattered wave field. This estimate allows us to approximately reconstruct the wave field at the measurement board using an inversion formula. This reduces the phaseless inverse scattering problem to the phased one. At the second stage, we apply a recently developed globally convergent numerical method to reconstruct the desired refractive index from the total wave obtained at the first stage. Unlike the optimization approach, the two-stage method described above is global in the sense that it does not require a good initial guess of the true solution. We test our numerical method on both computationally simulated and experimental data. Although experimental data are noisy, our method produces quite accurate numerical results.

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Section: Discussionsupporting

confidence: 58%

Section: Introductionmentioning

confidence: 99%

A Numerical Method to Solve a Phaseless Coefficient Inverse Problem from a Single Measurement of Experimental Data

Klibanov¹,

Koshev²,

Nguyen³

et al. 2018

SIAM J. Imaging Sci.

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show abstract

“…Uniqueness theorems were proved and reconstruction procedures were proposed in [44][45][46]. In [6,7] a phaseless coefficient inverse scattering problem for Helmholtz equation was solved numerically using Kirchhoff migration and Born approximation. While coefficients of PDEs are subjects of interests in the above cited works, there is also a significant interest in the reconstruction of surfaces of scatterers from phaseless data.…”

Section: Introductionmentioning

confidence: 99%

A Coefficient Inverse Problem with a Single Measurement of Phaseless Scattering Data

Klibanov¹,

Nguyen²,

Nguyen³

2019

SIAM J. Appl. Math.

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This paper is concerned with a numerical method for a 3D coefficient inverse problem with phaseless scattering data. These are multi-frequency data generated by a single direction of the incident plane wave. Our numerical procedure consists of two stages. The first stage aims to reconstruct the (approximate) scattered field at the plane of measurements from its intensity. We present an algorithm for the reconstruction process and prove a uniqueness result of this reconstruction. After obtaining the approximate scattered field, we exploit a newly developed globally convergent numerical method to solve the coefficient inverse problem with the phased scattering data. The latter is the second stage of our algorithm. Numerical examples are presented to demonstrate the performance of our method. Finally, we present a numerical study which aims to show that, under a certain assumption, the solution of the scattering problem for the 3D scalar Helmholtz equation can be used to approximate the component of the electric field which was originally incident upon the medium.

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“…Proposition 5 is the electromagnetic analogue of a statistical stability result for scalar waves that was proven originally in [28] (see also [8]) and is proved in Appendix A.…”

Section: The Coherency Matrix From Time Domain Electric Field Autocor...mentioning

confidence: 72%

“…A key aspect of our imaging method is that we do not need to retrieve all phases from the data, since the imaging method we use does not require it. This is a feature that is also used for intensity only imaging in [8,9,[40][41][42]. For full aperture data, phase retrieval is done in [21][22][23] using a similar preprocessing of the intensity data as we present here.…”

Section: Related Workmentioning

confidence: 99%

Imaging small polarizable scatterers with polarization data

Bardsley¹,

Cassier²,

Vasquez³

2018

Inverse Problems

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We present a method for imaging small scatterers in a homogeneous medium from polarization measurements of the electric field at an array. The electric field comes from illuminating the scatterers with a point source with known location and polarization. We view this problem as a generalized phase retrieval problem with data being the coherency matrix or Stokes parameters of the electric field at the array. We introduce a simple preprocessing of the coherency matrix data that partially recovers the ideal data where all the components of the electric field are known for different source dipole moments. We prove that the images obtained using an electromagnetic version of Kirchhoff migration applied to the partial data are, for high frequencies, asymptotically identical to the images obtained from ideal data. We analyze the image resolution and show that polarizability tensor components in an appropriate basis can be recovered from the Kirchhoff images, which are tensor fields. A time domain interpretation of this imaging problem is provided and numerical experiments are used to illustrate the theory.

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Imaging with Power Controlled Source Pairs

Cited by 4 publications

References 36 publications

A Numerical Method to Solve a Phaseless Coefficient Inverse Problem from a Single Measurement of Experimental Data

A Numerical Method to Solve a Phaseless Coefficient Inverse Problem from a Single Measurement of Experimental Data

A Coefficient Inverse Problem with a Single Measurement of Phaseless Scattering Data

Imaging small polarizable scatterers with polarization data

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