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Radon-to-Helmholtz mappings and nonlinear diffraction tomography.

Gregory Samelsohn

    Journal of the Optical Society of America. A, Optics, Image Science, and Vision
    |March 10, 2021
    PubMed
    Summary
    This summary is machine-generated.

    This study introduces the Glauber approximation (GA) as a nonlinear Radon-to-Helmholtz (RtH) mapping for solving the Helmholtz equation. This method offers superior tomographic inversion compared to traditional Born approximations, especially for scattering potentials.

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    Area of Science:

    • Wave physics
    • Electromagnetics
    • Inverse problems

    Background:

    • The scalar Helmholtz equation is fundamental in wave propagation modeling.
    • Approximate analytical solutions are crucial for practical applications in imaging and scattering.
    • Existing methods like Born and Rytov approximations have limitations for certain scattering scenarios.

    Purpose of the Study:

    • To explore approximate, analytically invertible solutions for the scalar Helmholtz equation.
    • To investigate the Glauber approximation (GA) as a nonlinear Radon-to-Helmholtz (RtH) mapping.
    • To demonstrate the utility of the GA in diffraction tomography.

    Main Methods:

    • Formulating the Glauber approximation (GA) as a nonlinear Radon-to-Helmholtz (RtH) mapping.
    • Developing a position space counterpart for the GA.
    • Comparing GA with Mazar-Felsen propagator and Born/Rytov approximations.
    • Applying RtH mappings to nonlinear diffraction tomography of penetrable objects.

    Main Results:

    • The GA transforms scattering potential sinograms into approximate Helmholtz equation solutions.
    • A paraxial GA aligns with the Mazar-Felsen propagator for forward-scattered waves.
    • For weak scattering, solutions reduce to Born/Rytov approximations while retaining nonlinear model formats.
    • The GA provides efficient inversion of synthetic data, outperforming Born inversions.

    Conclusions:

    • The Glauber approximation (GA), viewed as a nonlinear Radon-to-Helmholtz (RtH) mapping, offers a powerful tool for solving the Helmholtz equation.
    • RtH mappings are analytically invertible and applicable to nonlinear diffraction tomography.
    • The GA demonstrates superior performance in tomographic reconstruction compared to conventional methods, even for moderately scattering objects.