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Microlocal analysis of non-linear operators arising in Compton CT.

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This study introduces a new mathematical method for Compton scattering tomography (CST) by analyzing a non-linear ray transform. The research clarifies singularity properties, enabling improved image reconstruction in CST.

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Area of Science:

  • Mathematical imaging and inverse problems
  • Microlocal analysis
  • Tomographic reconstruction

Background:

  • Compton scattering tomography (CST) involves a non-linear ray transform (R) due to target-dependent attenuation.
  • Standard linear Fourier integral operator (FIO) theory is inapplicable because the transform's weights are non-smooth.
  • The V-line (broken ray) transform (V) models ray attenuation, crucial for understanding CST's complexities.

Purpose of the Study:

  • To develop a novel microlocal analysis for the non-linear ray transform (R) in Compton scattering tomography (CST).
  • To characterize singularities in the ray transform weights using the V-line transform (V).
  • To establish theoretical foundations for improved image reconstruction in CST.

Main Methods:

  • Applied microlocal analysis to a non-linear ray transform (R) arising in Compton scattering tomography (CST).
  • Utilized the V-line transform (V) to model ray attenuation and analyze singularity properties of transform weights.
  • Quantified singularities of distributional products using Sobolev orders and combined with linear FIO theory.

Main Results:

  • Determined the Sobolev order of singularities for the non-linear ray transform (R) of a target function f.
  • Established a correspondence between the strongest singularities of (R)f and the wavefront set of the Radon transform of f.
  • Proved injectivity results for the non-linear ray transform (R) and developed novel reconstruction methods.

Conclusions:

  • The novel microlocal analysis provides a theoretical framework for understanding and reconstructing images in Compton scattering tomography (CST).
  • The developed methods, grounded in singularity analysis and FIO theory, offer improved accuracy and injectivity for the non-linear ray transform (R).
  • Simulated reconstructions validate the efficacy of the new theoretical approach and reconstruction techniques.