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

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.