Inversion of the broken ray transform in the case of energy-dependent attenuation

Area of Science:

• Medical Physics
• Image Reconstruction
• Computational Imaging

Background:

• The Broken Ray Transform (BRT) models X-ray propagation with single scattering.
• Existing BRT inversion algorithms assume a constant attenuation coefficient (μ).
• X-ray energy loss during Compton scattering necessitates accounting for energy-dependent attenuation.

Purpose of the Study:

• To develop and validate inversion algorithms for the BRT with N ≥ 3 detectors.
• To address the challenge of energy-dependent attenuation coefficients in X-ray scattering.
• To improve the accuracy and stability of X-ray imaging reconstruction.

Main Methods:

• Derivation of inversion formulas for BRT with N ≥ 3 detectors.
• Approximation of energy-dependent attenuation coefficients as linear functions.
• Development of iterative reconstruction algorithms utilizing global and local data.
• Optimization of inversion formulas for enhanced noise stability with N > 4 detectors.

Main Results:

• A family of inversion formulas is derived for N ≥ 4 detectors.
• An optimal, noise-stable formula is identified for N > 4 detectors.
• For N = 4, a single, non-optimizable formula is obtained.
• N = 3 detectors are sufficient for inversion when attenuation is energy-independent.
• Iterative algorithms demonstrate successful reconstruction capabilities.

Conclusions:

• The developed methods enable accurate BRT inversion with energy-dependent attenuation.
• The choice of detector number (N) influences the inversion formula and optimization possibilities.
• Iterative reconstruction algorithms offer robust solutions for complex X-ray imaging scenarios.