Parameter constraints and ill-conditioning in tensor tomography reconstruction: a theoretical and numerical approach

Area of Science:

• Material Science
• Medical Imaging
• Computational Imaging

Background:

• Tensor tomography enables anisotropic, non-invasive internal structure analysis in materials and tissues.
• Reconstructing anisotropic scattering properties is crucial for understanding microstructures but mathematically challenging.
• Accurate reconstruction necessitates optimized acquisition parameters for tensor tomography.

Purpose of the Study:

• To establish a theoretical framework for accurately approximating scattering functions in anisotropic X-ray dark-field tomography (AXDT).
• To determine the optimal number of parameters required for precise scattering tensor representation.
• To provide insights for improving tensor tomography system design and acquisition stability.

Main Methods:

• Utilized spherical harmonic decomposition to represent scattering properties.
• Employed singular value decomposition to analyze parameter requirements.
• Derived a theoretical upper limit for scattering function approximation in AXDT.

Main Results:

• Established a theoretical upper limit of 15 parameters for accurate scattering function approximation.
• Demonstrated the critical importance of selecting optimal acquisition parameters for reliable tensor tomography reconstruction.
• Provided a condition number for analyzing the stability of acquisition schemes.

Conclusions:

• The 15-parameter limit offers a pathway for efficient and accurate tensor tomography.
• Optimized parameter selection is key to enhancing the reliability of material and medical diagnostics.
• These findings contribute to improved tensor tomography system design and application efficiency.