Tomographic reconstruction and estimation based on multiscale natural-pixel bases

Area of Science:

• Image reconstruction
• Computational imaging
• Applied mathematics

Background:

• Tomographic imaging often faces challenges with incomplete, irregular, and noisy projection data.
• Existing reconstruction methods struggle with ill-conditioning arising from imaging system geometry.
• Multiscale representations are crucial for handling complex data and achieving robust reconstructions.

Purpose of the Study:

• To develop a multiscale reconstruction technique for tomographic imaging from incomplete data.
• To address ill-conditioning and noise in projection data for improved object reconstruction.
• To enable object reconstruction at multiple scales or resolutions.

Main Methods:

• Utilizing a natural pixel-type representation to construct nearly orthonormal multiscale basis functions.
• Exploiting the sparsity of the system matrix for efficient reconstruction.
• Developing a multiscale reconstruction procedure by partitioning the system matrix by scales.
• Extending the technique for noisy data using maximum a posteriori probability (MAP) estimation with a self-similar prior.

Main Results:

• Achieved extremely sparse system matrices due to nearly orthonormal basis functions.
• Successfully addressed ill-conditioning by isolating it in the coarsest scale elements.
• Developed a reconstruction procedure involving inversion of only well-conditioned, sparse matrices.
• Demonstrated the capability to reconstruct objects at multiple scales from incomplete and noisy data.

Conclusions:

• The proposed multiscale reconstruction framework offers a robust solution for incomplete and noisy tomographic data.
• The technique effectively regularizes imaging problems and facilitates object feature recognition directly from projection data.
• This approach enhances the applicability of tomography in various scientific and engineering domains requiring high-resolution imaging.