Deep learning-based solvability of underdetermined inverse problems in medical imaging

Area of Science:

• Medical Imaging
• Deep Learning
• Inverse Problems

Background:

• Deep learning significantly advances medical imaging by addressing underdetermined inverse problems.
• Applications include undersampled MRI, interior tomography, and sparse-view CT, aiming for high-resolution images with minimal data.
• Current limitations include a lack of mathematical understanding for deep learning's success in these areas.

Purpose of the Study:

• To provide a mathematical explanation for deep learning's effectiveness in solving highly underdetermined inverse problems.
• To analyze the causal relationships learned from training data structures.
• To compare deep learning with conventional methods using a low-dimensional solution model.

Main Methods:

• Development of a low-dimensional solution model to contrast deep learning with conventional approaches.
• Analysis of how deep learning methods learn reconstruction maps from training data across MRI, CT, and tomography.
• Investigation into the nonlinearity of underdetermined linear systems and the M-RIP condition for learning.

Main Results:

• Demonstration of deep learning's advantage over conventional methods in specific underdetermined scenarios.
• Analysis of the conditions under which deep learning can successfully learn desired reconstruction maps.
• Insights into the nonlinear structure of underdetermined systems and learning requirements.

Conclusions:

• Deep learning's success in underdetermined inverse problems is linked to learning causal relationships within data.
• Mathematical analysis, including the M-RIP condition, is crucial for understanding and optimizing these methods.
• This work provides a foundation for developing more robust and interpretable deep learning solutions in medical imaging.