Computing a projection operator onto the null space of a linear imaging operator: tutorial.

Area of Science:

• Medical Imaging
• Computational Imaging
• Machine Learning in Imaging

Background:

• Imaging systems use linear operators, but can be "blind" to object components.
• This "blindness" is mathematically explained by the imaging operator's null space.
• Objects within the null space are invisible to the imaging system, leading to information loss.

Purpose of the Study:

• To survey computational procedures for characterizing the null space of imaging operators.
• To present a novel machine learning-based approach using linear autoencoders.
• To facilitate the design of regularization strategies for image reconstruction.

Main Methods:

• Survey of computational procedures for establishing projection operators.
• Implementation of a machine learning approach utilizing linear autoencoders.
• Demonstration using biomedical imaging examples.

Main Results:

• Comparison of computational complexities and memory requirements for different procedures.
• Validation of the machine learning approach in characterizing null spaces.
• Insights into the trade-offs between computational demands and accuracy.

Conclusions:

• Characterizing the null space is crucial for imaging system design and comparison.
• The surveyed methods and the new machine learning approach offer practical tools for null space analysis.
• Understanding null spaces improves image reconstruction and reduces information loss.