Learning a stable approximation of an existing but unknown inverse mapping: application to the half-time circular...

Area of Science:

• Medical Imaging
• Computational Imaging
• Applied Mathematics

Background:

• Supervised deep learning (DL) offers implicit regularization for image reconstruction but faces robustness concerns in ill-posed problems.
• Existing DL methods struggle with instability when solving problems lacking unique, stable inverse mappings.
• A novel application of DL is explored for image reconstruction problems where a stable, albeit unknown, inverse mapping exists.

Purpose of the Study:

• Investigate the performance of supervised DL for image reconstruction in a scenario with a known stable inverse mapping.
• Develop a DL-based method to approximate an unknown inverse mapping for improved generalization.
• Explore the potential of DL to reveal insights into unknown analytic inverse formulas.

Main Methods:

• A learned filtered backprojection method using a convolutional neural network (CNN) was developed.
• The CNN approximates the unknown filtering operation in image reconstruction from radially truncated circular Radon transform (CRT) data.
• The method was specifically designed for 'half-time' measurement data.

Main Results:

• The developed learned filtered backprojection method demonstrated stable performance.
• The method exhibited robust generalization capabilities, performing well on data significantly different from the training set.
• The DL approach successfully approximated the inverse mapping without requiring optimization-based reconstruction.

Conclusions:

• Supervised deep learning can effectively learn stable inverse mappings for image reconstruction problems with known stable solutions.
• The developed CNN-based method offers a stable and generalizable alternative for specific inverse problems.
• This approach holds promise for wave-based imaging modalities, including photoacoustic computed tomography.