Equivariant neural networks for inverse problems.

Area of Science:

• Deep Learning
• Computational Imaging
• Applied Mathematics

Background:

• Convolutional neural networks (CNNs) effectively use translational equivariance for inductive bias.
• Research is expanding to incorporate other symmetries into deep learning via group equivariant CNNs.
• Existing work often focuses on roto-translational or spherical symmetries.

Purpose of the Study:

• To integrate group equivariant convolutional operations into learned reconstruction methods for inverse problems.
• To leverage variational regularization principles and group symmetry invariance.
• To develop novel iterative reconstruction algorithms using these equivariant operations.

Main Methods:

• Incorporating group equivariant convolutional operations into proximal operators for inverse problems.
• Modeling proximal operators as group equivariant convolutional neural networks.
• Applying roto-translationally equivariant operations to low-dose CT and subsampled MRI reconstruction.

Main Results:

• Demonstrated that group equivariant operations can be naturally integrated into learned reconstruction.
• Showcased improved reconstruction quality in low-dose CT and subsampled MRI.
• Achieved these improvements with minimal extra training cost and no additional test time cost.

Conclusions:

• Learned iterative methods using group equivariant CNNs are effective for inverse problems.
• The proposed methodology enhances reconstruction quality in medical imaging applications.
• This approach offers a computationally efficient way to improve learned reconstruction techniques.