Total Deep Variation: A Stable Regularization Method for Inverse Problems.

Area of Science:

• Computer Vision
• Medical Imaging
• Applied Mathematics
• Deep Learning

Background:

• Inverse problems are prevalent in computer vision and medical imaging.
• Variational methods, minimizing data fidelity and regularization terms, are common solutions.
• Handcrafted regularizers are often surpassed by deep learning techniques.

Purpose of the Study:

• To introduce a general-purpose, data-driven regularizer for inverse problems.
• To integrate deep learning within the variational framework for enhanced problem-solving.
• To provide mathematical analysis and experimental validation of the proposed method.

Main Methods:

• Developed a "total deep variation" regularizer using convolutional neural networks for multi-scale feature extraction.
• Combined variational formulation with deep learning for inverse problem solving.
• Conducted mathematical analysis, including optimal control and stability analysis, and experimental verification.

Main Results:

• Achieved state-of-the-art performance on various imaging tasks.
• Demonstrated robustness against adversarial attacks.
• Derived numerical upper bounds for generalization error.

Conclusions:

• The proposed total deep variation regularizer effectively enhances inverse problem solutions.
• Deep learning integration within variational methods offers significant advantages.
• The approach provides a robust and mathematically sound framework for advanced imaging applications.