A Convex Variational Model for Learning Convolutional Image Atoms from Incomplete Data

Area of Science:

• Computational Mathematics
• Image Processing
• Machine Learning

Background:

• Image reconstruction from corrupted or incomplete data is a significant challenge in inverse problems.
• Learning effective image representations (atoms) is crucial for robust reconstruction.
• Existing methods may struggle with simultaneous reconstruction and representation learning.

Purpose of the Study:

• To introduce and analyze a novel variational model for learning convolutional image atoms.
• To enable simultaneous image reconstruction and atom learning within a general inverse problems framework.
• To provide theoretical guarantees for well-posedness and stability.

Main Methods:

• Development of a convex variational model based on lifting and relaxation strategies.
• Analysis in function space to establish theoretical properties.
• Numerical implementation and experimentation, including a semi-convex variant for improved performance.

Main Results:

• The proposed model is convex, allowing for simultaneous image reconstruction and atom learning.
• Analytical properties ensuring well-posedness and stability for inverse problems are proven.
• Numerical computations demonstrate globally optimal solutions for incomplete, noisy, and blurry data.

Conclusions:

• The developed variational model offers a robust approach for learning image atoms from degraded data.
• The model provides theoretical guarantees and demonstrates practical effectiveness in various inverse problem applications.
• The inclusion of a semi-convex variant enhances numerical performance.