Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

Area of Science:

• Applied Mathematics
• Image Processing
• Optimization Theory

Background:

• Higher-order total variation (TV) models are crucial for image reconstruction.
• Parameter learning in these models is essential for optimal performance.
• Existing bilevel learning approaches often rely on least squares cost functionals.

Purpose of the Study:

• To develop and analyze a bilevel optimization approach for parameter learning in higher-order TV image reconstruction.
• To introduce and evaluate an alternative cost functional based on a Huber-regularized TV seminorm.
• To provide a comprehensive comparison of different regularizers within the bilevel framework.

Main Methods:

• Bilevel optimization framework for parameter learning.
• Analysis of differentiability properties of the solution operator.
• Derivation of a first-order optimality system.
• Development of a combined quasi-Newton/semismooth Newton algorithm.
• Numerical experiments for performance evaluation.

Main Results:

• Verification of differentiability properties and derivation of optimality conditions.
• Successful implementation of a combined quasi-Newton/semismooth Newton algorithm.
• Demonstration of the suitability of the proposed bilevel optimization approach.
• Evidence of improved performance with the new Huber-regularized TV seminorm cost functional.
• Detailed comparison of $$ and $$ regularizers, highlighting their respective advantages and limitations based on image characteristics and noise levels.

Conclusions:

• The proposed bilevel optimization approach effectively handles parameter learning in higher-order TV image reconstruction.
• The novel Huber-regularized TV seminorm cost functional enhances reconstruction performance.
• The framework facilitates a nuanced understanding of different regularizers for image processing applications.