Structural Changes in Nonlocal Denoising Models Arising Through Bi-Level Parameter Learning.

Area of Science:

• Image Processing
• Optimization Theory
• Computational Mathematics

Background:

• Parameter learning in image processing often faces challenges with non-compact domains.
• Existing methods struggle when common constraints like box constraints are removed.

Purpose of the Study:

• To introduce a unified framework for parameter learning in image processing using bi-level optimization.
• To address challenges posed by non-compact parameter domains.
• To identify optimal regularizers within a parameterized family.

Main Methods:

• Development of a unified framework based on bi-level optimization schemes.
• Extension of the upper-level functional to the closure of the parameter domain via Gamma-convergence.
• Analysis under assumptions of Mosco-convergence and uniqueness of lower-level problem minimizers.

Main Results:

• The proposed extension of the upper-level functional coincides with the relaxation.
• The framework admits minimizers relevant to the parameter optimization problem.
• Application to four image denoising models with nonlocality and diverse parameter dependencies.

Conclusions:

• The framework provides a robust approach for parameter learning in image processing, even with non-compact domains.
• Theoretical conditions for model stability are established for practical image reconstruction.
• The study offers insights into parameter dependence and its impact on model stability.