Explore bilevel optimization approaches for variational image denoising in this 40-minute lecture by Juan Carlos De los Reyes. Delve into the mathematical foundations of determining noise models in corrupted images using supervised machine learning techniques. Examine differentiability properties of solution operators, derive Karush-Kuhn-Tucker systems for optimality conditions, and learn about fast Newton-type algorithms for numerical problem-solving. Compare the performance of different cost functionals and local regularizers, and investigate the extension to non-local variational models. Gain insights into the challenges of implementing these approaches and evaluate the effectiveness of local and non-local denoising models through test image comparisons.
Overview
Syllabus
Intro
Outline
A generic inverse problem in imaging
The variational approach..
Modelling
Total variation (TV) denoising Least squares minimization
Modified non-local means Giboa Osher (2007)
State of the art in optimal model design
Bilevel optimal reconstruction model Assumptions
Learning from training sets
Learning by optimisation in imaging
Learning in function space
A generic TV denoising model
Learning TV denoising model
State of the art on optimality systems
In this setting we can prove
Optimality system for the regularized problems
Optimality system for bilevel problem
Numerical notes
Mixed Gauss & Poisson noise
Impulse noise
Partial conclusions
Nonhomogeneous noise
Ingredients for optimality conditions
Experiments
Motivation
Forward denoising problem
The kernel
Different kernels, different results
Bilevel optimization problem Optimal weight
What we are trying to do..
Conclusions and outlook
Taught by
Hausdorff Center for Mathematics
