From Phase Separation in Heterogeneous Media to Learning Training Schemes for Image Denoising

Explore a comprehensive lecture on two interconnected themes: phase separation in heterogeneous media and learning training schemes for image denoising. Delve into the variational approaches, energy dimensionalities, and geometric measure theory concepts that unite these topics. Examine modern technologies and biological systems utilizing engineered inclusions and natural heterogeneities to create novel composite materials. Investigate the gradient theory of phase transitions, considering critical, supercritical, and subcritical cases with fixed and moving wells. Learn about the interaction between homogenization and phase transition processes, leading to anisotropic and isotropic interfacial energies. Discover the role of thermal fluctuations in nanodomain formation. Shift focus to image denoising techniques, exploring total variation (TV) and total generalized variation (TGV) regularizations. Understand the importance of tuning parameters and automatic selection through multilevel approaches. Examine space-dependent parameters on dyadic grids and their optimization. Gain insights into existence of minimizers for discontinuous parameters and finite optimal partitions. Compare the performance of optimized parameters against constant parameters on representative test images.