In this talk we will use variational models involving density measures of different dimensionality to study training/learning schemes for a novel class of image-processing operators that provides a unified approach to the standard regularizers and PDE-based approaches to image denoising.
To illustrate the relevance of similar bulk-surface energy models in the study of novel materials, we will analyze the onset of man-made nanocrystals of semiconducting materials (quantum dots). Their formation and assembly patterns play a central role in nanotechnology, and in particular in the optoelectronic properties of semiconductors. Changing the dots' size and shape gives rise to many applications that permeate our daily lives. As the creation of quantum dots evolves with time, materials defects appear and these may strongly influence material properties, including rigidity and conductivity. The regularity and evolution of the quantum dots shapes, and the nucleation and motion of dislocations will be addressed.
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