Higher-Order Variational Techniques for Image Reconstruction and Enhancement

Abstract: 

Restoring the original image contents from distorted measurements is one of the most important tasks in image processing. It comprises the enhancement and reconstruction of images distorted by noise or blur (image denoising/deblurring), the filling-in of gaps in images (image inpainting) and the reconstruction of an image from noisy (and possible undersampled) Fourier/Radon measurements. Within various standard methodologies for the solution of these tasks, variational approaches constitute a rich toolbox of methodologies for image reconstruction and enhancement. These techniques are interesting from both an applicational viewpoint - because they are able to produce qualitatively good visual results and can be captured within automatable processing algorithms - but also from a mathematical analysis point of view - because they show some beautiful mathematical concepts and pose interesting analytical problems.

In this presentation we shall concentrate on a specific class of variational techniques, namely higher-order approaches, i.e., second- and third-order. After spending some time on introducing the concept of such methods and giving a historical overview of some important contributions in this area, we will get to know some recently proposed higher-order methods, their mathematical properties and applications. The presentation will be furnished by various numerical examples and applications for image restoration, surface interpolation and MRI.