The resolution of ill-posed inverse problems is addressed by the study of regularization methods that require the solution of convex and non-convex, constrained, optimization problems.
Variational approaches to the solution of ill-posed imaging problems are currently the most successful and widely used methods.
Various imaging problems result from discretization of continuous inverse problems. Advanced numerical techniques are needed for their resolution due to the ill-conditioning and large dimension of the problem. To treat the ill-conditioning efficiently one needs regularization models, in which an a priori knowledge about the sought solution is integrated in the variational model.
To process considerable amounts of data it is necessary to study computationally efficient numerical optimization algorithms.
The applications considered are in the industrial field (barcodes), in medical imaging, in microscopy, and in the non-destructive investigation of materials. The following areas are specifically considered:
inversion of NMR relaxometry and spectroscopy data;
reconstruction of 3D computer tomography images using gradient and quasi-Newton methods applied to linear least squares problem with TV regularization;
identification of parameters in differential diffusion-transport and nonlinear reaction models;
deblurring/denoising 2D/3D images;
reconstruction of 2D/3D images from tomography to electrical impedance (EIT), cone-beam computer tomography;
analysis of signals/images with compressed sensing techniques;
segmentation of structures into 2D images and scalar fields on manifolds.