Variational and learning models for image and time series inverse problems

Inverse problems are at the core of many challenging applications. Variational and learning models provide estimated solutions of inverse problems as the outcome of specific reconstruction maps. In the variational approach, the result of the reconstruction map is the solution of a regularized minimization problem encoding information on the acquisition process and prior knowledge on the solution. In the learning approach, the reconstruction map is a parametric function whose parameters are identified by solving a minimization problem depending on a large set of data. In this thesis, we go beyond this apparent dichotomy between variational and learning models and we show they can be harmoniously merged in unified hybrid frameworks preserving their main advantages. We develop several highly efficient methods based on both these model-driven and data-driven strategies, for which we provide a detailed convergence analysis. The arising algorithms are applied to solve inverse problems involving images and time series. For each task, we show the proposed schemes improve the performances of many other existing methods in terms of both computational burden and quality of the solution. In the first part, we focus on gradient-based regularized variational models which are shown to be effective for segmentation purposes and thermal and medical image enhancement. We consider gradient sparsity-promoting regularized models for which we develop different strategies to estimate the regularization strength. Furthermore, we introduce a novel gradient-based Plug-and-Play convergent scheme considering a deep learning based denoiser trained on the gradient domain. In the second part, we address the tasks of natural image deblurring, image and video super resolution microscopy and positioning time series prediction, through deep learning based methods. We boost the performances of supervised, such as trained convolutional and recurrent networks, and unsupervised deep learning strategies, such as Deep Image Prior, by penalizing the losses with handcrafted regularization terms.

Plasticity and neuromodulation of the extended recurrent visual network

The extended visual network, which includes occipital, temporal and parietal posterior cortices, is a system characterized by an intrinsic connectivity consisting of bidirectional projections. This network is composed of feedforward and feedback projections, some hierarchically arranged and others bypassing intermediate areas, allowing direct communication across early and late stages of processing. Notably, the early visual cortex (EVC) receives considerably more feedback and lateral inputs than feedforward thalamic afferents, placing it at the receiving end of a complex cortical processing cascade, rather than just being the entrance stage of cortical processing of retinal input. The critical role of back-projections to visual cortices has been related to perceptual awareness, amplification of neural activity in lower order areas and improvement of stimulus processing. Recently, significant results have shown behavioural evidence suggesting the importance of reentrant projections in the human visual system, and demonstrated the feasibility of inducing their reversible modulation through a transcranial magnetic stimulation (TMS) paradigm named cortico-cortical paired associative stimulation (ccPAS). Here, a novel research line for the study of recurrent connectivity and its plasticity in the perceptual domain was put forward. In the present thesis, we used ccPAS with the aim of empowering the synaptic efficacy, and thus the connectivity, between the nodes of the visuocognitive system to evaluate the impact on behaviour. We focused on driving plasticity in specific networks entailing the elaboration of relevant social features of human faces (Chapters I & II), alongside the investigation of targeted pathways of sensory decisions (Chapter III). This allowed us to characterize perceptual outcomes which endorse the prominent role of the EVC in visual awareness, fulfilled by the activity of back-projections originating from distributed functional nodes.