Several aspects of modern signal analysis: optimal tuning parameters in MRI inverse problems and symplectic time-frequency analysis of modulation spaces

In this thesis, we explore two prominent aspects of the mathematics involved in MRI signal analysis. First, we focus on compressed sensing MRI, and we introduce an iterative algorithm (ALMA) inspired by convex analysis that efficiently determines the tuning parameter for Total Variation (TV) regularized LASSO (TV-LASSO). The selection of the tuning parameter is crucial for minimizing noise and artifacts in the reconstructed image, and this choice is often made manually, which is time-consuming. The results of ALMA show near-optimality of our reconstructions in terms of multiscale structural similarity, peak signal-to-noise ratio and coefficient of joint variation.Second, we generalize the most well-known and widely used time-frequency representations through the so-called metaplectic operators. This generalization enables us to explain the properties of what we called metaplectic Wigner distributions in terms of the symplectic group, to which they are closely related. Thanks to these techniques it is possible to construct metaplectic Wigner distributions with the right properties, according to the applications.