In Ultrasound Non-Destructive Testing, we analyse the echoes of ultrasound impulses inserted into a specimen to detect flaws. The echoes alone do not suffice for this task, they need to be reconstructed to become usable: in this thesis, we modelled the Reconstruction using a Linear Problem called Basis Pursuit. We will explain how this Linear Problem can be solved using Column Generation, a Linear Optimization technique exploiting sparsity, with regard to the modelling of the required subproblem.