In the first part of this thesis, we focus on the Poisson summation formula in the non-periodic setting. We introduce the concept of quasicrystal and construct new types of Poisson summation formulas on it. We then develop the theory of sampling and interpolation on model sets. We will prove some uniqueness results regarding exact reconstruction on model sets. In the second part of this thesis, we focus on compressed sensing and image super-resolution. We develop a technique to reconstruct an image with few measurements

but high resolution. Experimental results will be shown.