In this thesis, signal sampling and processing techniques are developed for magnetic resonance applications, to improve the estimation of magnetic resonance parameters and to reduce experimental acquisition times.

Two processing techniques are developed for Nuclear Magnetic Resonance (NMR) relaxation and diffusion experiments. $L1$ regularization is recommended for extracting parameter distributions which are a priori known to be composed of sparse features. $L1$ regularization is shown to be stable at signal-to-noise ratios < 20 and capable of resolving relaxation time constants and diffusion coefficients which differ by as little as 10%, such as in relaxation and diffusion studies of hexane/dodecane in porous media. Modified Total Generalized Variation (MTGV) regularization is recommended for extracting parameter distributions for which there is no prior knowledge of whether they are composed of sparse or smooth features. MTGV regularization is shown to perform better than conventional processing techniques and $L1$ regularization over a range of simulated distributions.

A method for optimising sampling patterns for relaxation and diffusion experiments, based on the Cramér-Rao Lower Bound theory, is presented. The method is validated against pulsed field gradient NMR diffusion data of two experimental systems. In the first experimental system, the sampling pattern is optimised for the most accurate estimation of the lognormal distribution parameters of an emulsion droplet size distribution of toluene in water. In the second experimental system, the sampling pattern is optimised for the most accurate estimation of the bi-exponential model parameters of a binary mixture of methane/ethane adsorbed in a zeolite. The proposed method predicts an uncertainty in estimating the model parameters which is < 10% different from the uncertainty estimated from the experimental data sampled using the same sampling pattern.

Signal sampling and processing techniques are subsequently combined to reduce experimental acquisition times, which opens opportunities for studying unsteady systems over a long acquisition time and investigating fast-changing phenomena. A 32-fold decrease in the experimental acquisition time is achieved in extracting 3D spatially resolved spin spin relaxation maps. This is expected to be useful in investigating porous media systems. Three-component velocity maps on a 2D image, acquired every 4 ms, are used to capture, for the first time, the hydrodynamics of a bubble burst event. The experimental data are used to validate the predictions of numerical works.