2D sparse sampling algorithm for ND Fredholm equations with applications to NMR relaxometry

In [1], Cloninger, Czaja, Bai, and Basser developed an algorithm for compressive sampling based data acquisition for the solution of 2D Fredholm equations. We extend the algorithm to N dimensional data, by randomly sampling in 2 dimensions and fully sampling in the remaining N-2 dimensions. This new algorithm has direct applications to 3-dimensional nuclear magnetic resonance relaxometry and related experiments, such as T₁-D-T₂ or T₁-T_1,ρ-T₂. In these experiments, the first two parameters are time-consuming to acquire, so sparse sampling in the first two parameters can provide significant experimental time savings, while compressive sampling is unnecessary in the third parameter.