A greedy variational approach for generating sparse T₁-T₂ NMR relaxation time distributions

We present a nonlinear inversion method for generating sparse solutions to the Fredholm Integral equation describing two-dimensional distributions of nuclear magnetic resonance (NMR) relaxation times or diffusion coefficients. Our greedy variational method approximates the distribution of exponential rate constants using a sum of Dirac delta functions, which constitute our dictionary elements. The greedy nature of the method promotes sparsity in the representation by iteratively increasing the number of terms. The variational component estimates the parameters of the Dirac delta functions from a continuum at each iteration by reducing the least squares misfit to the data. Unlike sparsity promoting linearized inversion methods, where the dictionary is fixed and can exponentially grow in the case of multiple variables or when searching for higher resolution, the greedy component of our method aims to keep the dictionary small while the variational component keeps the dictionary dynamic. We demonstrate our method with synthetic data and experimental measurements of T₁-T₂ correlations of liquid-saturated porous rocks. The sparsity of the approximate solutions is ideal for real-time processing and transmission in remote or mobile NMR applications such as well logging.