An estimation method known as least absolute shrinkage and selection operator (LASSO) or ℓ1-regularized LS estimation has been found to perform well in a number of applications. In this paper, we use the majorize-minimize method to develop an algorithm for minimizing the LASSO objective function, which is the sum of a linear LS objective function plus an ℓ1 penalty term. The proposed algorithm, which we call the LASSO estimation via majorization-minimization (LMM) algorithm, is straightforward to implement, parallelizable, and guaranteed to produce LASSO objective function values that monotonically decrease. In addition, we formulate an extension of the LMM algorithm for reconstructing ground penetrating radar (GPR) images, that is much faster than the standard LMM algorithm and utilizes significantly less memory. Thus, the GPR specific LMM (GPR-LMM) algorithm is able to accommodate the big data associated with GPR imaging. We compare our proposed algorithms to the state-of-the-art ℓ1-regularized LS algorithms using a time and space complexity analysis. The GPR-LMM greatly outperforms the competing algorithms in terms of the performance metrics we considered. In addition, the reconstruction results of the standard LMM and GPR-LMM algorithms are evaluated using both simulated and real GPR data.