An ℓ1 based post-processing method with an application to ground penetrating radar imaging

In this paper, we present a method for postprocessing ground penetrating radar (GPR) images that were reconstructed using the well known delay-and-sum (DAS) algorithm. The method generates improved GPR images from DAS images by minimizing a cost function where a DAS image is viewed as the data and regularization is achieved through an ℓ₁ penalty function. We use the majorize-minimize principle to develop an algorithm, which we refer to as the ℓ₁ sparsity improvement restoration (ℓ₁-SIR) algorithm, with the property that it produces a decreasing sequence of cost function values. The ℓ₁-SIR algorithm is computationally efficient and only takes approximately 3% of the time required by the DAS algorithm. In studies using simulated data, the ℓ₁-SIR image was a significant improvement over the DAS image in that it had reduced clutter and improved sparsity without a loss of known scatterers. A recently developed fast ℓ₁ least-squares estimation (ℓ₁-LS) algorithm had comparable performance as the ℓ₁-SIR algorithm but requires approximately 100 times the run-time. It should be mentioned that the popular "shooting algorithm" could not be used in the simulation study. The reason is because the high dimensionality of the GPR image reconstruction problem required the shooting algorithm to have a computing platform with memory resources that exceeds what is currently available in "standard computers".