The time-invariant Radon transform (RT) is commonly used to regularize and interpolate sparsely sampled or irregularly acquired prestack seismic data. The sparseness of the Radon model significantly influences the results of regularization. We have developed an effective and efficient method for the regularization and interpolation of 2D as well as 3D prestack seismic data. We used an accelerated sparse time-invariant RT in the mixed frequency-time domain to improve the performance of RT-based seismic data regularization. This 2D sparse RT incorporated the iterative 2D model shrinkage algorithm instead of the traditional iteratively reweighted least-squares (IRLS) algorithm in the time domain, and we computed the forward and inverse RTs in the frequency domain to solve the sparse inverse problem, which dramatically reduced the computational cost while obtaining a high-resolution result. The 2D synthetic and real data examples revealed that our 2D approach can better interpolate beyond aliasing a 2D prestack seismic record that contains a large gap, compared with the least-squares-based RT and the frequency-domain sparse RT methods. To extend the 2D technique to 3D more efficiently, we first formulate the 3D RT as a problem of solving a special matrix equation. Next, we use the iterative 3D model shrinkage algorithm to obtain a high-resolution 3D Radon model. The proposed 3D sparse RT method can be applied in the regularization of 3D prestack gathers, such as in the cable interpolation in a 3D marine survey. We achieved robustness and effectiveness with our 3D approach with successful applications to 3D synthetic and real data.