Least-squares reverse time migration (RTM) has become the method of choice for quantitative seismic imaging. The main drawback of such a scheme is that it requires many migration/modeling cycles. The convergence of least-squares RTM can be accelerated by using a suitable preconditioner. In the context of an extended domain in variable-density acoustic media, the pseudoinverse Born operator is the recommended preconditioner, providing quantitative results within a single iteration. This method consists of two steps: application of the pseudoinverse Born operator and inversion of two parameters using an efficient weighted least-squares approach based on the Radon transform. As expected, crosstalk artifacts are generated in the second step due to limited acquisition. We have developed a variable-density pseudoinverse Born operator constrained with the [Formula: see text]-norm for each model parameter to suppress the artifacts. The fast iterative shrinkage-thresholding algorithm is used to carry out the optimization problem. In classic iterative least-squares migration, the [Formula: see text]-norm constraints would affect the whole imaging process. Because the imaging method is split into two steps, only the Radon transform part is modified, where no wave-based operators are involved. Through numerical experiments, we verify the robustness of our method against different migration artifacts including the parameter crosstalk, interfaces with abrupt truncations, sparse shot acquisition geometry, noisy data, and high-contrast complex structures.