Anti-collision assistance, integral to the current drive towards increased vehicular autonomy, relies heavily on precise detection and localization of moving targets in the vehicle's vicinity. A crucial step towards achieving this is the removal of static objects from the scene, thereby enhancing the detection and localization of dynamic targets - a pivotal aspect in augmenting overall system performance. In this paper, we propose a static background removal algorithm tailored for automotive scenarios, designed for common frequency-modulated continuous wave (FMCW) radars. This algorithm effectively eliminates reflections corresponding to static backgrounds from radar images through a two-step process: 4-dimensional (4D) radar imaging and filtering in the azimuth-elevation-Doppler domain. Our proposed approach is underpinned by a model customized for FMCW radar signals, incorporating a time-division multiplexing-based multiple-input multiple-output scheme on the non-uniform radar array. Furthermore, our filtering process requires knowledge of the 3-dimensional (3D) radar ego-motion velocity, typically obtained from an external sensor. To address scenarios where such sensors are unavailable, we introduce a self-contained 3D ego-motion estimation approach. Finally, we evaluate the performance of our algorithm using both simulated and real-world data, analyzing its sensitivity and time complexity in comparison to established baselines.