In the surveillance of interested regions by unmanned aerial vehicle (UAV), system performance relies greatly on the motion control strategy of the UAV and the operation characteristics of the onboard sensors. This paper investigates the 2D path planning problem for the lightweight UAV synthetic aperture radar (SAR) system in an environment of multiple regions of interest (ROIs), the sizes of which are comparable to the radar swath width. Taking into account the special requirements of the SAR system on the motion of the platform, we model path planning for UAV SAR as a constrained multiobjective optimization problem (MOP). Based on the fact that the UAV route can be designed in the map image, an image-based path planner is proposed in this paper. First, the neighboring ROIs are merged by the morphological operation. Then, the parts of routes for data collection of the ROIs can be located according to the geometric features of the ROIs and the observation geometry of UAV SAR. Lastly, the route segments for ROIs surveillance are connected by a path planning algorithm named the sampling-based sparse A* search (SSAS) algorithm. Simulation experiments in real scenarios demonstrate that the proposed sensor-oriented path planner can improve the reconnaissance performance of lightweight UAV SAR greatly compared with the conventional zigzag path planner.