We develop a novel kind of single-pixel imaging (SPI) scheme to reconstruct high-quality images of the axially fast-moving target by utilizing the non-diffracting beam illumination. In this paper, a kind of cosine shaped source is elaborately designed and introduced for the first time in the SPI scheme to realize the reconstruction of the dynamic target. The imaging formula for the SPI system with the cosine shaped source is theoretically derived and numerical examples are given to see clearly the influences of system parameters and motion models of the target on the image quality. Our results show that the image quality and imaging efficiency of the SPI system for the static or dynamic target can be significantly improved by utilizing the cosine shaped source, due to its non-diffracting and orthogonal features. Even there exists strong scattering media, our SPI proposal can still work well for the static or dynamic target. Therefore, our work offers great promise for the SPI real application in forward-looking lidar detection.