A novel technique for Radon single-pixel imaging with projective sampling, which is based on the theorem of the Radon transform, is proposed. In contrast to current patterns in conventional single-pixel imaging systems, candy-striped patterns called Radon basis patterns, which are produced by projecting the 1D Hadamard functions along different angles, are employed in our proposed technique. Here, the patterns are loaded into a projection system and then illuminated onto an object. The light reflected from the object is detected by a single-pixel detector. An iterative reconstruction method is used to restore the object's 1D projection functions by summing the 1D Hadamard functions and detected intensities. Next, the Radon spectrum of the object is recovered by arranging the 1D projection functions along the projection angle. Finally, the image of the object can be recovered using a filtered back-projection algorithm with the Radon spectrum. Experiments demonstrate that the proposed technique can obtain the information of the Radon spectrum and image of the object. Recognition directly in the Radon spectrum domain, rather than in the image domain, is fast and yields robust and high classification rates. A recognition experiment is performed by detecting the lines in one scene by searching the singular peaks in the Radon spectrum domain. According to the results, the lines in the scene can be easily detected in the Radon spectrum domain. Other shapes can also be detected by the characteristics of those shapes in the Radon spectrum domain.