This article extends the Radon transform, a classical image-processing tool for fast tomography and denoising, to the quantum computing platform. A new kind of periodic discrete Radon transform (PDRT), called the quantum periodic discrete Radon transform (QPRT), is proposed. The quantum implementation of QPRT based on the amplitude encoding method is exponentially faster than the classical PDRT. We design an efficient quantum image denoising algorithm using QPRT. The simulation results show that QPRT preserves good denoising capability as in the classical PDRT. Also, a quantum algorithm for IDRT is proposed, which can be used for fast line detection. Both the quantum extension of IDRT and the line detection algorithm can provide polynomial speedups over the classical counterparts in certain cases.