This paper presents a new algorithm for the computation of the two-dimensional discrete Fourier transform and discrete Hartley transform. By using the discrete Radon transform, the algorithm essentially converts the two-dimensional transforms into a number of one-dimensional ones. However, the present algorithm is improved as compared to the previous propositions in that about 20% of the additions are saved. It is achieved by a new decomposition technique for the computation of the discrete Radon transform. In fact, the present algorithm has exactly the same arithmetic complexity as the respective fastest algorithms which use the polynomial transform for their decomposition. However, the present approach has the advantage over the ones using the polynomial transform on the point that it can be easily realized.