The ridgelet transform (Candes and Donoho, 1999) was introduced as a new multiscale representation for functions on continuous spaces that are smooth away from discontinuities along lines. In this paper, we present several discrete versions of the ridgelet transform that lead to algorithmic implementations. The resulting transforms are invertible, non-redundant and computed via fast algorithms. Furthermore, this construction leads to a family of directional orthonormal bases for digital images. Numerical results show that the new transforms are more effective than the wavelet transform in approximating and denoising images with straight edges. Keywords: wavelets, ridgelets, Radon transform, directional bases, discrete transforms, non-linear approximation, image denoising.