Although the ridgelet transform is introduced as a new multiscale representation for functions on continuous spaces, discrete versions of the ridgelet transform that lead to algorithmic implementations remains to be solved. In this paper, approximate digital implementation is described by using a new method. As an important tool, Bresenham algorithm is used to offer exact reconstruction. Compared with the nearest-neighbor interpolation method, the new method has better performance such as stability against perturbations, low computation complexity and easy implementation. Compared with the ridgelet transform based on the Z\+2\-p method, the numerical results show that the new transform is more effective in reconstruction, compression and denoising images with straight edges, which lays a solid foundation for further research.