In this work we describe a fast and stable algorithm for the computation of the orthogonal moments of an image. Indeed, orthogonal moments are characterized by a high discriminative power, but some of their possible formulations are characterized by a large computational complexity, which limits their real-time application. This paper describes in detail an approach based on recurrence relations, and proposes an optimized Matlab implementation of the corresponding computational procedure, aiming to solve the above limitations and put at the community's disposal an efficient and easy to use software. In our experiments we evaluate the effectiveness of the recurrence formulation, as well as its performance for the reconstruction task, in comparison to the closed form representation, often used in the literature. The results show a sensible reduction in the computational complexity, together with a greater accuracy in reconstruction. In order to assess and compare the accuracy of the computed moments in texture analysis, we perform classification experiments on six well-known databases of texture images. Again, the recurrence formulation performs better in classification than the closed form representation. More importantly, if computed from the GLCM of the image using the proposed stable procedure, the orthogonal moments outperform in some situations some of the most diffused state-of-the-art descriptors for texture classification.