The beneficial properties of the Radon transform make it an useful intermediate representation for the extraction of invariant features from pattern images for the purpose of indexing/matching. This paper revisits the problem of Radon image utilization with a generic view on a popular Radon transform-based transform and pattern descriptor, the R-transform and R-signature, bringing in a class of transforms and descriptors spatially describing patterns at all directions and at different levels, while maintaining the beneficial properties of the conventional R-transform and R-signature. The domain of this class, which is delimited due to the existence of singularities and the effect of sampling/quantization and additive noise, is examined. Moreover, the ability of the generic R-transform to encode the dominant directions of pattern is also discussed, adding to the robustness to additive noise of the generic R-signature. The stability of dominant direction encoding by the generic R-transform and the superiority of the generic R-signature over existing invariant pattern descriptors on grayscale and binary noisy datasets have been confirmed by experiments.