Optimal translational-rotational invariant dictionaries for images

We provide the construction of a set of square matrices whose translates and rotates provide a Parseval frame that is optimal for approximating a given dataset of images. Our approach is based on abstract harmonic analysis techniques. Optimality is considered with respect to the quadratic error of approximation of the images in the dataset with their projection onto a linear subspace that is invariant under translations and rotations. In addition, we provide an elementary and fully self-contained proof of optimality, and the numerical results from datasets of natural images.