Optimal translational-rotational invariant dictionaries for images

TL;DR

This paper introduces a method to construct square matrices whose translated and rotated versions form an optimal Parseval frame for image approximation, leveraging harmonic analysis and providing both theoretical proof and numerical validation.

Contribution

It presents a novel construction of invariant dictionaries for images that are optimal for approximation, with a self-contained proof and practical numerical results.

Findings

Constructed matrices form optimal invariant frames for image datasets.

The method achieves minimal quadratic approximation error.

Numerical experiments confirm effectiveness on natural images.

Abstract

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.