Geometrically Uniform Frames

TL;DR

This paper introduces geometrically uniform (GU) and compound GU (CGU) frames with symmetry properties, providing efficient computation methods, analysis of element removal, and conditions for optimal distance spectra in frame design.

Contribution

The paper defines GU and CGU frames, explores their properties, and develops methods for their construction and analysis, including efficient dual frame computation and distance spectrum conditions.

Findings

Dual and canonical tight frames are GU or CGU, enabling efficient computation.

Methods for constructing optimal GU frames from non-GU sets are provided.

Conditions for GU frames to have strictly positive distance spectra are derived.

Abstract

We introduce a new class of frames with strong symmetry properties called geometrically uniform frames (GU), that are defined over an abelian group of unitary matrices and are generated by a single generating vector. The notion of GU frames is then extended to compound GU (CGU) frames which are generated by an abelian group of unitary matrices using multiple generating vectors. The dual frame vectors and canonical tight frame vectors associated with GU frames are shown to be GU and therefore generated by a single generating vector, which can be computed very efficiently using a Fourier transform defined over the generating group of the frame. Similarly, the dual frame vectors and canonical tight frame vectors associated with CGU frames are shown to be CGU. The impact of removing single or multiple elements from a GU frame is considered. A systematic method for constructing optimal GU frames from a given set of frame vectors that are not GU is also developed. Finally, the Euclidean distance properties of GU frames are discussed and conditions are derived on the abelian group of unitary matrices to yield GU frames with strictly positive distance spectrum irrespective of the generating vector.