Deterministic construction of unimodular tight frames consisting orthogonal blocks via block preserving operators

Abstract

A highly redundant incoherent unimodular tight frame (UMTF) finds desirability across various fields, including quantum communications and signal processing. Incorporating a larger number of orthogonal blocks into such a UMTF yields advantages, especially in domains like image compression and wireless communications. Despite their valuable utility, constructions of these frames are exceedingly rare. Moreover, in most of such existing UMTFs, the number of orthogonal blocks is in the order of the frame vectors dimension. This paper focuses on a deterministic construction of incoherent UMTFs consisting of a large number of orthogonal blocks using combinatorial design. Further, we illustrate that the computational complexity for constructing the stated frame is lower compared to the existing methods. Additionally, we show that our proposed construction yields comparable results in sparse and block sparse signal recovery compared to the existing methods and outperforms the partial discrete Fourier transform matrix. Furthermore, empirical findings highlight that images reconstructed using our construction exhibit higher structural similarity index measure values in comparison with other existing methods.