Construction of Structured Incoherent Unit Norm Tight Frames

The exact recovery property of Basis pursuit (BP) and Orthogonal Matching Pursuit (OMP) has a relation with the coherence of the underlying frame. A frame with low coherence provides better guarantees for exact recovery. In particular, Incoherent Unit Norm Tight Frames (IUNTFs) play a significant role in sparse representations. IUNTFs with special structure, in particular those given by a union of several orthonormal bases, are known to satisfy better theoretical guarantees for recovering sparse signals. In the present work, we propose to construct structured IUNTFs consisting of large number of orthonormal bases. For a given $r, k, m$ with $k$ being less than or equal to the smallest prime power factor of $m$ and $r<k,$ we construct a CS matrix of size $mk \times (mk\times m^{r})$ with coherence at most $\frac{r}{k},$ which consists of $m^{r}$ number of orthonormal bases and with density $\frac{1}{m}$. We also present numerical results of recovery performance of union of orthonormal bases as against their Gaussian counterparts.