Construction of highly redundant incoherent unit norm tight frames as a union of orthonormal bases

Sasmal, Pradip and Jampana, Phanindra Varma and Sastry, Challa Subrahmanya (2019) Construction of highly redundant incoherent unit norm tight frames as a union of orthonormal bases. Journal of Complexity. ISSN 0885-064X (In Press)

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Abstract

Unit norm tight frames are useful in providing stable reconstruction. Incoherence, restricted isometry property and statistical restricted isometry property of a frame provide performance guarantees of sparse signal recovery algorithms in compressed sensing. Equiangular tight frames, mutually unbiased bases and chirp matrices are well known incoherent unit norm tight frames that exhibit near optimal sparse recovery guarantees. The dimension of frame elements belongs to some particular family of numbers and the number of elements in these frames is in the order of square of the dimension of the underlying space. In the present work, we construct incoherent unit norm tight frames for more general dimensions with higher redundancy than these frames. One of the important features of our construction is that it produces incoherent unit norm tight frames that are a union of orthonormal bases. In addition, the incoherent unit norm tight frames that we construct possess the statistical restricted isometry property and statistical incoherence property. As a result they exhibit near optimal theoretical guarantees in recovering sparse signals obtained from a generic random signal model. Using simulations, we demonstrate the efficacy of the constructed tight frames as good candidates for recovering sparse signals as against partial Fourier matrices and with existing constructions of incoherent unit norm tight frames.

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