This paper demonstrates that if the restricted isometry constant δK+1 of the measurement matrix A satisfies δK+1 < 1 √ K + 1 , then a greedy algorithm called Orthogonal Matching Pursuit (OMP) can recover every K-sparse signal x in K iterations from Ax. By contrast, a matrix is also constructed with the restricted isometry constant δK+1 = 1 √ K such that OMP can not recover some K-sparse signal x in K iterations. This result positively verifies the conjecture given by Dai and Milenkovic in 2009. Index Terms-compressed sensing, restricted isometry property, orthogonal matching pursuit, sparse signal reconstruction.

Abstract-Support recovery of sparse signals from noisy measurements with orthogonal matching pursuit (OMP) has been extensively studied. In this paper, we show that for any Ksparse signal x, if a sensing matrix A satisfies the restricted isometry property (RIP) with restricted isometry constant (RIC) δK+1 < 1/ √ K + 1, then under some constraints on the minimum magnitude of nonzero elements of x, OMP exactly recovers the support of x from its measurements y = Ax + v in K iterations, where v is a noise vector that is ℓ2 or ℓ∞ bounded. This sufficient condition is sharp in terms of δK+1 since for any given positive integer K and any 1/ √ K + 1 ≤ δ < 1, there always exists a matrix A satisfying the RIP with δK+1 = δ for which OMP fails to recover a K-sparse signal x in K iterations. Also, our constraints on the minimum magnitude of nonzero elements of x are weaker than existing ones. Moreover, we propose worst-case necessary conditions for the exact support recovery of x, characterized by the minimum magnitude of the nonzero elements of x.Index Terms-Compressed sensing (CS), restricted isometry property (RIP), restricted isometry constant (RIC), orthogonal matching pursuit (OMP), support recovery.

Let M be a 2 × 2 matrix of Laurent polynomials with real coefficients and symmetry. In this paper, we obtain a necessary and sufficient condition for the existence of four Laurent polynomials (or FIR filters) u 1 , u 2 , v 1 , v 2 with real coefficients and symmetry such that u 1 (z) v 1 (z) u 2 (z) v 2 (z) u 1 (1/z) u 2 (1/z) v 1 (1/z) v 2 (1/z) = M (z) ∀ z ∈ C\{0} and [Su 1 ](z)[Sv 2 ](z) = [Su 2 ](z)[Sv 1 ](z), where [Sp](z) = p(z)/p(1/z) for a nonzero Laurent polynomial p. Our criterion can be easily checked and a step-by-step algorithm will be given to construct the symmetric filters u 1 , u 2 , v 1 , v 2. As an application of this result to symmetric framelet filter banks, we present a necessary and sufficient condition for the construction of a symmetric MRA tight wavelet frame with two compactly supported generators derived from a given symmetric refinable function. Once such a necessary and sufficient condition is satisfied, an algorithm will be used to construct a symmetric framelet filter bank with two high-pass filters which is of interest in applications such as signal denoising and image processing. As an illustration of our results and algorithms in this paper, we give several examples of symmetric framelet filter banks with two high-pass filters which have good vanishing moments and are derived from various symmetric low-pass filters including some B-spline filters.

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Support recovery of sparse signals from noisy measurements with orthogonal matching pursuit (OMP) has been extensively studied in the literature. In this paper, we show that for any K-sparse signal x, if the sensing matrix A satisfies the restricted isometry property (RIP) of order K + 1 with restricted isometry constant (RIC) δK+1 < 1/ √ K + 1, then under some constraint on the minimum magnitude of the nonzero elements of x, the OMP algorithm exactly recovers the support of x from the measurements y = Ax + v in K iterations, where v is the noise vector. This condition is sharp in terms of δK+1 since for any given positive integer K ≥ 2 and any 1/ √ K + 1 ≤ t < 1, there always exist a K-sparse x and a matrix A satisfying δK+1 = t for which OMP may fail to recover the signal x in K iterations. Moreover, the constraint on the minimum magnitude of the nonzero elements of x is weaker than existing results.Index Terms-Compressed sensing (CS), restricted isometry property (RIP), orthogonal matching pursuit (OMP), support recovery. J. Wen is with1 √ K+1 , then OMP exactly recovers the K-sparse signal x in K iterations. On the other hand, it was conjectured in [19] that there exist a matrix A with δ K+1 ≤ 1 √ K and a K-sparse x such that OMP fails to recover x in K iterations. Examples provided in [15], [16] confirmed this conjecture. Later, the example in [20] showed that for any given positive integer K ≥ 2 and for any given t satisfying 1 √ K+1 ≤ t < 1, there always exist a K-sparse x and a matrix A satisfies the RIP of order K + 1 with δ K+1 = t such that