Quantization of compressed sensing measurements is typically justified by the robust recovery results of Candès, Romberg and Tao, and of Donoho. These results guarantee that if a uniform quantizer of step size δ is used to quantize m measurements y = Φx of a k-sparse signal x ∈ R N , where Φ satisfies the restricted isometry property, then the approximate recoveryThe simplest and commonly assumed approach is to quantize each measurement independently. In this paper, we show that if instead an rth order Σ∆ quantization scheme with the same output alphabet is used to quantize y, then there is an alternative recovery method via Sobolev dual frames which guarantees a reduction of the approximation error by a factor of (m/k) (r−1/2)α for any 0 < α < 1, if m r k(log N ) 1/(1−α) . The result holds with high probability on the initial draw of the measurement matrix Φ from the Gaussian distribution, and uniformly for all k-sparse signals x that satisfy a mild size condition on their supports.

Abstract. A new class of alternative dual frames is introduced in the setting of finite frames for R d . These dual frames, called Sobolev duals, provide a high precision linear reconstruction procedure for Sigma-Delta (Σ∆) quantization of finite frames. The main result is summarized as follows: reconstruction with Sobolev duals enables stable rth order Sigma-Delta schemes to achieve deterministic approximation error of order O(N −r ) for a wide class of finite frames of size N . This asymptotic order is generally not achievable with canonical dual frames. Moreover, Sobolev dual reconstruction leads to minimal mean squared error under the classical white noise assumption.

Duality principles in Gabor theory such as the Ron-Shen duality principle and the Wexler-Raz biorthogonality relations play a fundamental role for analyzing Gabor systems. In this article we present a general approach to derive duality principles in abstract frame theory. For each sequence in a separable Hilbert space we define a corresponding sequence dependent only on two orthonormal bases. Then we characterize exactly properties of the first sequence in terms of the associated one, which yields duality relations for the abstract frame setting. In the last part we apply our results to Gabor systems.

Abstract. We study an intriguing question in frame theory we call Weaving Frames that is partially motivated by preprocessing of Gabor frames. Two frames {ϕ i } i∈I and {ψ i } i∈I for a Hilbert space H are woven if there are constants 0 < A B so that for every subset σ ⊂ I , the family {ϕ i } i∈σ ∪ {ψ i } i∈σ c is a frame for H with frame bounds A,B . Fundamental properties of woven frames are developed and key differences between weaving Riesz bases and weaving frames are considered. In particular, it is shown that a Riesz basis cannot be woven with a redundant frame. We also introduce an apparently weaker form of weaving but show that it is equivalent to weaving. Weaving frames has potential applications in wireless sensor networks that require distributed processing under different frames, as well as preprocessing of signals using Gabor frames.Mathematics subject classification (2010): 42C15.

Recent results make it clear that the compressed sensing paradigm can be used effectively for dimension reduction. On the other hand, the literature on quantization of compressed sensing measurements is relatively sparse, and mainly focuses on pulse-code-modulation (PCM) type schemes where each measurement is quantized independently using a uniform quantizer, say, of step size 8. The robust recovery result of Candes et ale and Donoho guarantees that in this case, under certain generic conditions on the measurement matrix such as the restricted isometry property, £1 recovery yields an approximation of the original sparse signal with an accuracy of 0 (8). In this paper, we propose sigma-delta quantization as a more effective alternative to PCM in the compressed sensing setting. We show that if we use an rth order sigma-delta scheme to quantize m compressed sensing measurements of a k-sparse signal in }RN, the reconstruction accuracy can be improved by a factor of (m/k)(r-1/2)a for any 0 < a < 1 if m~r k(log N)l/(l-a) (with high probability on the measurement matrix). This is achieved by employing an alternative recovery method via rth-order Sobolev dual frames.