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We study recovery conditions of weighted 1 minimization for signal reconstruction from compressed sensing measurements when partial support information is available. We show that if at least 50% of the (partial) support information is accurate, then weighted 1 minimization is stable and robust under weaker sufficient conditions than the analogous conditions for standard 1 minimization. Moreover, weighted 1 minimization provides better upper bounds on the reconstruction error in terms of the measurement noise and the compressibility of the signal to be recovered. We illustrate our results with extensive numerical experiments on synthetic data and real audio and video signals. Index TermsCompressed sensing, weighted 1 minimization, adaptive recovery. I. INTRODUCTIONCompressed sensing (see, e.g., [1]-[3]) is a paradigm for effective acquisition of signals that admit sparse (or approximately sparse) representations in some transform domain. The approach can be used to reliably recover such signals from significantly fewer linear measurements than their ambient dimension.Because a wide range of natural and man-made signals-e.g., audio, natural and seismic images, video, and wideband radio frequency signals-are sparse or approximately sparse in appropriate transform domains, the potential applications of compressed sensing can be immense.

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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.

Summary Many critical perceptual judgments, from telling whether fruit is ripe, to determining whether the ground is slippery, involve estimating the material properties of surfaces. Very little is known about how the brain recognizes materials, even though the problem is likely as important for survival as navigating or recognizing objects. Though previous research has focused nearly exclusively on the properties of static images [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,15,16], recent evidence suggests that motion may affect the appearance of surface material [17, 18, 19]. However, what kind of information motion conveys and how this information may be used by the brain is still unknown. Here, we have identified three motion cues that the brain could rely on to distinguish between matte and shiny surfaces. We show that these motion measurements can override static cues, leading to dramatic changes in perceived material depending on the image motion characteristics. A classifier algorithm based on these cues correctly predicts both successes and some striking failures of human material perception. Together these results reveal a previously unknown use for optic flow in the perception of surface material properties.

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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.