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Unformatted text preview: Exact Signal Recovery from Sparsely Corrupted Measurements through the Pursuit of Justice Jason N. Laska, Mark A. Davenport, Richard G. Baraniuk Department of Electrical and Computer Engineering Rice University Houston, Texas, 77005 Abstract —Compressive sensing provides a framework for re covering sparse signals of length N from M N measurements. If the measurements contain noise bounded by , then standard algorithms recover sparse signals with error at most C . However, these algorithms perform suboptimally when the measurement noise is also sparse. This can occur in practice due to shot noise, malfunctioning hardware, transmission errors, or narrowband interference. We demonstrate that a simple algorithm, which we dub Justice Pursuit (JP), can achieve exact recovery from measurements corrupted with sparse noise. The algorithm han dles unbounded errors, has no input parameters, and is easily implemented via standard recovery techniques. I. INTRODUCTION The recently developed compressive sensing (CS) frame work enables acquisition of a signal x ∈ R N from a small set of M nonadaptive, linear measurements [1, 2]. This process can be represented as y = Φ x (1) where Φ is an M × N matrix that models the measurement system. The hope is that we can design Φ so that x can be accurately recovered even when M N . While this is not possible in general, when x is Ksparse, meaning that it has only K nonzero entries, it is possible to exactly recover x using Φ with M = O ( K log( N/K )) . Signal reconstruction can be performed using optimization techniques or greedy algorithms. The broad applicability of this framework has inspired research that extends the CS framework by proposing practical implementations for numerous applications, includ ing subNyquist sampling systems [3–5], compressive imaging architectures [6–8], and compressive sensor networks [9]. In practical settings, there may be many sources of noise, including noise present in the signal x , noise caused by the measurement hardware, quantization noise, and transmission errors in the case where the measurements are sent over a noisy channel. Thus, it is typically more realistic to represent the measurement process as y = Φ x + e , (2) This work was supported by the grants NSF CCF0431150, CCF0728867, CNS0435425, CNS0520280, and CCF0926127, DARPA/ONR N66001 0812065, ONR N000140710936, N000140811067, N000140811112, and N000140811066, AFOSR FA95500710301, ARO MURIs W311NF 0710185 and W911NF0910383, and the Texas Instruments Leadership University Program. Email: { laska, md, richb } @rice.edu. Web: dsp.rice.edu. where e is an M × 1 vector that represents noise in the measurements. It has been shown that it is possible to recon struct the signal with ‘ 2error that is at most C k e k 2 , where C > 1 is a small constant that depends on certain properties of Φ [10, 11]. Thus, CS systems are stable in the sense that if the measurement error is bounded, then the reconstruction...
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 Spring '10
 RunyiYu

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