Stochastic Systems Group
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Sparsity has been employed in a variety of signal and image processing applications, such as denoising, deconvolution, inpainting, and compression. The new acquisition paradigm of compressive sensing (CS) leverages signal sparsity for recovery from a small set of random measurements. The standard CS theory dictates that robust recovery of a K-sparse, N-length signal is possible from M=O(K log(N/K)) measurements. We show that it is possible to substantially decrease M without sacrificing robustness by leveraging more concise signal models that go beyond simple sparsity and compressibility. We review two frameworks to represent the additional structure. First, we present a model-based CS theory that exploits the dependencies between values and locations of the significant signal coefficients; we also provide concrete guidelines on how to create model-based recovery algorithms with provable performance guarantees that require only M=O(K) measurements. Second, we review manifold-based CS for applications where the signal acquired is governed by a small set of K parameters. Interestingly, we show that the information in an N-dimensional signal from the manifold can be recovered from M=O(K log N) measurements. We derive compressive estimation and classification algorithms that leverage the large amount of structure present in manifold models.

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