Compressive Sensing by Random Convolution

By Justin K. Romberg. SIAM Journal of Imaging Sciences, 2009

455 citations recorded by the source corpus.

Subfields: Biomedical Engineering, Computational Mechanics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/0563e12dbf715d529698e9fb7abbbc056babdfa2

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In Computational Mechanics

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Citations in the corpus

Robust 1-Bit Compressive Sensing via Binary Stable Embeddings of Sparse Vectors (2011, methodology): “Instead, we solve Basis Pursuit (BP) by relaxing the objective in (RCS) to the `1-norm; the result is a convex, polynomial-time algorithm [6].”
A Probabilistic and RIPless Theory of Compressed Sensing (2010, background): “In fact, certain random convolutions may be used universally to efficiently sample a signal that is sparsely represented in any orthonormal basis while still giving strong theoretical guarantees for signal recovery [43]—an appealing feature due to the fast application of a convolution matrix.”
Structured Compressed Sensing: From Theory to Applications (2011, methodology): “In fact, such an FFT-based procedure can also be exploited to generate good CS matrices [92].”
Fast and Efficient Compressive Sensing Using Structurally Random Matrices (2011, methodology): “From the best of our knowledge, the proposed sensing algorithm is distinct from currently existing methods such as random projection [16], random filters [17], structured Toe plitz [18] and random convolution [19] via the first step of prerandomization.”
Toeplitz Compressed Sensing Matrices With Applications to Sparse Channel Estimation (2010, background): “A related effort, [37], established guarantees for the recovery of sparse signals using observations that arise from circular convolution with a specially constructed probe sequence (designed to have exactly orthogonal circular shifts) followed by random downsampling or randomized “block averaging” ”
Restricted Eigenvalue Properties for Correlated Gaussian Designs (2010, background): “It has also been demonstrated that RIP is satisfied for matrices from unitary ensembles (e.g., Guédon et al., 2007, 2008; Romberg, 2009; Rudelson and Vershynin, 2008), for which the rows are generated based on independent draws from a set of uncorrelated basis functions.”
A Systematic Review of Compressive Sensing: Concepts, Implementations and Applications (2018, methodology): “[23] J. K. Romberg, ‘‘Compressive sensing by random convolution,’’ SIAM J. Imag.”
New and Improved Johnson-Lindenstrauss Embeddings via the Restricted Isometry Property (2010, unknown): “Other classes of structured random matrices known to have the RIP include partial circulant matrices [34, 30, 32].”
Accelerating magnetic resonance imaging via deep learning (2016, methodology): “The zero-filled MR image z is generated as the direct inverse transform of the observed data like z = FPFu (2) As stated in [19], in terms of linear algebra, the circular convolution of a signal u with a pulse p can be written as FPFu, where P is a diagonal matrix whose non-zero entries are the Four”
Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals (2009, methodology): “In fact, if we replace the integrator with an ideal lowpass filter, so that we are in essence taking consecutive samples of the Fourier transform of the demodulated signal, the architecture would be very similar to that analyzed in [40], with the roles of time and frequency reversed.”
Signal Processing With Compressive Measurements (2010, unknown): “We start with the simple case where we desire a -stable embedding of , where and are ﬁnite sets of points in .”