## FAST COMPRESSIVE SAMPLING WITH STRUCTURALLY RANDOM MATRICES

Citations: | 19 - 6 self |

### BibTeX

@MISC{Do_fastcompressive,

author = {Thong T. Do and Trac D. Tran and Lu Gan},

title = {FAST COMPRESSIVE SAMPLING WITH STRUCTURALLY RANDOM MATRICES},

year = {}

}

### OpenURL

### Abstract

This paper presents a novel framework of fast and efficient compressive sampling based on the new concept of structurally random matrices. The proposed framework provides four important features. (i) It is universal with a variety of sparse signals. (ii) The number of measurements required for exact reconstruction is nearly optimal. (iii) It has very low complexity and fast computation based on block processing and linear filtering. (iv) It is developed on the provable mathematical model from which we are able to quantify trade-offs among streaming capability, computation/memory requirement and quality of reconstruction. All currently existing methods only have at most three out of these four highly desired features. Simulation results with several interesting structurally random matrices under various practical settings are also presented to verify the validity of the theory as well as to illustrate the promising potential of the proposed framework. Index Terms — Fast compressive sampling, random projections, nonlinear reconstruction, structurally random matrices 1.

### Citations

1295 | Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information
- Candes, Romberg, et al.
- 2004
(Show Context)
Citation Context ...ial of the proposed framework. Index Terms— Fast compressive sampling, random projections, nonlinear reconstruction, structurally random matrices 1. INTRODUCTION In the compressive sampling framework =-=[1]-=-, if the signal is compressible, i.e., it has a sparse representation under some linear transformation, a small number of random projections of that signal contains sufficient information for exact re... |

738 |
Stable signal recovery from incomplete and inaccurate measurements
- Candes, Romberg, et al.
- 2006
(Show Context)
Citation Context ...l desirable features of these aforementioned methods while simultaneously eliminates or at least minimizes their significant drawbacks. A special case of our method was mentioned of its efficiency in =-=[5, 6]-=- (as the socalled Scrambled/Permuted FFT) but without an analysis of its performance. The remainder of the paper is organized as follow. Section 2 gives fundamental definitions and theoretical results... |

285 | Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems
- Figueiredo, Nowak, et al.
- 2007
(Show Context)
Citation Context ... are used as input signals of length 2 18 . The sparsifying operator Ψ is chosen as the popular Daubechies 9/7 wavelet transform. The l1 based linear programming solver is based on the GPSR algorithm =-=[10]-=-. Figure 1 and Figure 2 illustrate the PSNR results of reconstructed Lena and Boat images, respectively. Our main interest here is the performance of highly sparse sensing matrices. In this experiment... |

125 | Sparsity and incoherence in compressive sampling
- Candès, Romberg
(Show Context)
Citation Context ... they inherently have two major drawbacks in practical applications: huge memory buffering for storage of matrix elements and high computational complexity due to their completely unstructured nature =-=[3]-=-. The second family is partial Fourier [3] (or more generally, random rows of any orthonormal matrix). Partial Fourier exploits the fast computational property of Fast Fourier Transform (FFT) ∗ This w... |

63 | Random filters for compressive sampling and reconstruction - Tropp, Wakin, et al. - 2006 |

52 | Uniform uncertainty principle for Bernoulli and subgaussian ensembles
- Mendelson, Pajor, et al.
- 2009
(Show Context)
Citation Context ... the received measurements. The first family of sensing matrices for l1 based reconstruction algorithms consists of random Gaussian/Bernoulli matrices (or more generally, sub-Gaussian random matrices =-=[2]-=-). Their main advantage is that they are universally incoherent with any sparse signal and thus, the number of compressed measurements required for exact reconstruction is almost minimal. However, the... |

41 |
Fast reconstruction of piecewise smooth signals from random projections
- Duarte, Wakin, et al.
- 2005
(Show Context)
Citation Context ...l desirable features of these aforementioned methods while simultaneously eliminates or at least minimizes their significant drawbacks. A special case of our method was mentioned of its efficiency in =-=[5, 6]-=- (as the socalled Scrambled/Permuted FFT) but without an analysis of its performance. The remainder of the paper is organized as follow. Section 2 gives fundamental definitions and theoretical results... |

21 | Average performance analysis for thresholding
- Schnass, Vandergheynst
- 2007
(Show Context)
Citation Context ...erated by the local randomization model. Theorem 2.2.1: The coherence of Φ and Ψ is not larger than O( √ log n/s) with probability at least 1 − O(1/n). Theorem 2.2.2: The 2-Babel cumulative coherence =-=[8]-=- of Φ and a uniformly √ random set of k columns of Ψ is not larger than O( k/n + √ k log n3/2 /s) with probability at least 1 − O(1/n). In the case that the sensing matrix Φ is generated by a global r... |

11 |
Concentration-of-Measure Inequalities,” Lecture notes, www.econ.upf.edu/ lugosi/anu.ps
- Lugosi
- 2006
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Citation Context ...chnical tools are large deviation inequalities of sum of independent random variables. In particular, the Bernstein’s and Hoeffding’s concentration inequalities of sum of independent random variables =-=[9]-=- are used very frequently. Key arguments are as follows. (i) Of two models, the global randomization is harder to analyze due to its combinatorial nature. We approximate it by the following propositio... |

1 |
Fast compressive sampling using structurally random matrices
- Do, Tran, et al.
- 2007
(Show Context)
Citation Context ...d in Section 5. Due to lack of space, only heuristic arguments and proof sketches are provided. Detail proofs of these theorems and associated lemmas are provided in the journal version of this paper =-=[7]-=-.2. COHERENCE OF STRUCTURALLY RANDOM MATRICES 2.1. Basic Definitions Definition 2.1.1: Given a unit-length vector x ∈ R n and a random seed vector π ∈ R n , define a new random vector y as y = π(x). ... |