## MAKING DO WITH LESS: AN INTRODUCTION TO COMPRESSED SENSING

### BibTeX

@MISC{Bryan_makingdo,

author = {Kurt Bryan and Tanya Leise},

title = {MAKING DO WITH LESS: AN INTRODUCTION TO COMPRESSED SENSING},

year = {}

}

### OpenURL

### Abstract

Abstract. This article offers an accessible but rigorous and essentially self-contained account of the main ideas in compressed sensing (also known as compressive sensing or compressive sampling),

### Citations

1865 | Compressed sensing
- Donoho
- 2006
(Show Context)
Citation Context ...liably recover all of the masses. The field of CS emerged as a hot topic with the publication of two seminal papers in 2006 by Emmanuel Candès, Justin Romberg, and Terence Tao [9] and by David Donoho =-=[17]-=-. Traditional signal processing based on Shannon’s information theory focuses on uniform sampling, that is, systematically collecting data at evenly spaced points on a grid to achieve some desired res... |

1778 | Atomic decomposition by basis pursuit
- Chen, Donoho, et al.
- 1998
(Show Context)
Citation Context ...numerically exploring some examples, [8] provides helpful Matlab code. For background on linear programming see [21], and for development of basis pursuit as a means of efficient ℓ 1 minimization see =-=[13]-=-. Exercise 21. Why didn’t we assume Φ satisfies the RIP of order 2 in the simple 1-sparse solution scenario above? Redo the analysis under the assumption of the RIP of order 2, with |Tj| = 1, and show... |

1401 | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information
- Candès, Romberg, et al.
(Show Context)
Citation Context ...enough information to reliably recover all of the masses. The field of CS emerged as a hot topic with the publication of two seminal papers in 2006 by Emmanuel Candès, Justin Romberg, and Terence Tao =-=[9]-=- and by David Donoho [17]. Traditional signal processing based on Shannon’s information theory focuses on uniform sampling, that is, systematically collecting data at evenly spaced points on a grid to... |

888 | Near-optimal signal recovery from random projections: Universal encoding strategies
- Candès, Tao
(Show Context)
Citation Context ...2, 23, 25, 26, 27], as well as [22] which includes a historical overview. For a wonderful audio demonstration of CS, see the website [2]. More complete mathematical developments of CS can be found in =-=[11, 7, 3, 15]-=-, to mention a few examples from the rapidly expanding CS literature. A fairly comprehensive listing of CS-related articles and other materials can be found at http://dsp.rice.edu/cs. 1.3. The General... |

705 | Decoding by linear programming
- Candès, Tao
- 2005
(Show Context)
Citation Context ....2) for the rescaled system (Φ and b both rescaled by a factor 2/(c1 + c2)). Note that c1 > 0 excludes the possibility δ = 1. This motivates the following key definition, introduced by Candés and Tao =-=[10]-=-: Definition 2.4. An n × N matrix Φ satisfies the restricted isometry property (RIP) of order m if there is some constant δm ∈ (0, 1) such that 1 − δm ≤ ∥Φu∥2 ≤ 1 + δm (2.3)8 K. BRYAN AND T. LEISE fo... |

690 |
An introduction to compressive sampling
- Candes, Wakin
- 2008
(Show Context)
Citation Context ...cal CS devices is extremely challenging, requiring new sensing technology able to physically mimic the action of random measurement matrices. For overviews of CS accessible to a general audience, see =-=[12, 23, 25, 26, 27]-=-, as well as [22] which includes a historical overview. For a wonderful audio demonstration of CS, see the website [2]. More complete mathematical developments of CS can be found in [11, 7, 3, 15], to... |

362 | 2004) For Most Large Underdetermined Systems of Linear Equations the Minimal l1 norm Solution is also the sparsest Solution. URL : http://stat.stanford.edu/˜donoho/Reports/2004
- Donoho
(Show Context)
Citation Context ...(which simply counts the number of nonzero elements in x, and isn’t really a norm) subject to the condition Φx = b. Unfortunately, ℓ0 minimization turns out to be extremely difficult, in fact NP-hard =-=[18]-=-. Another possibility is to use the ℓ1 norm: ∥x∥1 = N∑ |xi|. (1.2) Minimizing ∥x∥1 subject to linear constraints like Φx = b may look difficult, since |xi| is not differentiable, but it turns out that... |

331 | The restricted isometry property and its implications for compressed - Candès - 2008 |

320 | A Simple Proof of the Restricted Isometry Property for Random Matrices
- Baraniuk, Davenport, et al.
- 2007
(Show Context)
Citation Context ...2, 23, 25, 26, 27], as well as [22] which includes a historical overview. For a wonderful audio demonstration of CS, see the website [2]. More complete mathematical developments of CS can be found in =-=[11, 7, 3, 15]-=-, to mention a few examples from the rapidly expanding CS literature. A fairly comprehensive listing of CS-related articles and other materials can be found at http://dsp.rice.edu/cs. 1.3. The General... |

244 |
Sparse MRI: The application of compressed sensing for rapid
- Lustig, Donoho, et al.
- 2007
(Show Context)
Citation Context ...lysis, compressive radar, and genetic screening, among others. For example, CS can greatly20 K. BRYAN AND T. LEISE reduce scan times and potentially increase resolution of magnetic resonance imaging =-=[30]-=-. CS microarrays combine group testing and CS principles to accurately identify genetic sequences, e.g, to detect pathogens in a water sample [19]. CS has also been used to image the rupture process o... |

175 |
Uncertainty principles and signal recovery
- Donoho, Stark
(Show Context)
Citation Context ... analysis, the classic uncertainty principle says that if a signal’s support is concentrated on a small set, then its Fourier transform must be “spread out” over a relatively large set of frequencies =-=[19]-=-. In the CS context, we have a similar paradigm in that we want the sensing matrix Φ to have properties such that the measurement vector b = Φx of a sparse signal x has broad support. The RIP gives us... |

158 | Compressed sensing and best k-term approximation
- Cohen, Dahmen, et al.
(Show Context)
Citation Context ...2, 23, 25, 26, 27], as well as [22] which includes a historical overview. For a wonderful audio demonstration of CS, see the website [2]. More complete mathematical developments of CS can be found in =-=[11, 7, 3, 15]-=-, to mention a few examples from the rapidly expanding CS literature. A fairly comprehensive listing of CS-related articles and other materials can be found at http://dsp.rice.edu/cs. 1.3. The General... |

155 |
Database-friendly random projections: Johnson-Lindenstrauss with binary coins
- Achlioptas
(Show Context)
Citation Context ... the estimate in the lemma (noting that e nϵ3 /6 > 1 and 2 + nϵ > 2).□ For a proof of an identical result where Bernoulli random variables are used to form Φ (like we did for the marble problem), see =-=[1]-=-. Exercise 16. Let u be a unit vector in RN and Φ an n×N matrix whose entries are sampled from independent normal random variables with mean 0 and variance 1/n. Prove that the expected value of ∥Φu∥2 ... |

154 | Single-pixel imaging via compressive sampling
- Duarte, Davenport, et al.
- 2008
(Show Context)
Citation Context ...al megapixel digital camera, a CS-designed single-pixel camera takes relatively few measurements in a manner that directly gathers compressed information equivalent to sums of randomly located pixels =-=[20]-=-, analogous to our example of using a balance to measure the total mass of randomly selected subsets of marbles. The key mathematical ideas in CS, as we will show, can be easily understood and form an... |

126 | Quantitative robust uncertainty principles and optimally sparse decompositions - Candès, Romberg |

92 | Deterministic Constructions of Compressed Sensing Matrices
- DeVore
- 2007
(Show Context)
Citation Context ...inary encoding strategy we used earlier will work. How should we design the sensing matrix? Structured methods for choosing subsets to test in order to detect multiple deviations have been developed (=-=[3, 20]-=-), but we’ll take a different tack: we’ll choose the subsets randomly! In the present example we’ll create a sensing matrix Φ by selecting n = 20 random subsets of coins to weigh. Specifically, let Φ ... |

84 | Introduction to the non-asymptotic analysis of random matrices
- Vershynin
(Show Context)
Citation Context ...ecified in the lemma.□ COMPRESSED SENSING 15 The following lemma tells us how to obtain a specific suitable subset AQ ⊂ UT for our purposes, through a bound for the covering number of the unit sphere =-=[28]-=-: Lemma 3.4. For each ϵ ∈ (0, 1) and positive integer m ≥ 2, there exists a set AQ(ϵ) with at most Q(ϵ) := (3/ϵ) m points on the unit sphere S m−1 = {u ∈ R m : ∥u∥2 = 1} that satisfies the following p... |

50 | Imaging via Compressive Sampling
- Romberg
- 2008
(Show Context)
Citation Context ...cal CS devices is extremely challenging, requiring new sensing technology able to physically mimic the action of random measurement matrices. For overviews of CS accessible to a general audience, see =-=[12, 23, 25, 26, 27]-=-, as well as [22] which includes a historical overview. For a wonderful audio demonstration of CS, see the website [2]. More complete mathematical developments of CS can be found in [11, 7, 3, 15], to... |

34 |
Sparsity and Incoherence
- Candès, Romberg
- 2007
(Show Context)
Citation Context |

34 | Sparse recovery using sparse matrices
- Gilbert, Indyk
(Show Context)
Citation Context .... 1.1. Left: Actual deviations xi in mass. Center: Result of ℓ 2 -regularization. Right: Result of ℓ 1 -regularization. on a biological target with the goal of identifying a few active compounds (see =-=[27]-=- for more on this and other applications). To illustrate, let us consider the counterfeit coin problem on a somewhat larger scale. Suppose we have N = 100 coins, numbered from 1 to 100, of which a sma... |

24 | Compressive sensing
- Fornasier, Rauhut
- 2011
(Show Context)
Citation Context ...enging, requiring new sensing technology able to physically mimic the action of random measurement matrices. For overviews of CS accessible to a general audience, see [12, 23, 25, 26, 27], as well as =-=[22]-=- which includes a historical overview. For a wonderful audio demonstration of CS, see the website [2]. More complete mathematical developments of CS can be found in [11, 7, 3, 15], to mention a few ex... |

17 | A negative result concerning explicit matrices with the restricted isometry property. preprint
- Chandar
- 2008
(Show Context)
Citation Context ...us to Theorems 3.1 and 4.1 also hold; see [1]. We should point out, however, that for 0 − 1 Bernoulli matrices like those used in the coin problem the RIP estimates are not quite as good, as shown in =-=[15]-=- and discussed in [6]. Exercise 23. Why didn’t we assume Φ satisfies the RIP of order 2 in the simple 1-sparse solution scenario above? Redo the analysis under the assumption of the RIP of order 2, wi... |

5 | A simple proof that random matrices are democratic
- Davenport, Laska, et al.
- 2009
(Show Context)
Citation Context ...e recovery step (the ℓ 1 minimization), rather than at the sensing step (weighing the marbles). In addition, CS is “democratic” and robust in the sense that all measurements are equally (un)important =-=[16]-=-. To illustrate, suppose we expect around 3 defective4 K. BRYAN AND T. LEISE marbles but aren’t sure of the exact number, so we sequentially put all 100 marbles on the balance. If we forget to write ... |

4 |
Compressed sensing makes every pixel count. What’s Happening
- Mackenzie
(Show Context)
Citation Context ...cal CS devices is extremely challenging, requiring new sensing technology able to physically mimic the action of random measurement matrices. For overviews of CS accessible to a general audience, see =-=[12, 23, 25, 26, 27]-=-, as well as [22] which includes a historical overview. For a wonderful audio demonstration of CS, see the website [2]. More complete mathematical developments of CS can be found in [11, 7, 3, 15], to... |

4 | Practical compressed sensing: Modern data acquisition and signal processing” Ph.D. dissertation, California Inst
- Becker
- 2011
(Show Context)
Citation Context ... 4.1 also hold; see [1]. We should point out, however, that for 0 − 1 Bernoulli matrices like those used in the coin problem the RIP estimates are not quite as good, as shown in [15] and discussed in =-=[6]-=-. Exercise 23. Why didn’t we assume Φ satisfies the RIP of order 2 in the simple 1-sparse solution scenario above? Redo the analysis under the assumption of the RIP of order 2, with |Tj| = 1, and show... |

3 |
Inequalities for the Gamma function
- Batir
- 2008
(Show Context)
Citation Context ...It follows that for α ≥ 1/ϵ we have ∫ 1−ϵ x 0 α−1 e −αx dx ≤ (1 − ϵ)M = (1 − ϵ) α e −α(1−ϵ) . (3.6) From (3.4) and by using (3.6) with α = n/2 we then have, for n ≥ 2/ϵ, ∫ 1−ϵ Applying the bound (see =-=[5]-=-) 0 fn(x) dx ≤ (n/2)n/2 Γ(n/2) e−n(1−ϵ)/2 (1 − ϵ) n/2 . (3.7) α α Γ(α) ≤ eα√ α √ 2π (3.8) with α = n/2 ≥ 1 in (3.7) yields ∫ 1−ϵ 0 fn(x) dx ≤ √ n 2 √ π enϵ/2 (1 − ϵ) n/2 . (3.9) From Exercise 20(d) we... |

2 |
Magic reconstruction: Compressed sensing
- Moler
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Citation Context |

1 |
Compressed sensing audio demonstration, http://sunbeam.ece.wisc.edu/csaudio/ (accessed 5/21/11
- Balzano, Nowak, et al.
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Citation Context ...rices. For overviews of CS accessible to a general audience, see [12, 23, 25, 26, 27], as well as [22] which includes a historical overview. For a wonderful audio demonstration of CS, see the website =-=[2]-=-. More complete mathematical developments of CS can be found in [11, 7, 3, 15], to mention a few examples from the rapidly expanding CS literature. A fairly comprehensive listing of CS-related article... |

1 |
Elementary Probability Theory,” 4th ed
- Chung, Aitsahlia
- 2003
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Citation Context ...8) holds. Thus (3.19) also holds with probability greater than 1 − p(n, δ), or equivalently, fails to hold with probability less than p(n, δ). Recall Boole’s inequality, also called the “union bound” =-=[14]-=-: P (E1 ∪ E2 ∪ · · · ∪ EQ) ≤ P (E1) + · · · + P (EQ), (3.20) where the Ek are any events, which need not be independent. Let Ek denote the event that the inequality (3.19) fails to hold for the point ... |

1 |
The best bits, American Scientist 97
- Hayes
- 2009
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Citation Context |

1 |
Deterministic Construction of Binary
- Amini, Marvasti
(Show Context)
Citation Context ...inary encoding strategy we used earlier will work. How should we design the sensing matrix? Structured methods for choosing subsets to test in order to detect multiple deviations have been developed (=-=[3, 20]-=-), but we’ll take a different tack: we’ll choose the subsets randomly! In the present example we’ll create a sensing matrix Φ by selecting n = 20 random subsets of coins to weigh. Specifically, let Φ ... |

1 |
Mecklenbrauker,Compressive sensing of the Tohoku-Oki Mw 9.0 earthquake: Frequency-dependent rupture modes
- Yao, Gerstoft, et al.
(Show Context)
Citation Context ...esting and CS principles to accurately identify genetic sequences, e.g, to detect pathogens in a water sample [19]. CS has also been used to image the rupture process of a main shock in an earthquake =-=[35]-=-. Many other applications are being developed (see http://dsp.rice.edu/cs). All in all, CS is proving to be a powerful and flexible paradigm and will continue to be an exciting field of mathematical r... |