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## SPIKE DETECTION FROM INACCURATE SAMPLINGS

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4203 | Regression shrinkage and selection via the lasso
- Tibshirani
- 1996
(Show Context)
Citation Context ... essentially the ℓ 1-norm of the ℓ2-norms of the finite differences at any point. In particular, our model has nothing to do with the Rudin-Osher-Fatemi (ROF) model [ROF92]. By analogy with the LASSO =-=[Tib96]-=-, Beurling LASSO (BLASSO) is the process of reconstructing a discrete measure ∆ from the samples y by finding a solution to: (BLASSO) ˆ∆ 1 ∈ arg min µ∈M 2 ‖ ∫ Φ dµ−y‖ 2 2 +λ‖µ‖TV , where λ is a tuning... |

2267 | Nonlinear total variation based noise removal algorithms
- Rudin, Osher, et al.
- 1992
(Show Context)
Citation Context ...-norm of signal processing which is essentially the ℓ 1-norm of the ℓ2-norms of the finite differences at any point. In particular, our model has nothing to do with the Rudin-Osher-Fatemi (ROF) model =-=[ROF92]-=-. By analogy with the LASSO [Tib96], Beurling LASSO (BLASSO) is the process of reconstructing a discrete measure ∆ from the samples y by finding a solution to: (BLASSO) ˆ∆ 1 ∈ arg min µ∈M 2 ‖ ∫ Φ dµ−y... |

698 | Regression shrinkage and selection via the - Tibshirani - 1996 |

366 |
Super-resolution image reconstruction, a technical overview
- Park, Park, et al.
- 2003
(Show Context)
Citation Context ...estions in applied harmonic analysis such as the problem of source separation. Many companion applications in astronomy, medical imaging and single molecule imaging in 3D microscopy are at stake, see =-=[16, 15, 21]-=- and references therein. Hence, theoretical guarantees of source detection are of crucial importance in practice. In this paper, we prove quantitative detection guarantees from noisy observations (Fou... |

207 |
The Markov moment problem and extremal problems
- Krein, Nudel’man
- 1973
(Show Context)
Citation Context ...such that for all k = 1, . . . , s, P(T k) = exp(−iθ k), and ∀x ∈ T , 1−|P(x)|≥ min T∈S {Cam 2 d(x, T) 2 , C b} . Before stating the theorem, we would like to point out that, under general conditions =-=[14]-=-, the solutions of the convex program (BLASSO) always contain an atomic solution with support of size less than m+ 2. Moreover, this solution can be computed from the solution of the convex dual probl... |

127 |
Convex Analysis in General Vector Spaces. World Scientific,
- Zălinescu
- 2002
(Show Context)
Citation Context ... t∈[0,1] ‖Z(t)‖ > u} ≤ 4( exp(− u2 2(2 fc + 1) ) + √ fc( fc + 1) 3 exp (− u2 4(2 fc + 1) )) . 5. Numerical experiments Convex analysis shows that (BLASSO) can be viewed as a Fenchel dual problem, see =-=[24, 3]-=- for a definition. As a matter of fact, any solution to (BLASSO) can be faithfully computed from a companion program that builds a dual certificate. Proposition 5.1 ([24, 3]). The problem (5.1) min a∈... |

118 |
de Geer. Statistics for high-dimensional data
- Bühlmann, van
- 2011
(Show Context)
Citation Context ... total number of spikes s. This property is of great importance in actual practice. Only the minimal distance between any pair of atoms is relevant to BLASSO. Remark. Unlike the finite dimension case =-=[BVDG11]-=- (where T should be viewed as Rn and BLASSO is simply LASSO), we have numerically witnessed to the following fact: the solutions ˆ∆ to BLASSO have generally less atoms than the target ∆ (i.e. the size... |

84 |
Towards a mathematical theory of superresolution,” arXiv:1203.5871
- Candès, Granda
(Show Context)
Citation Context ...asure; Semidefinite programming; Compressed Sensing. 12 JEAN-MARC AZAÏS, YOHANN DE CASTRO, AND FABRICE GAMBOA hal-00780808, version 2 - 25 Feb 2014 In the super-resolution frame, the important paper =-=[6]-=- shows that if the spikes are well “seperated” then there exists a dual certificate, i.e. an ℓ∞-constrained trigonometric polynomial that interpolates the phase of the weights at the spikes locations.... |

72 | Convergence rates of convex variational regularization
- Burger, Osher
(Show Context)
Citation Context ... that k spikes trains can be faithfully resolved from m = 2k+1 samples (Fourier samples, Stieltjes transformation, Laplace transform...) by using an ℓ 1-minimization method. Following recent proposal =-=[3, 4, 17, 19]-=- on inverse problems regularization in Banach spaces, we consider convergence rates in Bregman divergence. On a more general note, inverse problems on the space of measures are now well understood, se... |

69 |
Superresolution via sparsity constraints.
- Donoho
- 1992
(Show Context)
Citation Context ...antitative estimates can be computed using a tractable algorithm, called BLASSO. 1.2. Previous works. The theoretical analysis of theℓ1-regularization in the space of measures was initiated by Donoho =-=[8]-=-. Few years after, rates of convergence in super-resolution have been investigated by P. Doukhan, E. Gassiat and one author of this present paper [9, 12]. They considered the exact reconstruction of a... |

56 |
Polynomials and Polynomial Inequalities,
- Borwein, Erdelyi
- 1995
(Show Context)
Citation Context ... not greater than m/2 are tight separable measures with respect to the standard moments, i.e. ϕ k(x) = x k . Indeed, let ∆ be a nonnegative measure and S = {T 1, . . . , Ts} be its support. Following =-=[BE95]-=-, set P(x) = 1−c s ∏(x−T i) i=1 2 . hal-00780808, version 1 - 24 Jan 2013 Then, for a sufficiently small value of the parameter c, the polynomial P has supremum norm not greater than 1. The existence ... |

50 | A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators
- Hofmann, Kaltenbacher, et al.
(Show Context)
Citation Context ...n inverse problems regularization in Banach spaces, we consider convergence rates in Bregman divergence. On a more general note, inverse problems on the space of measures are now well understood, see =-=[13, 20]-=- for instance. We capitalize on these earlier works to construct our analysis. In particular, we use them to give quantitative localizations of the recovered spikes, which is new. Date: February 25, 2... |

42 |
Positive Trigonometric Polynomials and Signal Processing Applications
- Dumitrescu
- 2007
(Show Context)
Citation Context ... seems difficult to solve due to the ℓ∞ norm in the hard constraint. As pointed out by [CFG12b], this difficulty can be circumvented using the following lemma. Lemma 6.3 (Corollary to Theorem 4.24 in =-=[Dum07]-=-). A trigonometric polynomial: P = fc ∑ ak exp(i 2πkt) k=− fc is bounded by one in magnitude if and only if there exists a Hermitian matrix Q ∈ C (m+1)×(m+1) satisfying: ( Q a (6.4) a∗ ) { m+1−j 1 , i... |

36 |
Functional Analysis. McGraw-Hill Series in Higher Mathematics
- Rudin
- 1973
(Show Context)
Citation Context ...e unit circle S1 (which is identified as R mod (2π) via the mapping z = eit ). Let ∆ be a complex measure on T with discrete support of (unknown) size s. In particular, ∆ has polar decomposition (see =-=[Rud87]-=- for a definition): s ∆ = ∑ ∆k exp(i θk) δT , k k=1 where ∆k > 0, θk ∈ R, Tk ∈ T for k = 1, . . . , s and δx denotes the Dirac measure at point x. Let m be a positive integer and F = {ϕ0, ϕ1, . . . , ... |

31 |
Error estimates for non-quadratic regularization and the relation to enhancement
- Resmerita, Scherzer
(Show Context)
Citation Context ... that k spikes trains can be faithfully resolved from m = 2k+1 samples (Fourier samples, Stieltjes transformation, Laplace transform...) by using an ℓ 1-minimization method. Following recent proposal =-=[3, 4, 17, 19]-=- on inverse problems regularization in Banach spaces, we consider convergence rates in Bregman divergence. On a more general note, inverse problems on the space of measures are now well understood, se... |

25 |
Sur les integrales de Fourier absolument convergentes et leur application a une transformation fonctionnelle,
- Beurling
- 1939
(Show Context)
Citation Context ...y‖ 2 2 +λ‖µ‖TV , where λ is a tuning parameter. TSPIKE DETECTION FROM INACCURATE SAMPLINGS 3 Remark. For the case of Fourier coefficients and ε = 0, (BLASSO) is simply Beurling Minimal Extrapolation =-=[Beu38]-=-. Moreover, in the finite dimension framework (i.e. T should be viewed as R n ), BLASSO is nothing else than LASSO. BLASSO is named after this remark. Remark. The factor 1/2 in the definition of BLASS... |

21 |
A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail
- Azäıs, Wschebor
(Show Context)
Citation Context ...ZAÏS, YOHANN DE CASTRO, AND FABRICE GAMBOA Last but not least, our analysis involves an estimate of the magnitude of the noise perturbation in the signal domain using the Rice method, see for example =-=[AW08]-=-. In particular, we derive explicit bounds for the tunning parameter appearing in BLASSO. hal-00780808, version 1 - 24 Jan 2013 1.2. General model and notation. Let T be a compact set homeomorphic to ... |

18 |
Compressive fluorescence microscopy for biological and hyperspectral imaging.
- Studer, Bobin, et al.
- 2012
(Show Context)
Citation Context ...estions in applied harmonic analysis such as the problem of source separation. Many companion applications in astronomy, medical imaging and single molecule imaging in 3D microscopy are at stake, see =-=[16, 15, 21]-=- and references therein. Hence, theoretical guarantees of source detection are of crucial importance in practice. In this paper, we prove quantitative detection guarantees from noisy observations (Fou... |

13 | Exact reconstruction using Beurling Minimal Extrapolation. - Castro, Gamboa - 2012 |

12 | Exact support recovery for sparse spikes deconvolution.
- Duval, Peyre
- 2013
(Show Context)
Citation Context ...w of application, the recent works [5, 22] derive results in ℓ 1 and ℓ2 prediction, i.e. the estimation of the input frequencies. Moreover, note that noise robustness of support recovery is proved in =-=[10]-=-. A postdating paper [11] bounds the support detection error for a constrained formulation of theℓ1-minimization in the space of measures as in Theorem 2.2. Regarding unconstrained formulation, a seco... |

8 | Inverse problems in spaces of measures
- Bredies, Pikkarainen
- 2010
(Show Context)
Citation Context ... that k spikes trains can be faithfully resolved from m = 2k+1 samples (Fourier samples, Stieltjes transformation, Laplace transform...) by using an ℓ 1-minimization method. Following recent proposal =-=[3, 4, 17, 19]-=- on inverse problems regularization in Banach spaces, we consider convergence rates in Bregman divergence. On a more general note, inverse problems on the space of measures are now well understood, se... |

6 |
Sets of superresolution and the maximum entropy method on the mean
- Gamboa, Gassiat
- 1996
(Show Context)
Citation Context ...n in the space of measures was initiated by Donoho [8]. Few years after, rates of convergence in super-resolution have been investigated by P. Doukhan, E. Gassiat and one author of this present paper =-=[9, 12]-=-. They considered the exact reconstruction of a nonnegative measure and derived results when one only knows the values of a finite number of linear functionals at the target measure. Moreover, they st... |

6 |
Mathematical concepts of optical superresolution.
- Lindberg
- 2012
(Show Context)
Citation Context ...estions in applied harmonic analysis such as the problem of source separation. Many companion applications in astronomy, medical imaging and single molecule imaging in 3D microscopy are at stake, see =-=[16, 15, 21]-=- and references therein. Hence, theoretical guarantees of source detection are of crucial importance in practice. In this paper, we prove quantitative detection guarantees from noisy observations (Fou... |

5 |
Near minimax line spectral estimation. arxiv:1303.4348
- Tang, Bhaskar, et al.
- 2013
(Show Context)
Citation Context ...ures. In a predating paper [7], the authors investigates ℓ1-minimization with different types of measurements: trigonometric, polynomial, Laplace transform... In view of application, the recent works =-=[5, 22]-=- derive results in ℓ 1 and ℓ2 prediction, i.e. the estimation of the input frequencies. Moreover, note that noise robustness of support recovery is proved in [10]. A postdating paper [11] bounds the s... |

4 |
Superresolution from noisy data. Arxiv 1211.0290
- Candes, Fernandez-Granda
- 2012
(Show Context)
Citation Context ...s subsection, we mention the example of Fourier samples to illustrate our results. Recently, much emphasis has been put on the recovery of a spike train (discrete measure) from noisy bandlimited data =-=[CFG12a]-=-. In this setting, we observe noisy Fourier samples up until a frequency cut-off fc ∈ N ∗ . We shall specify notation: • The number of samples is 2 fc+ 1 hence m = 2 fc. • For sake of simplicity, we p... |

4 |
Support detection in super-resolution. arXiv preprint arXiv:1302.3921
- Fernandez-Granda
- 2013
(Show Context)
Citation Context ...ent works [5, 22] derive results in ℓ 1 and ℓ2 prediction, i.e. the estimation of the input frequencies. Moreover, note that noise robustness of support recovery is proved in [10]. A postdating paper =-=[11]-=- bounds the support detection error for a constrained formulation of theℓ1-minimization in the space of measures as in Theorem 2.2. Regarding unconstrained formulation, a second postdating paper [22] ... |

3 |
Inverse problems in spaces of measures,” ESAIM
- Bredies, Pikkarainen
- 2013
(Show Context)
Citation Context ...sform... Furthermore, the article [CFG12b] provides an explicit construction of a tight dual certificate P in the Fourier case and gives an upper bound on the magnitude of P at each point. We mention =-=[BP10]-=- which aim at approximating the solution by estimating the support of the signal in an iterative fashion. In the case of noisy measurements, [CFG12a] derives a stability result for a weighted ℓ 1 dist... |

3 |
Superresolution rates in Prokhorov metric
- Doukhan, Gamboa
- 1996
(Show Context)
Citation Context ...n in the space of measures was initiated by Donoho [8]. Few years after, rates of convergence in super-resolution have been investigated by P. Doukhan, E. Gassiat and one author of this present paper =-=[9, 12]-=-. They considered the exact reconstruction of a nonnegative measure and derived results when one only knows the values of a finite number of linear functionals at the target measure. Moreover, they st... |

3 |
and complex analysis, mcgraw&hill series in higher mathematics
- Real
- 1987
(Show Context)
Citation Context ...z = e i2πt . In this latter case, the distance d is taken around the circle. Let ∆ be a complex measure on T with discrete support of size s. In particular, the measure ∆ has polar decomposition, see =-=[18]-=- for a definition: (1.1) ∆ = s ∑ ∆k exp(i θk) δT , k k=1 where ∆k > 0, θk ∈ R, Tk ∈ T for k = 1, . . . , s and δx denotes the Dirac measure at point x. Let m be a positive integer and F = {ϕ0, ϕ1, . .... |

3 |
Variational methods in imaging, volume 167
- Scherzer
- 2009
(Show Context)
Citation Context |

3 |
Sparsity regularization for radon measures
- Scherzer, Walch
- 2009
(Show Context)
Citation Context ...n inverse problems regularization in Banach spaces, we consider convergence rates in Bregman divergence. On a more general note, inverse problems on the space of measures are now well understood, see =-=[13, 20]-=- for instance. We capitalize on these earlier works to construct our analysis. In particular, we use them to give quantitative localizations of the recovered spikes, which is new. Date: February 25, 2... |

1 |
Sets of superresolution and themaximum entropymethod on themean
- Gamboa, Gassiat
- 1996
(Show Context)
Citation Context ...n in the space of measures was initiated by Donoho [8]. Few years after, rates of convergence in super-resolution have been investigated by P. Doukhan, E. Gassiat and one author of this present paper =-=[9, 12]-=-. They considered the exact reconstruction of a nonnegative measure and derived results when one only knows the values of a finite number of linear functionals at the target measure. Moreover, they st... |