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## SIAM J. IMAGING SCIENCES c © xxxx Society for Industrial and Applied Mathematics Vol. xx, pp. x x–x Signal Recovery and System Calibration from Multiple Compressive Poisson Measurements

### Citations

2253 | Nonlinear total variation based noise removal algorithms
- Rudin, Osher, et al.
- 1992
(Show Context)
Citation Context ...er scaling is applied in order to satisfy the Kraft inequality. Typical choices for the pen(·) include the l1 norm, of interest for sparse signals [4], and the total-variation norm for smooth signals =-=[26]-=-. In fact, the Kraft-compliant penalty is related to the prefix codes for estimators, and more concrete examples of this penalty functions are presented in [32]. As elaborated in the proof of Theorem ... |

1426 | Compressive sampling
- Candès
- 2006
(Show Context)
Citation Context ...gnated as many popular penalty functions, when a proper scaling is applied in order to satisfy the Kraft inequality. Typical choices for the pen(·) include the l1 norm, of interest for sparse signals =-=[4]-=-, and the total-variation norm for smooth signals [26]. In fact, the Kraft-compliant penalty is related to the prefix codes for estimators, and more concrete examples of this penalty functions are pre... |

1208 | Algorithms for non-negative matrix factorization
- Lee, Seung
- 2000
(Show Context)
Citation Context ... image-deblurring method. It has also been noticed in [36] that (4.4) is closely connected to nonnegative matrix factorization guided by a KL-divergence-minimization criterion, i.e., minΦ,f KL(y||Φf) =-=[17]-=-. In addition to these methods, the PG model is able to accommodate a sparsity assumption on {f̂k}Kk=1 as previously explained, thereby exhibiting more flexibility. Compared with a recently proposed g... |

603 |
Extension of lipschitz maps into a hilbert space
- Johnson, Lindenstrauss
- 1984
(Show Context)
Citation Context ...r-distributed random matrix has been derived, and we now present several theoretical applications of this result. Specifically, we formulate a modified version of the Johnson-Lindenstrauss (JL) Lemma =-=[13]-=- for block-diagonal matrices. Recall the definition of the stable embedding [5] Definition 3.3. For U, V ⊂ Rn, a map Φ : Rn → Rm is called an ǫ-stable embedding of (U, V ) if (3.8) (1− ǫ)‖x− y‖22 ≤ ‖Φ... |

584 |
The concentration of measure phenomenon
- Ledoux
- 2001
(Show Context)
Citation Context ...entration-of-measure Inequalities. Concentration of measure is a phenomenon describing the tendency of certain functions of a high-dimensional random process to concentrate sharply around their means =-=[16]-=-. Our first result is a concentration-of-measure inequality for the block-diagonal Rademacher-distributed matrix Ã0, which serves as a key ingredient in the proof of the performance bounds. Theorem 3... |

386 |
Bayesian-based iterative method of image restoration
- Richardson
- 1972
(Show Context)
Citation Context ...j · ∑m i=1 Φ̂i,jyk,i ∑ j Φ̂i,j f̂k,j∑m i=1 Φ̂i,j , where the superscript t indexes the iteration of the inference. Equation (4.4) is the same multiplicative update as in the Richardson–Lucy algorithm =-=[25]-=-, a classical image-deblurring method. It has also been noticed in [36] that (4.4) is closely connected to nonnegative matrix factorization guided by a KL-divergence-minimization criterion, i.e., minΦ... |

353 |
On a measure of divergence between two statistical populations defined by their probability distributions
- Bhattacharyya
- 1943
(Show Context)
Citation Context ...bed in Section 3.1, and a significant amount of dark-current may be present. We generate Φ0 ∈ R50×100+ by randomly setting half the entries to nonzero, with non-zero values distributed uniformly over =-=[0, 1]-=-, and generate ΦE via δ = 0.3. Each {fk}30k=1 is generated by concatenating several bell-shaped curves of different bandwidths (examples Signal Recovery and System Calibration from Multiple Compressiv... |

347 | Introduction to the non-asymptotic analysis of random matrices
- Vershynin
- 2012
(Show Context)
Citation Context ... φ(wi) is a sub-Gaussian random variable by its definition. Moreover, si, i = 1, . . . ,m are independent sub-Gaussian random variables. Via the Hoeffding inequality for sub-Gaussian random variables =-=[28]-=-, we have P (∣∣∣∣∣ m∑ i=1 (si − E[si]) ∣∣∣∣∣ > ǫ‖f‖22 ) ≤ e · exp ( −c3ǫ 2‖f‖42 B2m ) ,(A.8) where c3 > 0 is a constant. B def = maxi ‖si‖O and ‖ · ‖O is the Orlicz norm [28]. By the property of Orlic... |

295 | Single-pixel imaging via compressive sampling
- Duarte, Davenport, et al.
(Show Context)
Citation Context ...pressive sensing (CS) in actual physical systems, with the goal of efficiently (compressively) measuring the information characteristic of an entity under test. Examples include a single-pixel camera =-=[6]-=-, hyperspectral imaging [14], and compressive video [12, 19, 34]. The Gaussian measurement model is assumed in each of these examples, and is widely employed in existing theory and applications. A Gau... |

101 | Signal processing with compressive measurements
- Davenport, Boufounos, et al.
- 2010
(Show Context)
Citation Context ...ical applications of this result. Specifically, we formulate a modified version of the Johnson-Lindenstrauss (JL) Lemma [13] for block-diagonal matrices. Recall the definition of the stable embedding =-=[5]-=- Definition 3.3. For U, V ⊂ Rn, a map Φ : Rn → Rm is called an ǫ-stable embedding of (U, V ) if (3.8) (1− ǫ)‖x− y‖22 ≤ ‖Φ(x− y)‖22 ≤ (1 + ǫ)‖x− y‖22, ∀x ∈ U, y ∈ V. In other words, a map is a stable e... |

85 | Mixture Density Estimation
- Li, Barron
- 1999
(Show Context)
Citation Context ...·||·) denotes the Kullback-Leibler distance. The expectation is taken with respect to an arbitrary joint distribution on {A, f}. Proof. [Proof of Lemma D.2] The proof is based on the techniques as in =-=[18]-=-. Denote pA∗f∗ def = p(y|(A0 + A∗E)f∗ + λ) and pAf def = p(y|(A0 + AE)f + λ). Define H(A∗f∗,Af) def=∫ √ pA∗f∗pAfdν as the Hellinger affinity. We have (D.4) 2 log 1 H(A∗f∗, Âf̂) = 2 log √pÂf̂/pA∗f... |

48 | General deviants: An analysis of perturbations in compressed sensing
- Herman, Strohmer
- 2010
(Show Context)
Citation Context ...ng matrix [2], and hence perturbations to this matrix must be inferred when performing CS inversion. The problem of fluctuating sensing matrices has been studied for the Gaussian measurement model in =-=[11, 33, 37]-=-. To the authors’ knowledge, almost all previous work focuses on establishing various theoretical properties of both Gaussian and Poisson measurement models with a single compressive measurement. CS w... |

44 | Multiscale Poisson intensity and density estimation
- Willett, Nowak
- 2007
(Show Context)
Citation Context ...total-variation norm for smooth signals [26]. In fact, the Kraft-compliant penalty is related to the prefix codes for estimators, and more concrete examples of this penalty functions are presented in =-=[32]-=-. As elaborated in the proof of Theorem 3.1, we utilize a main result from [15], which requires the countability assumption as reflected in Γ. In practice, the signal energy level I and the perturbati... |

40 | Sparse Poisson intensity reconstruction algorithms,”
- Harmany, Marcia, et al.
- 2009
(Show Context)
Citation Context ...ement model have been studied from various perspectives. Algorithms for recovering the (sparse) Poisson rate function (i.e., the associated parameter of the Poisson distribution) have been studied in =-=[10]-=-, and performance bounds for inversion algorithms have been developed in [22, 24]. However, these algorithms assume perfect knowledge of the sensing matrix. In real physical systems that motivate this... |

37 | Concentration inequalities for dependent random variables via the martingale method
- Kontorovich, Ramanan
(Show Context)
Citation Context ...lty is related to the prefix codes for estimators, and more concrete examples of this penalty functions are presented in [32]. As elaborated in the proof of Theorem 3.1, we utilize a main result from =-=[15]-=-, which requires the countability assumption as reflected in Γ. In practice, the signal energy level I and the perturbation bound ǫ1 may not be known precisely. In order to increase the flexibility of... |

33 | Video from a single coded exposure photograph using a learned over-complete dictionary
- Hitomi, Gu, et al.
- 2011
(Show Context)
Citation Context ...the goal of efficiently (compressively) measuring the information characteristic of an entity under test. Examples include a single-pixel camera [6], hyperspectral imaging [14], and compressive video =-=[12, 19, 34]-=-. The Gaussian measurement model is assumed in each of these examples, and is widely employed in existing theory and applications. A Gaussian measurement model is not appropriate for many important ap... |

29 | Beta-negative binomial process and Poisson factor analysis
- Zhou, Hannah, et al.
- 2012
(Show Context)
Citation Context ...reasonable” settings). If an estimate of the mean of u is available, it may be used in the prior, but such prior knowledge of u has been found unnecessary in practice. We propose a Poisson-Gamma (PG) =-=[36]-=- Bayesian model: (4.1) yk ∼ Pois (Φfk + u) , fk ∼ ∏n j=1Gamma(fk,j;αf , βf ), Φi,j ∼ Gamma(Φi,j;βΦΦ0,i,j, βΦ), u ∼ ∏mi=1Gamma(ui;αu,i, βu,i), where {αf , βf , βΦ, αu,i, βu,i} are hyper-parameters and ... |

29 | Sparsity-cognizant total least-squares for perturbed compressive sampling
- Zhu, Leus, et al.
(Show Context)
Citation Context ...ng matrix [2], and hence perturbations to this matrix must be inferred when performing CS inversion. The problem of fluctuating sensing matrices has been studied for the Gaussian measurement model in =-=[11, 33, 37]-=-. To the authors’ knowledge, almost all previous work focuses on establishing various theoretical properties of both Gaussian and Poisson measurement models with a single compressive measurement. CS w... |

27 | Compressed sensing performance bounds under poisson noise
- Raginsky, Willett, et al.
- 2010
(Show Context)
Citation Context ...ering the (sparse) Poisson rate function (i.e., the associated parameter of the Poisson distribution) have been studied in [10], and performance bounds for inversion algorithms have been developed in =-=[22, 24]-=-. However, these algorithms assume perfect knowledge of the sensing matrix. In real physical systems that motivate this paper, it is usually impossible to build a device that perfectly matches the des... |

26 |
Optical imaging and spectroscopy
- Brady
- 2009
(Show Context)
Citation Context ...Wang, Huang, Yuan, Krishnamurthy, Greenberg, Cevher, Rodrigues, Brady, Calderbank and Carin sensing matrix. There are other situations for which the target to be sensed may perturb the sensing matrix =-=[2]-=-, and hence perturbations to this matrix must be inferred when performing CS inversion. The problem of fluctuating sensing matrices has been studied for the Gaussian measurement model in [11, 33, 37].... |

20 | Bayesian latent variable models for mixed discrete outcomes. Biostatistics,
- Dunson, Herring
- 2005
(Show Context)
Citation Context ...case it is appropriate to set αu,i = 1 and βu,i = 1/u0,i, where u0,i is the i-th element of the dark current u0 estimated (i.e., E(u) = u0). Inference for the PG model involves an augmentation scheme =-=[7]-=-, introducing latent variable ξ ∈ RK×m×(n+1)+ , with ξk,i,j = Φ̂i,j f̂k,jyk,i ∑n j=1 Φ̂i,j f̂k,j+ûi , j = 1, . . . , n; ξk,i,n+1 = ûiyk,i ∑n j=1 Φ̂i,j f̂k,j+ûi . We have developed both Markov Chain... |

20 | Robustly stable signal recovery in compressed sensing with structured matrix perturbation
- Yang, Zhang, et al.
- 2012
(Show Context)
Citation Context ...ng matrix [2], and hence perturbations to this matrix must be inferred when performing CS inversion. The problem of fluctuating sensing matrices has been studied for the Gaussian measurement model in =-=[11, 33, 37]-=-. To the authors’ knowledge, almost all previous work focuses on establishing various theoretical properties of both Gaussian and Poisson measurement models with a single compressive measurement. CS w... |

18 | Performance bounds for expander-based compressed sensing in Poisson noise,” - Raginsky, Jafarpour, et al. - 2011 |

18 | Negative binomial process count and mixture modeling
- Zhou, Carin
(Show Context)
Citation Context ...j Φi,j + (βΦΦ0,i,j − 1) ∑ i,j log Φi,j − ∑ i βu,iui + ∑ i (αu,i − 1) log ui + const. Note in (4.3) that the ℓ1 penalty is inherently imposed on fk, and the third term accounts for the over-dispersion =-=[35]-=- effect of count data in a variety of real applications. Interestingly, when αf = 1, (4.3) provides the same formulation as CMLE with the ℓ1 penalty, and βf plays the role of τ1 in (3.15). However, in... |

17 | Multiframe image estimation for coded aperture snapshot spectral imagers
- Kittle, Choi, et al.
- 2010
(Show Context)
Citation Context ...tual physical systems, with the goal of efficiently (compressively) measuring the information characteristic of an entity under test. Examples include a single-pixel camera [6], hyperspectral imaging =-=[14]-=-, and compressive video [12, 19, 34]. The Gaussian measurement model is assumed in each of these examples, and is widely employed in existing theory and applications. A Gaussian measurement model is n... |

16 | Concentration of measure for block diagonal matrices with applications to compressive sensing
- Park, Yap, et al.
- 2011
(Show Context)
Citation Context ... focuses on establishing various theoretical properties of both Gaussian and Poisson measurement models with a single compressive measurement. CS with multiple measurements has only been addressed in =-=[8, 21]-=-, and there for the Gaussian measurement model. Furthermore, randomly constituted sensing matrices employed in the aforementioned work violate the nonnegativity constraint of the Poisson measurement m... |

9 |
Concentration of measure inequalities in information theory, communications and coding
- Raginsky, Sason
- 2013
(Show Context)
Citation Context ... (0, 1), where c1 > 0 is a constant. Proof. The proof is presented in Appendix A. In contrast to many previous concentration-of-measure results for matrices populated with i.i.d. sub-Gaussian entries =-=[23]-=-, the decay rate indicated by Theorem 3.1 depends on the signal being measured. In particular, for the estimator candidates set Γ introduced in Section 3.1, we have the following corollary which will ... |

8 | The restricted isometry property for random block diagonal matrices
- Eftekhari, Yap, et al.
(Show Context)
Citation Context ... focuses on establishing various theoretical properties of both Gaussian and Poisson measurement models with a single compressive measurement. CS with multiple measurements has only been addressed in =-=[8, 21]-=-, and there for the Gaussian measurement model. Furthermore, randomly constituted sensing matrices employed in the aforementioned work violate the nonnegativity constraint of the Poisson measurement m... |

7 | Composite self-concordant minimization.,”
- Tran-Dinh, Kyrillidis, et al.
- 2013
(Show Context)
Citation Context ...1 = argmin Φ0 〈 ∇Φℓ(F̂t+1, Φ̂t),Φ− Φ̂t 〉 + τ2‖Φ−Φ0‖F + Ltφ 2 ‖Φ− Φ̂t‖2F , where Ltf and L t φ are local Lipschitz constants. Notice that by fixing either F or Φ, ℓ(F,Φ) is a self-concordant function =-=[27]-=- of the other variable, for which an optimal step-size [27] is available. Details of the alternating minimization steps are summarized in Algorithm 1. 4. Bayesian Model for Real Systems: Poisson-Gamma... |

5 |
Coded aperture compressive temporal imaging,” Optics Express
- Llull, Liao, et al.
- 2013
(Show Context)
Citation Context ...the goal of efficiently (compressively) measuring the information characteristic of an entity under test. Examples include a single-pixel camera [6], hyperspectral imaging [14], and compressive video =-=[12, 19, 34]-=-. The Gaussian measurement model is assumed in each of these examples, and is widely employed in existing theory and applications. A Gaussian measurement model is not appropriate for many important ap... |

5 | Low-cost compressive sensing for color video and depth
- Yuan, Llull, et al.
- 2014
(Show Context)
Citation Context ...the goal of efficiently (compressively) measuring the information characteristic of an entity under test. Examples include a single-pixel camera [6], hyperspectral imaging [14], and compressive video =-=[12, 19, 34]-=-. The Gaussian measurement model is assumed in each of these examples, and is widely employed in existing theory and applications. A Gaussian measurement model is not appropriate for many important ap... |

3 |
A Bregman matrix and the gradient of mutual information for vector Poisson and Gaussian channels
- Wang, Carlson, et al.
- 2014
(Show Context)
Citation Context ...ese examples, and is widely employed in existing theory and applications. A Gaussian measurement model is not appropriate for many important applications, including X-ray [9, 20] and chemical imaging =-=[29, 30, 31]-=-. The observed data in these applications are characterized by counts, typically under Poisson statistics. The properties of the Poisson measurement model have been studied from various perspectives. ... |

2 |
Snapshot molecular imaging using coded energy-sensitive detection,” Optics express
- Greenberg, Krishnamurthy, et al.
- 2013
(Show Context)
Citation Context ...odel is assumed in each of these examples, and is widely employed in existing theory and applications. A Gaussian measurement model is not appropriate for many important applications, including X-ray =-=[9, 20]-=- and chemical imaging [29, 30, 31]. The observed data in these applications are characterized by counts, typically under Poisson statistics. The properties of the Poisson measurement model have been s... |

2 |
Snapshot 2D tomography via coded aperture x-ray scatter imaging
- MacCabe, Holmgren, et al.
- 2013
(Show Context)
Citation Context ...odel is assumed in each of these examples, and is widely employed in existing theory and applications. A Gaussian measurement model is not appropriate for many important applications, including X-ray =-=[9, 20]-=- and chemical imaging [29, 30, 31]. The observed data in these applications are characterized by counts, typically under Poisson statistics. The properties of the Poisson measurement model have been s... |

2 |
Photon level chemical classification using digital compressive detection, Analytica Chimica Acta
- Wilcox, Buzzard, et al.
(Show Context)
Citation Context ...ese examples, and is widely employed in existing theory and applications. A Gaussian measurement model is not appropriate for many important applications, including X-ray [9, 20] and chemical imaging =-=[29, 30, 31]-=-. The observed data in these applications are characterized by counts, typically under Poisson statistics. The properties of the Poisson measurement model have been studied from various perspectives. ... |

1 |
Designed measurements for vector count data, in Advances in neural information processing systems
- Wang, Carlson, et al.
- 2013
(Show Context)
Citation Context ...ese examples, and is widely employed in existing theory and applications. A Gaussian measurement model is not appropriate for many important applications, including X-ray [9, 20] and chemical imaging =-=[29, 30, 31]-=-. The observed data in these applications are characterized by counts, typically under Poisson statistics. The properties of the Poisson measurement model have been studied from various perspectives. ... |