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## Non-Parametric Bayesian Dictionary Learning for Sparse Image Representations

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5036 |
Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images
- Geman, Geman
- 1984
(Show Context)
Citation Context ... dictionaries for analysis of imagery, with applications in denoising, interpolation, and CS. The inference is performed based on a Gibbs sampler, as is increasingly common in modern image processing =-=[16]-=-. Here, we demonstrate how generalizations of the beta-Bernoulli process allow one to infer the dictionary elements directly based on the underlying degraded image, without any a priori training data,... |

3998 | Regression shrinkage and selection via the lasso,”
- Tibshirani
- 1996
(Show Context)
Citation Context ...s been significant recent interest in sparse signal expansions in several settings. For example, such algorithms as the support vector machine (SVM) [1], the relevance vector machine (RVM) [2], Lasso =-=[3]-=- and many others have been developed for sparse regression (and classification). A sparse representation has several advantages, including the fact that it encourages a simple model, and therefore ove... |

3543 | Compressed sensing
- Donoho
- 2006
(Show Context)
Citation Context ...s on applying the algorithms to new compressive measurement techniques that have been developed recently. Specifically, we consider dictionary learning in the context of compressive sensing (CS) [5], =-=[10]-=-, in which the measurements correspond to projections of typical image pixels. We consider dictionary learning performed “offline” based on representative (training) images, with the learned dictionar... |

1477 | Signal recovery from random projections
- Candés, JK
- 2005
(Show Context)
Citation Context ...interpretation [4]. Of relevance for the current paper, there has recently been significant interest in sparse representations in the context of denoising, inpainting [5–10], compressive sensing (CS) =-=[11, 12]-=-, and classification [13]. All of these applications exploit the fact that most images may be sparsely represented in an appropriate dictionary. Most of the CS literature assumes “off-the-shelf” wavel... |

1270 |
An Introduction to Support Vector Machines
- Cristianini, Shawe-Taylor
- 2000
(Show Context)
Citation Context ...tate-of-the-art approaches. 1 Introduction There has been significant recent interest in sparse signal expansions in several settings. For example, such algorithms as the support vector machine (SVM) =-=[1]-=-, the relevance vector machine (RVM) [2], Lasso [3] and many others have been developed for sparse regression (and classification). A sparse representation has several advantages, including the fact t... |

1179 |
A bayesian analysis of some nonparametric problems
- Ferguson
- 1973
(Show Context)
Citation Context ...s inferred within the inversion). The noise variance can also be non-stationary. • The spatial inter-relationships between different components in images are exploited by use of the Dirichlet process =-=[18]-=- and a probit stick-breaking process [19]. • Using learned dictionaries, inferred off-line or in situ, the proposed approach yields CS perfor1mance that is markedly better than existing standard CS m... |

947 | Sparse Bayesian learning and the relevance vector machine
- Tipping
- 2001
(Show Context)
Citation Context ...on There has been significant recent interest in sparse signal expansions in several settings. For example, such algorithms as the support vector machine (SVM) [1], the relevance vector machine (RVM) =-=[2]-=-, Lasso [3] and many others have been developed for sparse regression (and classification). A sparse representation has several advantages, including the fact that it encourages a simple model, and th... |

941 | Sparse coding with an overcomplete basis set: A strategy employed by V1
- Olshausen, Field
- 1997
(Show Context)
Citation Context ...act that it encourages a simple model, and therefore over-training is often avoided. The inferred sparse coefficients also often have biological/physical meaning, of interest for model interpretation =-=[4]-=-. Of relevance for the current paper, there has recently been significant interest in sparse representations in the context of denoising, inpainting [5–10], compressive sensing (CS) [11, 12], and clas... |

919 | The K-SVD: An algorithm for designing of overcomplete dictionaries for sparse representation
- Aharon, Elad, et al.
- 2006
(Show Context)
Citation Context ...nd the model also yields effective inference of α even when x is partially or indirectly observed via a small number of measurements (of interest for inpainting, interpolation and compressive sensing =-=[5, 7]-=-). To the authors’ knowledge, all previous work in this direction has been performed in the following manner: (i) if D is given, the sparse vector α is estimated via a point estimate (without a poster... |

919 | Hierarchical Dirichlet processes
- Teh, Jordan, et al.
(Show Context)
Citation Context ...ture components (i.e., popular across different buffets in the franchise). One can impose this additional structure via a hierarchical BP (HBP) construction [37], related to the hierarchical DP (HDP) =-=[36]-=-. Briefly, in an HBP construction, one may draw the via the hierarchical construction, i.e., Beta Beta (10) where is the th component of . Vector constitutes “global” probabilities of using each of th... |

905 | Robust Face Recognition via Sparse Representation
- Wright, Yang, et al.
- 2009
(Show Context)
Citation Context ...nce for the current paper, there has recently been significant interest in sparse representations in the context of denoising, inpainting [5–10], compressive sensing (CS) [11, 12], and classification =-=[13]-=-. All of these applications exploit the fact that most images may be sparsely represented in an appropriate dictionary. Most of the CS literature assumes “off-the-shelf” wavelet and DCT bases/dictiona... |

832 | Exact matrix completion via convex optimization
- Candès, Recht
(Show Context)
Citation Context ...ith singular value of Z. Defining n = max(n1, n2), Candès and Tao [6] showed that with probability exceeding 1−n −3 , if m ≥ Cµ 2 nr(log n) 6 then Z = M, thereby recovering the missing entries of M ( =-=[8]-=- considers related issues). The parameter µ represents the coherence, a measure of how “spread out” the singular vectors of M are, and ideally the coherence is near one. For this result to be useful, ... |

671 | Compressive sensing
- Baraniuk
- 2007
(Show Context)
Citation Context ...e number of projections, and Φ ∈ ℜNp×64 (assuming that xi is represented by a 64-dimensional vector). There are many (typically random) ways in which Φ may be constructed, with the reader referred to =-=[24]-=-. Our goal is to have Np ≪ 64, thereby yielding compressive measurements. Based on the CS measurements {vi}i=1,NB, our objective is to recover {xi}i=1,NB. Consider a potential dictionary Ψ, as discuss... |

592 |
Image denoising via sparse and redundant representations over learned dictionaries
- Elad, Aharon
(Show Context)
Citation Context ...y×Nx with additive noise and missing pixels; we here assume a monochrome image for simplicity, but color images are also readily handled, as demonstrated when presenting results. As is done typically =-=[6, 7]-=-, we partition the image into NB = (Ny − B + 1) × (Nx − B + 1) overlapping blocks {xi}i=1,NB, for each of which xi ∈ ℜB2 (B = 8 is typically used). If there is only additive noise but no missing pixel... |

561 |
A constructive definition of dirichlet priors
- Sethuraman
- 1994
(Show Context)
Citation Context ..., G ∼ DP(β, ∏K k=1 Bernoulli(πk)), π ∼ ∏K k=1 Beta(a/K, b(K − 1)/K), where the zi are drawn i.i.d. from G. In practice we implement such DP constructions via a truncated stick-breaking representation =-=[25]-=-, again retaining the conjugate-exponential structure of interest for analytic VB or Gibbs inference. In such an analysis we place a non-informative gamma prior on the precision β. The construction in... |

527 | A Singular Value Thresholding Algorithm for Matrix Completion
- Cai, Candès, et al.
(Show Context)
Citation Context ...GB images, we also attempted a direct application of matrix completion based on the incomplete matrix X ∈ RP ×N, with columns defined by the image patches. We specifically considered the algorithm in =-=[19]-=-, using software from Prof. Candès’ website. For most of the examples considered above, even after very careful tuning of the April 17, 2010 DRAFT18 Fig. 4. Expected variance of each pixel for the (M... |

428 | Variational Algorithms for Approximate Bayesian Inference
- Beal
- 2003
(Show Context)
Citation Context ... are typically placed on γw and γɛ. Consecutive elements in the above hierarchical model are in the conjugate exponential family, and therefore inference may be implemented via a variational Bayesian =-=[22]-=- or Gibbs-sampling analysis, with analytic update equations (all inference update equations, and the software, will be referenced in a technical report, if the paper is accepted). After performing suc... |

421 | From sparse solutions of systems of equations to sparse modeling of signals and images
- Bruckstein, Donoho, et al.
- 2009
(Show Context)
Citation Context ...d ARO.2 DCT bases/dictionaries [20], but recent research has demonstrated the significant utility of learning an often over-complete dictionary matched to the signals of interest (e.g., images) [1], =-=[3]-=-, [12], [13], [24]– [26], [28], [29], [31], [33], [41]. Many of the existing methods for learning dictionaries are based on solving an optimization problem [1], [13], [24]–[26], [28], [29], in which o... |

419 | Image denoising by sparse 3-D transform-domain collaborative filtering
- Dabov, Foi, et al.
(Show Context)
Citation Context ...es. C. Denoising The BPFA denoising algorithm is compared with the original KSVD [13], for both grey-scale and color images. Newer denoising algorithms include block matching with 3D filtering (BM3D) =-=[9]-=-, the multiscale April 17, 2010 DRAFT14 KSVD [29], and KSVD with the non-local mean constraints [26]. These algorithms assume the noise variance is known, while the proposed model automatically infer... |

345 | The Power of Convex Relaxation: Near-Optimal Matrix Completion. ArXiv e-prints
- Candes, Tao
- 2009
(Show Context)
Citation Context ...s the vector of samples of the associated matrix that are in the set Ω. The nuclear norm ‖Z‖∗ = ∑ i γi(Z), where γi(Z) represents the ith singular value of Z. Defining n = max(n1, n2), Candès and Tao =-=[6]-=- showed that with probability exceeding 1−n −3 , if m ≥ Cµ 2 nr(log n) 6 then Z = M, thereby recovering the missing entries of M ( [8] considers related issues). The parameter µ represents the coheren... |

323 | Bayesian compressive sensing
- Ji, Xue, et al.
- 2008
(Show Context)
Citation Context ... of these applications exploit the fact that most images may be sparsely represented in an appropriate dictionary. Most of the CS literature assumes “off-the-shelf” wavelet and DCT bases/dictionaries =-=[14]-=-, but recent denoising and inpainting research has demonstrated the significant advantages of learning an often over-complete dictionary matched to the signals of interest (e.g., images) [5–10, 12, 15... |

292 | Single-pixel imaging via compressive sampling
- Duarte, Davenport, et al.
- 2008
(Show Context)
Citation Context ...nterest for its potential to reduce the number of required measurements, it has the disadvantage of requiring the development of new classes of cameras. Such cameras are revolutionary and interesting =-=[11]-=-, [37], but there have been decades of previous research performed on development of pixel-based cameras, and it would be desirable if such cameras could be modified simply to perform compressive meas... |

290 | Self-taught learning: Transfer learning from unlabeled data
- Raina, Battle, et al.
- 2007
(Show Context)
Citation Context ... recent research has demonstrated the significant utility of learning an often over-complete dictionary matched to the signals of interest (e.g., images) [1], [3], [12], [13], [24]– [26], [28], [29], =-=[31]-=-, [33], [41]. Many of the existing methods for learning dictionaries are based on solving an optimization problem [1], [13], [24]–[26], [28], [29], in which one seeks to match the dictionary to the im... |

270 | Infinite latent feature models and the indian buffet process
- Griffiths, Ghahramani
(Show Context)
Citation Context ...em, with the factor loadings corresponding to the dictionary elements (atoms). Utilizing nonparametric Bayesian methods like the beta process (BP) [30], [38], [42] and the Indian buffet process (IBP) =-=[18]-=-, [21], one may for example infer the number of factors (dictionary elements) needed to fit the data itself. Further, one may place a prior on the noise or residual variance, with this inferred from t... |

245 | Matrix completion with noise
- Candès, Plan
- 2010
(Show Context)
Citation Context ...t manifested by the model (constituted via ∑N i=1 ‖Pφi (xi −D(si ◦zi))‖2 2 ). Therefore, this construction is closely linked to the optimization approach advocated for near-low-rank matrix completion =-=[7]-=-, which also uses a Frobenius April 17, 2010 DRAFT13 norm. The key distinction, however, is manifested here via f({zi} N i=1 ; H), which moves beyond linear (low-rank) models to nonlinear constructio... |

237 | Online dictionary learning for sparse coding
- Mairal, Bach, et al.
(Show Context)
Citation Context ...proach is undesirable. To address this issue one may partition the data as D = D1 ∪ D2 ∪ . . . DJ−1 ∪ DJ, with the data processed sequentially. This issue has been considered for point estimates of D =-=[8]-=-, in which considerations are required to assure algorithm convergence. It is of interest to briefly note that sequential inference is handled naturally via the proposed Bayesian analysis. 3Specifica... |

222 | Learning midlevel features for recognition
- Boureau, Bach, et al.
- 2010
(Show Context)
Citation Context ...guille}@umn.edu Abstract There has been significant recent interest in dictionary learning and sparse coding, with applications in denoising, interpolation, feature extraction, and classification [1]–=-=[3]-=-. Increasingly it has been recognized that these models may be improved by imposing additional prior information, beyond sparseness. For example, a locality constraint has been used successfully in th... |

214 | Sparse representation for color image restoration
- Mairal, Elad, et al.
- 2007
(Show Context)
Citation Context ...he B 2 = 64 rounds discussed above. For Gibbs rounds 16, 32 and 64 the corresponding PSNR values were 27.66 dB, 28.22 dB and 28.76 dB. For this example we used K = 256. This example was considered in =-=[7]-=- (we obtained similar results for the “New Orleans” image, also considered in [7]); the best results reported there were a PSNR of 29.65 dB. However, to achieve those results a training data set was e... |

214 | Robust recovery of signals from a structured union of subspaces
- Eldar, Mishali
- 2009
(Show Context)
Citation Context ... modified simply such that it is useful. Specifically, because natural images manifest segments and self-similarity, one may view the dictionary-learning framework within a union-of-subspaces setting =-=[15]-=-, [23]. Each subspace, defined by a subset of the dictionary, represents a class of local structure within an image, and each subspace and associated data may be viewed as a low-rank matrix. The BP, D... |

213 | Efficient learning of sparse representations with an energy-based model
- Ranzato, Poultney, et al.
- 2006
(Show Context)
Citation Context ... methods in the literature. I. INTRODUCTION There has been significant recent interest in sparse image representations, in the context of denoising and interpolation [1], [13], [24]–[26], [28], [29], =-=[33]-=-, compressive sensing (CS) [5], [12], and classification [40]. All of these applications exploit the fact that images may be sparsely represented in an appropriate dictionary. Most of the denoising, i... |

192 |
Non-local sparse models for image restoration
- Mairal, Bach, et al.
(Show Context)
Citation Context ...mparisons to other methods in the literature. I. INTRODUCTION There has been significant recent interest in sparse image representations, in the context of denoising and interpolation [1], [13], [24]–=-=[26]-=-, [28], [29], [33], compressive sensing (CS) [5], [12], and classification [40]. All of these applications exploit the fact that images may be sparsely represented in an appropriate dictionary. Most o... |

189 | Supervised dictionary learning
- Mairal, Bach, et al.
- 2008
(Show Context)
Citation Context ...tion, both goals should be accounted for when designing D. For simplicity, we assume that the number of classes is NC = 2 (binary classification), with this readily extended [23] to NC > 2. Following =-=[9]-=-, we may define a linear or bilinear classifier based on the sparse weights α and the associated data x (in the bilinear case), with this here implemented in the form of a probit classifier. We focus ... |

187 | Image super-resolution via sparse representation
- Yang, Wright, et al.
- 2010
(Show Context)
Citation Context ...arch has demonstrated the significant utility of learning an often over-complete dictionary matched to the signals of interest (e.g., images) [1], [3], [12], [13], [24]– [26], [28], [29], [31], [33], =-=[41]-=-. Many of the existing methods for learning dictionaries are based on solving an optimization problem [1], [13], [24]–[26], [28], [29], in which one seeks to match the dictionary to the imagery of int... |

173 | Bayesian probabilistic matrix factorization using markov chain monte carlo
- Salakhutdinov, Mnih
- 2008
(Show Context)
Citation Context ...m, recovery of the missing pixels corresponds to the matrix-completion problem. However, the matrix-completion literature is based on the assumption that the matrix of interest is low rank [6], [22], =-=[35]-=-. Because the underlying dictionaries associated with natural images are typically over-complete, the assumption of a single low-rank matrix of pixel values is often inappropriate. While a direct appl... |

143 | Robust principal component analysis: Exact recovery of corrupted low-rank matrices via convex optimization
- Wright, Ganesh, et al.
(Show Context)
Citation Context ...mark-based dependent HBP, or Landmark-dHBP. The proposed model employs a robustness term to model sparse spiky noise or localized data anomalies, related to robust principal component analysis (RPCA) =-=[15]-=-, [16]. However, our model differs from RPCA in that the low-rank assumption on data is replaced with the richer covariatedependent union-of-subspace assumption, which is realized with a sparse factor... |

129 | Hierarchical beta processes and the Indian buffet process
- Thibaux, Jordan
- 2007
(Show Context)
Citation Context ...ten over-complete dictionary matched to the signals of interest (e.g., images) [5–10, 12, 15]. The purpose of this paper is to perform dictionary learning using new non-parametric Bayesian technology =-=[16,17]-=-, that offers several advantages not found in earlier approaches, which have generally sought point estimates. This paper makes four main contributions: • The dictionary is learned using a beta proces... |

126 | Learning with structured sparsity
- Huang, Zhang, et al.
(Show Context)
Citation Context ...nd sparseness. For example, a locality constraint has been used successfully in the context of feature learning and image classification [4]. Structured sparsity has been used for compressive sensing =-=[5]-=-. Other examples include hierarchical tree-based dictionary learning [6], submodular dictionary selection [7], and exploitation of self-similarity in images [8]. We propose a landmark-dependent hierar... |

124 | Proximal methods for sparse hierarchical dictionary learning
- Jenatton, Mairal, et al.
- 2010
(Show Context)
Citation Context ...ully in the context of feature learning and image classification [4]. Structured sparsity has been used for compressive sensing [5]. Other examples include hierarchical tree-based dictionary learning =-=[6]-=-, submodular dictionary selection [7], and exploitation of self-similarity in images [8]. We propose a landmark-dependent hierarchical beta process to address dictionary learning for data that are end... |

118 | Nonlinear learning using local coordinate coding,” NIPS
- Yu, Zhang, et al.
- 2009
(Show Context)
Citation Context ... may be improved by imposing additional prior information, beyond sparseness. For example, a locality constraint has been used successfully in the context of feature learning and image classification =-=[4]-=-. Structured sparsity has been used for compressive sensing [5]. Other examples include hierarchical tree-based dictionary learning [6], submodular dictionary selection [7], and exploitation of self-s... |

111 |
A theory for sampling signals from a union of subspaces
- Lu, Do
(Show Context)
Citation Context ...ied simply such that it is useful. Specifically, because natural images manifest segments and self-similarity, one may view the dictionary-learning framework within a union-of-subspaces setting [15], =-=[23]-=-. Each subspace, defined by a subset of the dictionary, represents a class of local structure within an image, and each subspace and associated data may be viewed as a low-rank matrix. The BP, DP and ... |

103 | Learning Multiscale Sparse Representations for Image and Video Restoration,” Multiscale Modeling
- Mairal, Sapiro, et al.
- 2008
(Show Context)
Citation Context ...ODUCTION A. Sparseness and Dictionary Learning Recently, there has been significant interest in sparse image representations, in the context of denoising and interpolation [1], [13], [24]–[26], [27], =-=[28]-=-, [31], compressive sensing (CS) [5], [12], and classification [40]. All of these applications exploit the fact that images may be sparsely represented in an appropriate dictionary. Most of the denois... |

89 | Exploiting structure in Wavelet-based Bayesian compressive sampling
- He, Carin
- 2009
(Show Context)
Citation Context ...ructure between the wavelet coefficients associated with natural images. Recently, researchers have utilized such structure to move beyond sparseness and achieve even better CS-inversion quality [2], =-=[19]-=-; in this paper, structural relationships and correlations between basis-function coefficients are accounted for. Additionally, there has been recent statistical research that has moved beyond sparsit... |

84 |
Sparsity and smoothness via the fused
- Tibshirani, Saunders, et al.
- 2005
(Show Context)
Citation Context ... between basis-function coefficients are accounted for. Additionally, there has been recent statistical research that has moved beyond sparsity and that are of interest in the context of CS inversion =-=[38]-=-. In the tests, we omit, for brevity, the algorithms in [2] and [19] yield performance that is comparable to the best results to the right in Fig. 7. Consequently, the imposition of structure in the f... |

78 | Nonparametric factor analysis with beta process priors
- Paisley, Carin
- 2009
(Show Context)
Citation Context ...ten over-complete dictionary matched to the signals of interest (e.g., images) [5–10, 12, 15]. The purpose of this paper is to perform dictionary learning using new non-parametric Bayesian technology =-=[16,17]-=-, that offers several advantages not found in earlier approaches, which have generally sought point estimates. This paper makes four main contributions: • The dictionary is learned using a beta proces... |

73 | Kernel stick-breaking processes
- Dunson, Park
- 2008
(Show Context)
Citation Context ... kernels are localized via learned “landmarks,” establishing links between data and landmark-dependent sparseness properties. The proposed model is related to the kernel stick breaking process (KSBP) =-=[13]-=- and Bayesian density regression (BDR) [14], although it is distinct from both. For example, the original KSBP construction focused primarily on covariate-dependent mixture models, and here we extend ... |

72 | Non-linear matrix factorization with Gaussian processes
- Lawrence, Urtasun
- 2009
(Show Context)
Citation Context ... random, recovery of the missing pixels corresponds to the matrix-completion problem. However, the matrix-completion literature is based on the assumption that the matrix of interest is low rank [6], =-=[22]-=-, [35]. Because the underlying dictionaries associated with natural images are typically over-complete, the assumption of a single low-rank matrix of pixel values is often inappropriate. While a direc... |

71 |
Wavelet shrinkage
- Donoho, Johnstone, et al.
- 1995
(Show Context)
Citation Context ... infers, as part of the same model, the noise variance from the image under test. There are existing methods for estimation of the noise variance, as a preprocessing step, e.g., via wavelet shrinkage =-=[10]-=-. However, it was shown in [42] that the denoising accuracy of methods like that in [13] can be sensitive to small errors in the estimated variance, which are likely to occur in practice. Additionally... |

69 | Bayesian Density Regression
- Dunson, Pillai, et al.
- 2007
(Show Context)
Citation Context ...ks,” establishing links between data and landmark-dependent sparseness properties. The proposed model is related to the kernel stick breaking process (KSBP) [13] and Bayesian density regression (BDR) =-=[14]-=-, although it is distinct from both. For example, the original KSBP construction focused primarily on covariate-dependent mixture models, and here we extend such ideas to a sparse factor analysis (SFA... |

68 | Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization
- Duarte-Carvajalino, Sapiro
- 2009
(Show Context)
Citation Context ...interpretation [4]. Of relevance for the current paper, there has recently been significant interest in sparse representations in the context of denoising, inpainting [5–10], compressive sensing (CS) =-=[11, 12]-=-, and classification [13]. All of these applications exploit the fact that most images may be sparsely represented in an appropriate dictionary. Most of the CS literature assumes “off-the-shelf” wavel... |

60 | Infinite sparse factor analysis and infinite independent components analysis
- Knowles, Ghahramani
- 2007
(Show Context)
Citation Context ...nary D ∈ ℜ n×K , with K → ∞; by inferring the number of columns of D that are required for accurate representation of x, the appropriate value of M is implicitly inferred (work has been considered in =-=[20, 21]-=- for the related but distinct application of factor analysis). We wish to also impose that α ∈ ℜ K is sparse, and therefore only a small fraction of the columns of D are used for representation of a g... |

57 | Variational Bayesian multinomial probit regression
- Girolami, Rogers
- 2006
(Show Context)
Citation Context ...compression and classification, both goals should be accounted for when designing D. For simplicity, we assume that the number of classes is NC = 2 (binary classification), with this readily extended =-=[23]-=- to NC > 2. Following [9], we may define a linear or bilinear classifier based on the sparse weights α and the associated data x (in the bilinear case), with this here implemented in the form of a pro... |

33 | Nonparametric Bayes conditional distribution modeling with variable selection
- Chung, Dunson
- 2009
(Show Context)
Citation Context ...ated to that discussed below, in [32], the concepts of learned dictionaries and beta-Bernoulli priors were not considered. Another related model, which employs a probit link function, is discussed in =-=[7]-=-. We augment the data as , where again represents pixel values from the th image patch, and represents the 2-D location of each patch. We wish to impose that proximate patches are more likely to be co... |

32 | The Infinite Hierarchical Factor Regression Model. Arxiv Preprint arXiv:0908.0570
- Rai, Daumé
- 2009
(Show Context)
Citation Context ...nary D ∈ ℜ n×K , with K → ∞; by inferring the number of columns of D that are required for accurate representation of x, the appropriate value of M is implicitly inferred (work has been considered in =-=[20, 21]-=- for the related but distinct application of factor analysis). We wish to also impose that α ∈ ℜ K is sparse, and therefore only a small fraction of the columns of D are used for representation of a g... |

26 | Submodular dictionary selection for sparse representation
- Krause, Cevher
- 2010
(Show Context)
Citation Context ...ng and image classification [4]. Structured sparsity has been used for compressive sensing [5]. Other examples include hierarchical tree-based dictionary learning [6], submodular dictionary selection =-=[7]-=-, and exploitation of self-similarity in images [8]. We propose a landmark-dependent hierarchical beta process to address dictionary learning for data that are endowed with an associated covariate. We... |

23 | Dependent hierarchical beta process for image interpolation and denoising - Zhou, Yang, et al. - 2011 |

22 | Nonparametric Bayesian models through probit stickbreaking processes
- Rodriguez, Dunson
(Show Context)
Citation Context ...se variance can also be non-stationary. • The spatial inter-relationships between different components in images are exploited by use of the Dirichlet process [18] and a probit stick-breaking process =-=[19]-=-. 1 • Using learned dictionaries, inferred off-line or in situ, the proposed approach yields CS performance that is markedly better than existing standard CS methods as applied to imagery. 2 Dictionar... |

15 | Self-similarity driven color demosaicking
- Buades, Coll, et al.
(Show Context)
Citation Context ...riate for image-processing problems, the framework may be modified to make it applicable. As one way in which the model may be modified, recall that images tend to possess significant self-similarity =-=[4]-=- and segments, implying that many B × B patches have similar structure. This suggests that there may be a clustering of the B ×B blocks, and that within each cluster the associated data can constitute... |

14 |
Robust principal component analysis? Submitted
- Candès, Li, et al.
- 2009
(Show Context)
Citation Context ...ased dependent HBP, or Landmark-dHBP. The proposed model employs a robustness term to model sparse spiky noise or localized data anomalies, related to robust principal component analysis (RPCA) [15], =-=[16]-=-. However, our model differs from RPCA in that the low-rank assumption on data is replaced with the richer covariatedependent union-of-subspace assumption, which is realized with a sparse factor analy... |

13 | Dependent Indian buffet processes
- Williamson, Orbanz, et al.
- 2010
(Show Context)
Citation Context ...on ignores relational information provided by covariates. A dependent IBP (dIBP) model has been introduced recently, with a hierarchical Gaussian process (GP) used to account for covariate dependence =-=[12]-=-. In the proposed model, rather than imposing relational information via a parametric covariance matrix, as in GP, we do so by employing a kernel-based construction. We introduce “landmarks” in the co... |

10 | The logistic stick breaking process
- Ren, Du, et al.
(Show Context)
Citation Context ...kely to be constituted in terms of similar columns of D 1 . To impose this information, we employ the probit stick-breaking process (PSBP). A logistic stick-breaking process is discussed in detail in =-=[34]-=-. We employ the closely related probit version here because it may be easily implemented in a Gibbs sampler. We note that while the method in [34] is related to that discussed below, in [34] the conce... |

9 | A Weighted Average of Sparse Representations is Better than the Sparsest One Alone - Elad, Yavheh - 2008 |

8 |
Nonparametric Bayesian models through probit stick-breaking processes
- A, Dunson
- 2009
(Show Context)
Citation Context ...se variance can also be non-stationary. • The spatial inter-relationships between different components in images are exploited by use of the Dirichlet process [18] and a probit stick-breaking process =-=[19]-=-. • Using learned dictionaries, inferred off-line or in situ, the proposed approach yields CS perfor1mance that is markedly better than existing standard CS methods as applied to imagery. 2 Dictionar... |

6 |
Compressive video sensors using multichannel imagers
- Shankar, Pitsianis, et al.
(Show Context)
Citation Context ...t for its potential to reduce the number of required measurements, it has the disadvantage of requiring the development of new classes of cameras. Such cameras are revolutionary and interesting [11], =-=[37]-=-, but there have been decades of previous research performed on development of pixel-based cameras, and it would be desirable if such cameras could be modified simply to perform compressive measuremen... |

2 | Universal sparse modeling
- Ramirez, Sapiro
- 2010
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Citation Context ...entary and alternative framework with respect to the more standard variational formulations. These can also be interpreted via statistical models with solutions obtained via MAP estimation, e.g., see =-=[32]-=- for an overview and new interpretation of this. Such probabilistic interpretations use models different than the ones here exploited, and as mentioned above, have to estimate critical parameters and ... |

1 |
Blind Compressed Sensing Technion—Israel Inst
- Gleichman, Eldar
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Citation Context ...AYESIAN DICTIONARY LEARNING FOR ANALYSIS OF IMAGES 131 consider the case for which the underlying dictionary is simultaneously learned with inversion (reconstruction), with this related to “blind” CS =-=[17]-=-. Finally, we design the CS projection matrix to be matched to the learned dictionary (when this is done offline), and demonstrate, as in [12], that in practice, this yields performance gains relative... |