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12
Stable principal component pursuit
- In Proc. of International Symposium on Information Theory
, 2010
"... We consider the problem of recovering a target matrix that is a superposition of low-rank and sparse components, from a small set of linear measurements. This problem arises in compressed sensing of structured high-dimensional signals such as videos and hyperspectral images, as well as in the analys ..."
Abstract
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Cited by 94 (3 self)
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We consider the problem of recovering a target matrix that is a superposition of low-rank and sparse components, from a small set of linear measurements. This problem arises in compressed sensing of structured high-dimensional signals such as videos and hyperspectral images, as well as in the analysis of transformation invariant low-rank structure recovery. We analyze the performance of the natural convex heuristic for solving this problem, under the assumption that measurements are chosen uniformly at random. We prove that this heuristic exactly recovers low-rank and sparse terms, provided the number of observations exceeds the number of intrinsic degrees of freedom of the component signals by a polylogarithmic factor. Our analysis introduces several ideas that may be of independent interest for the more general problem of compressed sensing and decomposing superpositions of multiple structured signals. 1
Simultaneously Structured Models with Application to Sparse and Low-rank Matrices
, 2014
"... The topic of recovery of a structured model given a small number of linear observations has been well-studied in recent years. Examples include recovering sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and low-rank matrices, among others. In various applications in signal p ..."
Abstract
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Cited by 41 (5 self)
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The topic of recovery of a structured model given a small number of linear observations has been well-studied in recent years. Examples include recovering sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and low-rank matrices, among others. In various applications in signal processing and machine learning, the model of interest is known to be structured in several ways at the same time, for example, a matrix that is simultaneously sparse and low-rank. Often norms that promote each individual structure are known, and allow for recovery using an order-wise optimal number of measurements (e.g., `1 norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to minimize a combination of such norms. We show that, surprisingly, if we use multi-objective optimization with these norms, then we can do no better, order-wise, than an algorithm that exploits only one of the present structures. This result suggests that to fully exploit the multiple structures, we need an entirely new convex relaxation, i.e. not one that is a function of the convex relaxations used for each structure. We then specialize our results to the case of sparse and low-rank matrices. We show that a nonconvex formulation of the problem can recover the model from very few measurements, which is on the order of the degrees of freedom of the matrix, whereas the convex problem obtained from a combination of the `1 and nuclear norms requires many more measurements. This proves an order-wise gap between the performance of the convex and nonconvex recovery problems in this case. Our framework applies to arbitrary structure-inducing norms as well as to a wide range of measurement ensembles. This allows us to give performance bounds for problems such as sparse phase retrieval and low-rank tensor completion.
Joint trace/tv norm minimization: a new efficient approach for spectral compressive imaging
- in IEEE International Conference on Image Processing
, 2012
"... In this paper we propose a novel and efficient model for compressed sensing of hyperspectral images. A large-size hyperspectral image can be subsampled by retaining only 3 % of its original size, yet robustly recovered using the new approach we present here. Our reconstruction approach is based on m ..."
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Cited by 7 (0 self)
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In this paper we propose a novel and efficient model for compressed sensing of hyperspectral images. A large-size hyperspectral image can be subsampled by retaining only 3 % of its original size, yet robustly recovered using the new approach we present here. Our reconstruction approach is based on minimizing a convex functional which penalizes both the trace norm and the TV norm of the data matrix. Thus, the solution tends to have a simultaneous low-rank and piecewise smooth structure: the two important priors explaining the underlying correlation structure of such data. Through simulations we will show our approach significantly enhances the conventional compression rate-distortion tradeoffs. In particular, in the strong undersampling regimes our method outperforms the standard TV denoising image recovery scheme by more than 17dB in the reconstruction MSE. Index Terms — Hyperspectral images, Compressed sensing,
A New Pansharpening Method Based on Spatial and Spectral Sparsity Priors
, 2014
"... Abstract — The development of multisensor systems in recent years has led to great increase in the amount of available remote sensing data. Image fusion techniques aim at inferring high quality images of a given area from degraded versions of the same area obtained by multiple sensors. This paper fo ..."
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Cited by 4 (2 self)
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Abstract — The development of multisensor systems in recent years has led to great increase in the amount of available remote sensing data. Image fusion techniques aim at inferring high quality images of a given area from degraded versions of the same area obtained by multiple sensors. This paper focuses on pansharpening, which is the inference of a high spatial resolution multispectral image from two degraded versions with complementary spectral and spatial resolution characteristics: 1) a low spatial resolution multispectral image and 2) a high spatial resolution panchromatic image. We introduce a new variational model based on spatial and spectral sparsity priors for the fusion. In the spectral domain, we encourage low-rank structure, whereas in the spatial domain, we promote sparsity on the local differences. Given the fact that both panchromatic and multispectral images are integrations of the underlying
Tensor-based formulation and nuclear norm regularization for multi-energy computed tomography
, 2014
"... ..."
A NOVEL COMPRESSIVE SENSING APPROACH TO SIMULTANEOUSLY ACQUIRE COLOR AND NEAR-INFRARED IMAGES ON A SINGLE SENSOR
"... Sensors of most digital cameras are made of silicon that is inherently sensitive to both the visible and near-infrared parts of the electromagnetic spectrum. In this paper, we address the problem of color and NIR joint acquisition. We propose a framework for the joint acquisition that uses only a si ..."
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Cited by 1 (0 self)
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Sensors of most digital cameras are made of silicon that is inherently sensitive to both the visible and near-infrared parts of the electromagnetic spectrum. In this paper, we address the problem of color and NIR joint acquisition. We propose a framework for the joint acquisition that uses only a single silicon sensor and a slightly modified version of the Bayer color-filter array that is already mounted in most color cameras. Implementing such a design for an RGB and NIR joint acquisition system requires minor changes to the hardware of commercial color cameras. One of the important differences between this design and the conventional color camera is the post-processing applied to the captured values to reconstruct full resolution images. By using a CFA similar to Bayer, the sensor records a mixture of NIR and one color channel in each pixel. In this case, separating NIR and color channels in different pixels is equivalent to solving an under-determined system of linear equations. To solve this problem, we propose a novel algorithm that uses the tools developed in the field of compressive sensing. Our method results in high-quality RGB and NIR images (the average PSNR of more than 30 dB for the reconstructed images) and shows a promising path towards RGB and NIR cameras. Index Terms — Color filter array, demosaicing, the Bayer CFA, near-infrared, compressive sensing, sparse decomposition.
A non-local structure tensor based approach for multicomponent image recovery problems
- IEEE Trans. Image Process
"... Non-Local Total Variation (NLTV) has emerged as a useful tool in variational methods for image recovery problems. In this paper, we extend the NLTV-based regularization to multicomponent images by taking advantage of the Structure Tensor (ST) resulting from the gradient of a multicomponent image. Th ..."
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Cited by 1 (0 self)
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Non-Local Total Variation (NLTV) has emerged as a useful tool in variational methods for image recovery problems. In this paper, we extend the NLTV-based regularization to multicomponent images by taking advantage of the Structure Tensor (ST) resulting from the gradient of a multicomponent image. The proposed approach allows us to penalize the non-local variations, jointly for the different components, through various `1,p matrix norms with p ≥ 1. To facilitate the choice of the hyper-parameters, we adopt a constrained convex optimization approach in which we minimize the data fidelity term subject to a constraint involving the ST-NLTV regularization. The resulting convex optimization problem is solved with a novel epigraphical projection method. This formulation can be efficiently implemented thanks to the flexibility offered by recent primal-dual proximal algorithms. Experiments are carried out for multispectral and hyperspectral images. The results demonstrate the interest of introducing a non-local structure tensor regularization and show that the proposed approach leads to significant improvements in terms of convergence speed over current state-of-the-art methods. 1
Journal of Machine Learning Research (2014) Submitted; Published Model Consistency of Partly Smooth Regularizers
, 2014
"... HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Simultaneously Structured Models with Application to Sparse and Low-rank Matrices
, 2012
"... The topic of recovery of a structured model given a small number of linear observations has been well-studied in recent years. Examples include recovering sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and low-rank matrices, among others. In various applications in signal p ..."
Abstract
- Add to MetaCart
The topic of recovery of a structured model given a small number of linear observations has been well-studied in recent years. Examples include recovering sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and low-rank matrices, among others. In various applications in signal processing and machine learning, the model of interest is known to be structured in several ways at the same time, for example, a matrix that is simultaneously sparse and low-rank. An important application is the sparse phase retrieval problem, where the goal is to recover a sparse signal from phaseless measurements. In machine learning, the problem comes up when combining several regularizers that each promote a certain desired structure. Often penalties (norms) that promote each individual structure are known and yield an order-wise optimal number of measurements (e.g., ℓ1 norm for sparsity, nuclear norm for matrix rank), soit isreasonabletominimize acombinationofsuchnorms. We showthat, surprisingly,if we use multi-objective optimization with the individual norms, then we can do no better, orderwise, than an algorithm that exploits only one of the several structures. This result suggests that to fully exploit the multiple structures, we need an entirely new convex relaxation, i.e., not one that is a function of the convex relaxations used for each structure. We then specialize our results to the case of sparse and low-rank matrices. We show that a nonconvex formulation of the problem can recover the model from very few measurements, on the order of the degrees of freedom of the matrix, whereas the convex problem obtained from a combination of the ℓ1 and nuclear norms requires many more measurements. This proves an order-wise gap between the performance of the convex and nonconvex recovery problems in this case.
