The Split Bregman Method for L1-Regularized Problems

Goldstein, Tom; Osher, Stanley

doi:10.1137/080725891

Cited by 3,942 publications

(3,304 citation statements)

References 12 publications

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“…Incorporating sensitivity maps from multiple channels helps balance data fidelity and regularization. However, due to the presence of sensitivity map S j , the fast numerical algorithms developed in (12,16,(18)(19)(20)(21) for single-channel MRI reconstruction cannot be applied directly. This results in a much longer reconstruction time to solve model [2], which hinders its clinical applicability.…”

Section: Ppi With Sparsity Constraintsmentioning

confidence: 99%

A rapid and robust numerical algorithm for sensitivity encoding with sparsity constraints: Self‐feeding sparse SENSE

Huang

Chen

Yin

et al. 2010

Magnetic Resonance in Med

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The method of enforcing sparsity during magnetic resonance imaging reconstruction has been successfully applied to partially parallel imaging (PPI) techniques to reduce noise and artifact levels and hence to achieve even higher acceleration factors. However, there are two major problems in the existing sparsity-constrained PPI techniques: speed and robustness. By introducing an auxiliary variable and decomposing the original minimization problem into two subproblems that are much easier to solve, a fast and robust numerical algorithm for sparsity-constrained PPI technique is developed in this work. The specific implementation for a conventional Cartesian trajectory data set is named self-feeding Sparse Sensitivity Encoding (SENSE). The computational cost for the proposed method is two conventional SENSE reconstructions plus one spatially adaptive image denoising procedure. With reconstruction time approximately doubled, images with a much lower root mean square error (RMSE) can be achieved at high acceleration factors. Using a standard eight-channel head coil, a net acceleration factor of 5 along one dimension can be achieved with low RMSE. Furthermore, the algorithm is insensitive to the choice of parameters. This work improves the clinical applicability of SENSE at high acceleration factors. Magn Reson Med 64:1078-1088, 2010. V C 2010 WileyLiss, Inc.Key words: partially parallel imaging; g-factor; sparsity constraint; prior information; compressed sensing; numerical algorithm Partially parallel imaging (PPI) techniques (1,2) are being routinely used to achieve increased image resolution, decreased motion artifacts, and shorter scan time in magnetic resonance imaging (MRI). However, PPI techniques reduce acquisition time at the cost of a reduction in signal-to-noise ratio (SNR). With an increase in the acceleration factor, the increase in noise and artifact levels can become significant, thereby reducing the diagnostic quality of the image. To avoid significant artifacts and noise amplification, the acceleration factor, R, is typically restricted to values far below the theoretical limit (i.e., the number of coil elements). For example, acceleration factors are no more than 3 in most clinical exams for the widely available eight-channel head coil. To reduce the noise and artifact levels, techniques of enforcing sparsity in PPI reconstruction have recently been proposed (3-6), where ''sparsity'' means that the to-be-reconstructed image has a sparse representation in a known and fixed mathematical transform domain (7). These techniques enforce the sparsities of the to-be-reconstructed image in finite difference domain and wavelet transform domain by minimizing its total variation (TV) norm and the L 1 norm of its wavelet transform. The sparsity constraints play an important role in lowering the noise/artifact levels of the reconstructed images. Meanwhile, a data fidelity term is used to preserve image contrast and resolution. Sparsity constraints and data fidelity are balanced by the parameters that are the weigh...

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Section: Ppi With Sparsity Constraintsmentioning

confidence: 99%

A rapid and robust numerical algorithm for sensitivity encoding with sparsity constraints: Self‐feeding sparse SENSE

Huang

Chen

Yin

et al. 2010

Magnetic Resonance in Med

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“…Since the matrix K of these examples can be diagonalized by Fourier transform, therefore our algorithms can be effectively carried out. We compare our methods with 1 l [14], 12 ll − [3], and 12 ll α − [4]. In order to verify the proposed methods thoroughly, we use 14 test images from [15] including synthesis images and natural images which are shown in Fig.…”

Section: Methodsmentioning

confidence: 99%

Hybrid Model and Split Bregman Iteration Algorithm for Image Denoising

Yu¹,

Zhang²

2018

Proceedings of the 2nd International Forum on Management, Education and Information Technology Application (IFMEITA 2017)

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Keywords: hybrid model; difference of convex algorithm; split Bregman iteration; image denoising. Abstract.Although difference of convex model has attracted many research efforts due to its superior performance for image processing, no attention has focused on robust data fidelity for this model. In this paper, we propose a novel model, which combines the 1 l and 2 l fidelity terms with a weighted difference of anisotropic and isotropic total variation (TV). Since our model takes a new difference form of convex terms, we employ difference of convex algorithm (DCA). In this paper, we adopt split Bregman iteration (SBI) to solve each DCA subproblem of the proposed model. Image denoising verifies the convergence of optimal solution and monotone decreasing of objective function. Experimental results on image denoising demonstrate that the proposed methods outperform other competing methods in terms of quantitative criteria and perceptual quality.

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“…In light of such higher spatial resolution techniques, we develop an improved method for elastic stress source recovery using ideas developed for image segmentation [15]. This class of methods relies on optimization that uses "compressive" L 1 regularization terms in the objective function that favor solutions that are compactly supported [16,17].…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of Cell Focal Adhesions using Physical Constraints and Compressive Regularization

Chang

Liu²,

Chou³

2017

Biophysical Journal

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We develop a method to reconstruct, from measured displacements of an underlying elastic substrate, the spatially dependent forces that cells or tissues impart on it. Given newly available high-resolution images of substrate displacements, it is desirable to be able to reconstruct small scale, compactly supported focal adhesions which are often localized and exist only within the footprint of a cell. In addition to the standard quadratic data mismatch terms that define least-squares fitting, we motivate a regularization term in the objective function that penalizes vectorial invariants of the reconstructed surface stress while preserving boundaries. We solve this inverse problem by providing a numerical method for setting up a discretized inverse problem that is solvable by standard convex optimization techniques. By minimizing the objective function subject to a number of important physically motivated constraints, we are able to efficiently reconstruct stress fields with localized structure from simulated and experimental substrate displacements. Our method incorporates the exact solution for the stress tensor accurate to first-order finite-differences and motivates the use of distance-based cut-offs for data inclusion and problem sparsification.

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The Split Bregman Method for L1-Regularized Problems

Cited by 3,942 publications

References 12 publications

A rapid and robust numerical algorithm for sensitivity encoding with sparsity constraints: Self‐feeding sparse SENSE

A rapid and robust numerical algorithm for sensitivity encoding with sparsity constraints: Self‐feeding sparse SENSE

Hybrid Model and Split Bregman Iteration Algorithm for Image Denoising

Reconstruction of Cell Focal Adhesions using Physical Constraints and Compressive Regularization

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