Error analysis of reweighted<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math>greedy algorithm for noisy reconstruction

Zhu, Jiehua; Li, Xiezhang; Arroyo, Fangjun; Arroyo, Edward

doi:10.1016/j.cam.2015.02.038

Cited by 2 publications

(2 citation statements)

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“…As ||f || T V is the l 1 -norm of the gradient of f , the TV minimization is also known as the l 1 -minimization method. A generalized l 1 greedy algorithm in the compressed sensing framework [10] was introduced to incorporate the threshold feature of the l 1 greedy algorithm [11] and the inversely proportional weights used in the reweighted l 1 -minimization algorithm for CT. Error analysis of reweighted l 1 greedy algorithm for noisy reconstruction was studied in [12]. The reweighted l 1 -norm was incorporated into non-local TV minimization to streghten the structural details and the tissue contrast and thus to enhance the CT reconstructing performance [13].…”

Section: Introductionmentioning

confidence: 99%

A Reweighted Total Variation Algorithm with the Alternating Direction Method for Computed Tomography

Statesboro¹,

Ga²

2019

ACT

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Section: Introductionmentioning

confidence: 99%

A Reweighted Total Variation Algorithm with the Alternating Direction Method for Computed Tomography

Statesboro¹,

Ga²

2019

ACT

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“…Using v q q has advantages when seeking sparse solutions, see for example [14,24,19,36,44], but at the expense of making the minimization problem non-convex. We show that to solve (8), the penalty approach in Chen, Lu and Pong [13] with a starting point obtained from the re-weighted 1 minimization [10,15,18,51,49] is promising.…”

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Spherical Designs and Nonconvex Minimization for Recovery of Sparse Signals on the Sphere

Chen

Womersley

2018

SIAM J. Imaging Sci.

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This paper considers the use of spherical designs and non-convex minimization for recovery of sparse signals on the unit sphere S 2. The available information consists of low order, potentially noisy, Fourier coefficients for S 2. As Fourier coefficients are integrals of the product of a function and spherical harmonics, a good cubature rule is essential for the recovery. A spherical t-design is a set of points on S 2 , which are nodes of an equal weight cubature rule integrating exactly all spherical polynomials of degree ≤ t. We will show that a spherical t-design provides a sharp error bound for the approximation signals. Moreover, the resulting coefficient matrix has orthonormal rows. In general the 1 minimization model for recovery of sparse signals on S 2 using spherical harmonics has infinitely many minimizers, which means that most existing sufficient conditions for sparse recovery do not hold. To induce the sparsity, we replace the 1-norm by the q-norm (0 < q < 1) in the basis pursuit denoise model. Recovery properties and optimality conditions are discussed. Moreover, we show that the penalty method with a starting point obtained from the reweighted 1 method is promising to solve the q basis pursuit denoise model. Numerical performance on nodes using spherical t-designs and t-designs (extremal fundamental systems) are compared with tensor product nodes. We also compare the basis pursuit denoise problem with q = 1 and 0 < q < 1.

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Error analysis of reweightedl1greedy algorithm for noisy reconstruction

Cited by 2 publications

References 22 publications

A Reweighted Total Variation Algorithm with the Alternating Direction Method for Computed Tomography

A Reweighted Total Variation Algorithm with the Alternating Direction Method for Computed Tomography

Spherical Designs and Nonconvex Minimization for Recovery of Sparse Signals on the Sphere

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