Hessian Schatten-Norm Regularization for Linear Inverse Problems

Lefkimmiatis, Stamatios; Ward, John Paul; Unser, Michaël

doi:10.1109/tip.2013.2237919

Cited by 155 publications

(193 citation statements)

References 46 publications

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“…We hypothesize that this is because the TV regularization is relatively less effective on the rat brain because it is more complex (less piecewise constant) than the other datasets and because (compared to the Shepp3D dataset) it has a small amount of measurement noise. We believe these results could be improved by taking measurements with a smaller pixel size (thereby making the reconstruction more piecewise constant) or by trying another regularizer, e.g., the Hessian Schatten-norm [25], wavelets [40] …”

Section: Phantom Dataset Comparisonmentioning

confidence: 99%

“…For example, total variation (TV) regularization applied in a slice-by-slice way promotes each slice to be piecewise constant, while TV applied in 3D promotes the reconstruction volume to be piecewise constant. Or, the Hessian Schatten-norm regularization [25] can be used to penalize second-order derivatives. This should enable the 3D reconstruction to achieve a more favorable trade-off between dose reduction and reconstruction quality.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Fast 3D reconstruction method for differential phase contrast X-ray CT

McCann¹,

Nilchian²,

Stampanoni³

et al. 2016

Opt. Express

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We present a fast algorithm for fully 3D regularized X-ray tomography reconstruction for both traditional and differential phase contrast measurements. In many applications, it is critical to reduce the X-ray dose while producing high-quality reconstructions. Regularization is an excellent way to do this, but in the differential phase contrast case it is usually applied in a slice-by-slice manner. We propose using fully 3D regularization to improve the dose/quality trade-off beyond what is possible using slice-by-slice regularization. To make this computationally feasible, we show that the two computational bottlenecks of our iterative optimization process can be expressed as discrete convolutions; the resulting algorithms for computation of the X-ray adjoint and normal operator are fast and simple alternatives to regridding. We validate this algorithm on an analytical phantom as well as conventional CT and differential phase contrast measurements from two real objects. Compared to the slice-by-slice approach, our algorithm provides a more accurate reconstruction of the analytical phantom and better qualitative appearance on one of the two real datasets. (14), 2961-2964 (1996). 10. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature 384(6607), 335-338 (1996). #261788Received 6 11. U. Bonse and M. Hart, "An X-ray interferometer," Appl. Phys. Lett. 6(8), 155-156 (1965). 12. A. Momose, T. Takeda, Y. Itai, and K. Hirano, "Phase-contrast X-ray computed tomography for observing biological soft tissues," Nat. Med. 2(4), 473-475 (1996

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Section: Phantom Dataset Comparisonmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fast 3D reconstruction method for differential phase contrast X-ray CT

McCann¹,

Nilchian²,

Stampanoni³

et al. 2016

Opt. Express

Self Cite

View full text Add to dashboard Cite

show abstract

“…where · S ∞ is the ∞ -norm of the singular values of the corresponding matrix (for more details, we refer the reader to [11]). …”

Section: Image Reconstructionmentioning

confidence: 99%

“…For our regularization scheme λ max (RR T ) ≤ γ where γ = 8 for the TV regularization for two-dimensional problems, and its value is 64 for the HS regularization as computed in [11].…”

Section: Parameter Settingmentioning

confidence: 99%

Constrained regularized reconstruction of X-ray-DPCI tomograms with weighted-norm

Nilchian¹,

Vonesch²,

Lefkimmiatis³

et al. 2013

Opt. Express

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Abstract:In this paper we introduce a new reconstruction algorithm for X-ray differential phase-contrast Imaging (DPCI). Our approach is based on 1) a variational formulation with a weighted data term and 2) a variable-splitting scheme that allows for fast convergence while reducing reconstruction artifacts. In order to improve the quality of the reconstruction we take advantage of higher-order total-variation regularization. In addition, the prior information on the support and positivity of the refractive index is considered, which yields significant improvement. We test our method in two reconstruction experiments involving real data; our results demonstrate its potential for in-vivo and medical imaging.

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“…High order variational models thus can be applied to remedy these side effects. Among these is the second order total variation (SOTV) model [1,2,22,26,27]. Unlike the high order variational models, such as the Gaussian curvature [28], mean curvature [23,29], Euler's elastica [21] etc., the SOTV is a convex high order extension of the FOTV, which guarantees a global solution.…”

Section: Introductionmentioning

confidence: 99%

Introducing diffusion tensor to high order variational model for image reconstruction

Duan

Ward

Sibbett

et al. 2017

Digital Signal Processing

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The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.For more information, please contact eprints@nottingham.ac.uk JID:YDSPR AID:2153 /FLA [m5G; v1.219; Prn:17/07/2017; 11:02] Second order total variation (SOTV) models have advantages for image restoration over their first order counterparts including their ability to remove the staircase artefact in the restored image. However, such models tend to blur the reconstructed image when discretised for numerical solution [1][2][3][4][5]. To overcome this drawback, we introduce a new tensor weighted second order (TWSO) model for image restoration. Specifically, we develop a novel regulariser for the SOTV model that uses the Frobenius norm of the product of the isotropic SOTV Hessian matrix and an anisotropic tensor. We then adapt the alternating direction method of multipliers (ADMM) to solve the proposed model by breaking down the original problem into several subproblems. All the subproblems have closed-forms and can be solved efficiently. The proposed method is compared with state-of-the-art approaches such as tensor-based anisotropic diffusion, total generalised variation, and Euler's elastica. We validate the proposed TWSO model using extensive experimental results on a large number of images from the Berkeley BSDS500. We also demonstrate that our method effectively reduces both the staircase and blurring effects and outperforms existing approaches for image inpainting and denoising applications.

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Hessian Schatten-Norm Regularization for Linear Inverse Problems

Cited by 155 publications

References 46 publications

Fast 3D reconstruction method for differential phase contrast X-ray CT

Fast 3D reconstruction method for differential phase contrast X-ray CT

Constrained regularized reconstruction of X-ray-DPCI tomograms with weighted-norm

Introducing diffusion tensor to high order variational model for image reconstruction

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