Total Generalized Variation in Diffusion Tensor Imaging

Valkonen, Tuomo; Bredies, Kristian; Knoll, Florian

doi:10.1137/120867172

Cited by 125 publications

(110 citation statements)

References 32 publications

(26 reference statements)

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“…One proposed approach for the satisfaction of this constraint is that of log-Euclidean metrics [3]. This approach has several theoretically desirable aspects, but some practical shortcomings [57]. Special Perona-Malik-type constructions on Riemannian manifolds can also be used to maintain the structure of the tensor field [14,54].…”

Section: N )mentioning

confidence: 99%

“…Namely, we follow up on the work in [55,[57][58][59] on the application of total generalised variation regularisation [9] to DTI. We should note that in all of these works, the fidelity function was the ROF-type [45] L 2 fidelity.…”

Section: N )mentioning

confidence: 99%

“…Either way, significant complexity is introduced into the model, and for the present work, we are content with the simple L 2 model. We may also consider, as is often the case, and as was done with TGV in [57], the linearised model…”

Section: The Forward Modelmentioning

confidence: 99%

“…Following [11,57], the second-order variant may be defined with the differentiation cascade formulation for symmetric tensor fields u ∈…”

Section: Choice Of the Regulariser Rmentioning

confidence: 99%

“…and R = α TV in the discretised setting, we refer to [57,58]; here it suffices to note that for R = α TV, we have K 0 = α E and R * 0 (φ) is the indicator function of the dual ball {φ | sup x φ(x) F ≤ 1}. Thus the resolvent (I + τ ∂ R * 0 ) −1 becomes a projection to the dual ball.…”

Section: (βα)mentioning

confidence: 99%

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Diffusion Tensor Imaging with Deterministic Error Bounds

2016

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Errors in the data and the forward operator of an inverse problem can be handily modelled using partial order in Banach lattices. We present some existing results of the theory of regularisation in this novel framework, where errors are represented as bounds by means of the appropriate partial order. We apply the theory to diffusion tensor imaging, where correct noise modelling is challenging: it involves the Rician distribution and the non-linear Stejskal-Tanner equation. Linearisation of the latter in the statistical framework would complicate the noise model even further. We avoid this using the error bounds approach, which preserves simple error structure under monotone transformations.

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Section: N )mentioning

confidence: 99%

Section: N )mentioning

confidence: 99%

Section: The Forward Modelmentioning

confidence: 99%

“…Following [11,57], the second-order variant may be defined with the differentiation cascade formulation for symmetric tensor fields u ∈…”

Section: Choice Of the Regulariser Rmentioning

confidence: 99%

Section: (βα)mentioning

confidence: 99%

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Diffusion Tensor Imaging with Deterministic Error Bounds

2016

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Ultrafast 3D Bloch–Siegert B‐mapping using variational modeling

Lesch

Schlöegl

Höller

et al. 2018

Magnetic Resonance in Med

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Purpose Highly accelerated B ‐mapping based on the Bloch–Siegert shift to allow 3D acquisitions even within a brief period of a single breath‐hold. Theory and Methods The B dependent Bloch–Siegert phase shift is measured within a highly subsampled 3D‐volume and reconstructed using a two‐step variational approach, exploiting the different spatial distribution of morphology and B ‐field. By appropriate variable substitution the basic non‐convex optimization problem is transformed in a sequential solution of two convex optimization problems with a total generalized variation (TGV) regularization for the morphology part and a smoothness constraint for the B ‐field. The method is evaluated on 3D in vivo data with retro‐ and prospective subsampling. The reconstructed B ‐maps are compared to a zero‐padded low resolution reconstruction and a fully sampled reference. Results The reconstructed B ‐field maps are in high accordance to the reference for all measurements with a mean error below 1% and a maximum of about 4% for acceleration factors up to 100. The minimal error for different sampling patterns was achieved by sampling a dense region in k ‐space center with acquisition times of around 10–12 s for 3D‐acquistions. Conclusions The proposed variational approach enables highly accelerated 3D acquisitions of Bloch–Siegert data and thus full liver coverage in a single breath hold.

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A model‐based reconstruction for undersampled radial spin‐echo DTI with variational penalties on the diffusion tensor

et al. 2015

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Radial spin echo diffusion imaging allows motion-robust imaging of tissues with very low T2 values like articular cartilage with high spatial resolution and signal-to-noise ratio (SNR). However, in vivo measurements are challenging due to the significantly slower data acquisition speed of spin-echo sequences and the less efficient k-space coverage of radial sampling, which raises the demand for accelerated protocols by means of undersampling. This work introduces a new reconstruction approach for undersampled DTI. A model-based reconstruction implicitly exploits redundancies in the diffusion weighted images by reducing the number of unknowns in the optimization problem and compressed sensing is performed directly in the target quantitative domain by imposing a Total Variation (TV) constraint on the elements of the diffusion tensor. Experiments were performed for an anisotropic phantom and the knee and brain of healthy volunteers (3 and 2 volunteers, respectively). Evaluation of the new approach was conducted by comparing the results to reconstructions performed with gridding, combined parallel imaging and compressed sensing, and a recently proposed model-based approach. The experiments demonstrated improvement in terms of reduction of noise and streaking artifacts in the quantitative parameter maps as well as a reduction of angular dispersion of the primary eigenvector when using the proposed method, without introducing systematic errors into the maps. This may enable an essential reduction of the acquisition time in radial spin echo diffusion tensor imaging without degrading parameter quantification and/or SNR.

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Total Generalized Variation in Diffusion Tensor Imaging

Cited by 125 publications

References 32 publications

Diffusion Tensor Imaging with Deterministic Error Bounds

Diffusion Tensor Imaging with Deterministic Error Bounds

Ultrafast 3D Bloch–Siegert B‐mapping using variational modeling

A model‐based reconstruction for undersampled radial spin‐echo DTI with variational penalties on the diffusion tensor

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