Anisotropy Preserving DTI Processing

Collard, Anne; Bonnabel, Silvère; Phillips, Christophe; Sepulchre, Rodolphe

doi:10.1007/s11263-013-0674-4

Cited by 13 publications

(32 citation statements)

References 56 publications

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“…The validity of the specific eigenvector homogeneity criterion for assessing the perturbation of cardiac tissue directional regularity due to disease has been tested elsewhere (Giannakidis et al, 2012). (ii) There exist few studies (Frindel et al, 2009; Yang et al, 2012; Collard et al, 2014; Zhou et al, 2015) that used fractional anisotropy (FA) to compare the various diffusion tensor distance functions. However, such a criterion is unfairly prejudiced, as it represents the Euclidean distance of each tensor from the fully isotropic tensor.…”

Section: Introductionmentioning

confidence: 99%

On the averaging of cardiac diffusion tensor MRI data: the effect of distance function selection

Giannakidis¹,

Melkus²,

Yang³

et al. 2016

Phys. Med. Biol.

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Diffusion tensor magnetic resonance imaging (DT-MRI) allows a unique insight into the microstructure of highly-directional tissues. The selection of the most proper distance function for the space of diffusion tensors is crucial in enhancing the clinical application of this imaging modality. Both linear and nonlinear metrics have been proposed in the literature over the years. The debate on the most appropriate DT-MRI distance function is still ongoing. In this paper, we presented a framework to compare the Euclidean, affine-invariant Riemannian and log-Euclidean metrics using actual high-resolution DT-MRI rat heart data. We employed temporal averaging at the diffusion tensor level of three consecutive and identically-acquired DT-MRI datasets from each of five rat hearts as a means to rectify the background noise-induced loss of myocyte directional regularity. This procedure is applied here for the first time in the context of tensor distance function selection. When compared with previous studies that used a different concrete application to juxtapose the various DT-MRI distance functions, this work is unique in that it combined the following: (i) Metrics were judged by quantitative –rather than qualitative– criteria, (ii) the comparison tools were non-biased, (iii) a longitudinal comparison operation was used on a same-voxel basis. The statistical analyses of the comparison showed that the three DT-MRI distance functions tend to provide equivalent results. Hence, we came to the conclusion that the tensor manifold for cardiac DT-MRI studies is a curved space of almost zero curvature. The signal to noise ratio dependence of the operations was investigated through simulations. Finally, the “swelling effect” occurrence following Euclidean averaging was found to be too unimportant to be worth consideration.

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Section: Introductionmentioning

confidence: 99%

On the averaging of cardiac diffusion tensor MRI data: the effect of distance function selection

Giannakidis¹,

Melkus²,

Yang³

et al. 2016

Phys. Med. Biol.

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“…This advantage in interpretation has not been found in popular geometric frameworks such as [24,11,3,10,6]. The approaches in [5,8,9,32] also explicitly use rotation of directions and many scaling-rotation curves are very similar to the deformation paths given in [9,32]. We defer the discussion on the difference between our framework and those in [9,32] to Section 5.2.…”

mentioning

confidence: 95%

“…For Sym + (3), other authors [9,32] have proposed dissimilarity-measures and interpolation schemes based on the same general idea as ours, i.e., separating the scaling and rotation of SPD matrices. Their deformations of SPD matrices can be similar to ours in many cases, thus enjoying similar interpretability.…”

mentioning

confidence: 99%

“…Their deformations of SPD matrices can be similar to ours in many cases, thus enjoying similar interpretability. But while [9,32] mainly focused on the p = 3 case, our work is more flexible by allowing unordered and equal eigenvalues. We discuss the importance of this later in Section 3.…”

mentioning

confidence: 99%

“…2.1 illustrates discretized trajectories of scaling-rotation curves in Sym + (3), visualized by the corresponding ellipsoids. These curves in general do not coincide with straight lines or geodesics in other geometric frameworks such as [31,21,24,17,11,3,10,9,32]. In section 3, we introduce a Riemannian metric which reproduces these scaling-rotation curves as images of geodesics.…”

mentioning

confidence: 99%

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Scaling-Rotation Distance and Interpolation of Symmetric Positive-Definite Matrices

Jung¹,

Schwartzman²,

Groisser³

2015

SIAM J. Matrix Anal. & Appl.

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We introduce a new geometric framework for the set of symmetric positive-definite (SPD) matrices, aimed to characterize deformations of SPD matrices by individual scaling of eigenvalues and rotation of eigenvectors of the SPD matrices. To characterize the deformation, the eigenvalue-eigenvector decomposition is used to find alternative representations of SPD matrices, and to form a Riemannian manifold so that scaling and rotations of SPD matrices are captured by geodesics on this manifold. The problems of non-unique eigen-decompositions and eigenvalue multiplicities are addressed by finding minimal-length geodesics, which gives rise to a distance and an interpolation method for SPD matrices. Computational procedures to evaluate the minimal scalingrotation deformations and distances are provided for the most useful cases of 2 × 2 and 3 × 3 SPD matrices. In the new geometric framework, minimal scaling-rotation curves interpolate eigenvalues at constant logarithmic rate, and eigenvectors at constant angular rate. In the context of diffusion tensor imaging, this results in better behavior of the trace, determinant and fractional anisotropy of interpolated SPD matrices in typical cases.

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Structure‐preserving invariant interpolation schemes for invertible second‐order tensors

Satheesh,

Schmidt,

Wall

et al. 2023

Numerical Meth Engineering

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Tensor interpolation is an essential step for tensor data analysis in various fields of application and scientific disciplines. In the present work, novel interpolation schemes for general, that is, symmetric or non‐symmetric, invertible square tensors are proposed. Critically, the proposed schemes rely on a combined polar and spectral decomposition of the tensor data , followed by an individual interpolation of the two rotation tensors and and the positive definite diagonal eigenvalue tensor resulting from this decomposition. Two different schemes are considered for a consistent rotation interpolation within the special orthogonal group , either based on relative rotation vectors or quaternions. For eigenvalue interpolation, three different schemes, either based on the logarithmic weighted average, moving least squares or logarithmic moving least squares, are considered. It is demonstrated that the proposed interpolation procedure preserves the structure of a tensor, that is, and remain orthogonal tensors and remains a positive definite diagonal tensor during interpolation, as well as scaling and rotational invariance (objectivity). Based on selected numerical examples considering the interpolation of either symmetric or non‐symmetric tensors, the proposed schemes are compared to existing approaches such as Euclidean, Log‐Euclidean, Cholesky and Log‐Cholesky interpolation. In contrast to these existing methods, the proposed interpolation schemes result in smooth and monotonic evolutions of tensor invariants such as determinant, trace, fractional anisotropy (FA), and Hilbert's anisotropy (HA). Moreover, a consistent spatial convergence behavior is confirmed for first‐ and second‐order realizations of the proposed schemes. The present work is mainly motivated by the frequently occurring necessity for remeshing or mesh adaptivity when applying the finite element method to complex problems of nonlinear continuum mechanics with inelastic constitutive behavior, which requires the consistent interpolation of tensor‐valued history data for the transfer between different meshes. However, the proposed schemes are very general in nature and suitable for the interpolation of general invertible second‐order square tensors independent of the specific application.

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Anisotropy Preserving DTI Processing

Cited by 13 publications

References 56 publications

On the averaging of cardiac diffusion tensor MRI data: the effect of distance function selection

On the averaging of cardiac diffusion tensor MRI data: the effect of distance function selection

Scaling-Rotation Distance and Interpolation of Symmetric Positive-Definite Matrices

Structure‐preserving invariant interpolation schemes for invertible second‐order tensors

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