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A Riemannian Bayesian Framework for Estimating Diffusion Tensor Images

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Abstract

Diffusion tensor magnetic resonance imaging (DT-MRI) is a non-invasive imaging technique allowing to estimate the molecular self-diffusion tensors of water within surrounding tissue. Due to the low signal-to-noise ratio of magnetic resonance images, reconstructed tensor images usually require some sort of regularization in a post-processing step. Previous approaches are either suboptimal with respect to the reconstruction or regularization step. This paper presents a Bayesian approach for simultaneous reconstruction and regularization of DT-MR images that allows to resolve the disadvantages of previous approaches. To this end, estimation theoretical concepts are generalized to tensor valued images that are considered as Riemannian manifolds. Doing so allows us to derive a maximum a posteriori estimator of the tensor image that considers both the statistical characteristics of the Rician noise occurring in MR images as well as the nonlinear structure of tensor valued images. Experiments on synthetic data as well as real DT-MRI data validate the advantage of considering both statistical as well as geometrical characteristics of DT-MRI.

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Notes

  1. Diffusion tensor images belong to the more general class of tensor-valued images where tensors might represent different information, e.g. orientation in case of structure tensors.

  2. cf. Helgason (1978) for an overview about Riemannian manifolds.

  3. Here we assume the curve to be in U. Otherwise the curve has to be divided into several parts each represented by its own chart and the overall curve length is obtained by summing up the curve length of its parts.

  4. If not otherwise stated we use the standard matrix elements as global coordinates such that the corresponding chart becomes the identity. In favor of uncluttered notation we make no difference between manifold elements and their matrix representation.

  5. We assume the exponential as well as the logarithmic map to be globally one to one as it is the case for the tensor manifold considered in this paper.

  6. Some medical NMR imaging systems apply multiple, e.g. \(\iota \) parallel working signal detectors to speed up the recording process (Roemer et al. 1990). As a result one obtains \(\iota \) complex valued NMR images corrupted by zero mean additive Gaussian noise having all the same standard deviation (Constantinides et al. 1997; Koay and Basser 2006). The NMR magnitude image is obtained by \(S_j(x_k)= \sqrt{\sum _{i=1}^\iota I_{j k i}^2+R_{j k i}^2}\) and follows a non-central Chi distribution (Constantinides et al. 1997) containing the Rician distribution as a special case (\(\iota =1\)). The likelihood model for multiple detector systems can straightforwardly be obtained from our model by exchanging the Rician distribution with the non-central Chi distribution.

  7. The Rician distribution becomes the Rayleigh distribution in that case.

  8. except for the pathologic cases, i.e. \(\hat{A}_{0 m}=0\) and \(S_{j m}=0\) for all m and j.

  9. Not to be confused with the elements of the diffusion tensor of the DTI.

  10. If an index variable occurs at least twice, it is understood to sum over its full range.

  11. The matrix entries parameterize the positive definite tensor.

  12. Lapack is freely-available at http://www.netlib.org/lapack/.

  13. The analytical solutions of matrix functions are in fact exact. However, due to inevitable numerical inaccuracies occurring in their practical computation its sensitiveness to these inaccuracies have to be taken under consideration.

  14. A detailed derivation of (84) can be found in the supplementary material.

  15. Note that there exists no energy formulation for nonlinear anisotropic diffusion, hence no MRF can be constructed.

  16. cmp. with Menzel (2002) for a detailed description of the imaging system and reconstruction method.

  17. We use the shorthand naming convention ‘noise-model’—‘regularization geometry’, thus e.g. ‘Gaussian–Riemannian’ is the Gaussian noise assumption based method of Lenglet et al. (2006) combined with our isotropic nonlinear Riemannian spatial regularization.

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  1. IBG-2: Plant Sciences, Forschungszentrum Jülich, 52425, Jülich, Germany

    Kai Krajsek & Hanno Scharr

  2. GE Global Research, Diagnostics & Biomedical Technologies, 85748, Garching, Germany

    Marion I. Menzel

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Krajsek, K., Menzel, M.I. & Scharr, H. A Riemannian Bayesian Framework for Estimating Diffusion Tensor Images. Int J Comput Vis 120, 272–299 (2016). https://doi.org/10.1007/s11263-016-0909-2

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