200 research outputs found
4TH ORDER DIFFUSION TENSOR ESTIMATION AND APPLICATION
International audienceHistorically Diffusion MRI started with Diffusion Tensor Imaging (DTI), which boosted the development of schemes for estimating positive definite tensors but were limited by their inability to detect fiber-crossings. Recent HARDI techniques have overcome that shortcoming with a plethora of new reconstruction schemes such as radial basis functions, Spherical Harmonics (SH), Higher Order Tensors (HOT), etc. It is appropriate, therefore, to explore HOT while leveraging the extensive framework already established for classical DTI. In this work, we propose a review and a comparison of the existing methods and an extension to the Riemannian framework to the space of 4 th order diffusion tensors
4th Order Symmetric Tensors and Positive ADC Modelling
International audienceHigh Order Cartesian Tensors (HOTs) were introduced in Generalized DTI (GDTI) to overcome the limitations of DTI. HOTs can model the apparent diffusion coefficient (ADC) with greater accuracy than DTI in regions with fiber heterogeneity. Although GDTI HOTs were designed to model positive diffusion, the straightforward least square (LS) estimation of HOTs doesn't guarantee positivity. In this chapter we address the problem of estimating 4th order tensors with positive diffusion profiles. Two known methods exist that broach this problem, namely a Riemannian approach based on the algebra of 4th order tensors, and a polynomial approach based on Hilbert's theorem on non-negative ternary quartics. In this chapter, we review the technicalities of these two approaches, compare them theoretically to show their pros and cons, and compare them against the Euclidean LS estimation on synthetic, phantom and real data to motivate the relevance of the positive diffusion profile constraint
Diffeomorphic Metric Mapping of High Angular Resolution Diffusion Imaging based on Riemannian Structure of Orientation Distribution Functions
In this paper, we propose a novel large deformation diffeomorphic
registration algorithm to align high angular resolution diffusion images
(HARDI) characterized by orientation distribution functions (ODFs). Our
proposed algorithm seeks an optimal diffeomorphism of large deformation between
two ODF fields in a spatial volume domain and at the same time, locally
reorients an ODF in a manner such that it remains consistent with the
surrounding anatomical structure. To this end, we first review the Riemannian
manifold of ODFs. We then define the reorientation of an ODF when an affine
transformation is applied and subsequently, define the diffeomorphic group
action to be applied on the ODF based on this reorientation. We incorporate the
Riemannian metric of ODFs for quantifying the similarity of two HARDI images
into a variational problem defined under the large deformation diffeomorphic
metric mapping (LDDMM) framework. We finally derive the gradient of the cost
function in both Riemannian spaces of diffeomorphisms and the ODFs, and present
its numerical implementation. Both synthetic and real brain HARDI data are used
to illustrate the performance of our registration algorithm
Non-Euclidean statistics for covariance matrices with applications to diffusion tensor imaging
The statistical analysis of covariance matrix data is considered and,
in particular, methodology is discussed which takes into account the nonEuclidean
nature of the space of positive semi-definite symmetric matrices.
The main motivation for the work is the analysis of diffusion tensors in medical
image analysis. The primary focus is on estimation of a mean covariance
matrix and, in particular, on the use of Procrustes size-and-shape space. Comparisons
are made with other estimation techniques, including using the matrix
logarithm, matrix square root and Cholesky decomposition. Applications
to diffusion tensor imaging are considered and, in particular, a new measure
of fractional anisotropy called Procrustes Anisotropy is discussed
Data augmentation in Rician noise model and Bayesian Diffusion Tensor Imaging
Mapping white matter tracts is an essential step towards understanding brain
function. Diffusion Magnetic Resonance Imaging (dMRI) is the only noninvasive
technique which can detect in vivo anisotropies in the 3-dimensional diffusion
of water molecules, which correspond to nervous fibers in the living brain. In
this process, spectral data from the displacement distribution of water
molecules is collected by a magnetic resonance scanner. From the statistical
point of view, inverting the Fourier transform from such sparse and noisy
spectral measurements leads to a non-linear regression problem. Diffusion
tensor imaging (DTI) is the simplest modeling approach postulating a Gaussian
displacement distribution at each volume element (voxel). Typically the
inference is based on a linearized log-normal regression model that can fit the
spectral data at low frequencies. However such approximation fails to fit the
high frequency measurements which contain information about the details of the
displacement distribution but have a low signal to noise ratio. In this paper,
we directly work with the Rice noise model and cover the full range of
-values. Using data augmentation to represent the likelihood, we reduce the
non-linear regression problem to the framework of generalized linear models.
Then we construct a Bayesian hierarchical model in order to perform
simultaneously estimation and regularization of the tensor field. Finally the
Bayesian paradigm is implemented by using Markov chain Monte Carlo.Comment: 37 pages, 3 figure
Statistical Analysis on Diffusion Tensor Estimation
A thesis submitted in partial fulfilment of the
requirements of the University of Wolverhampton for
the degree of Doctor of Philosophy.Diffusion tensor imaging (DTI) is a relatively new technology of magnetic resonance imaging,
which enables us to observe the insight structure of the human body in vivo and non-invasively.
It displays water molecule movement by a 3×3 diffusion tensor at each voxel. Tensor field
processing, visualisation and tractography are all based on the diffusion tensors. The accuracy
of estimating diffusion tensor is essential in DTI.
This research focuses on exploring the potential improvements at the tensor estimation
of DTI. We analyse the noise arising in the measurement of diffusion signals. We present
robust methods, least median squares (LMS) and least trimmed squares (LTS) regressions,
with forward search algorithm that reduce or eliminate outliers to the desired level. An
investigation of the criterion to detect outliers is provided in theory and practice. We compare
the results with the generalised non-robust models in simulation studies and applicants and
also validated various regressions in terms of FA, MD and orientations. We show that the
robust methods can handle the data with up to 50% corruption. The robust regressions have
better estimations than generalised models in the presence of outliers.
We also consider the multiple tensors problems. We review the recent techniques of
multiple tensor problems. Then we provide a new model considering neighbours’ information,
the Bayesian single and double tensor models using neighbouring tensors as priors, which
can identify the double tensors effectively. We design a framework to estimate the diffusion
tensor field with detecting whether it is a single tensor model or multiple tensor model.
An output of this framework is the Bayesian neighbour (BN) algorithm that improves the
accuracy at the intersection of multiple fibres. We examine the dependence of the estimators
on the FA and MD and angle between two principal diffusion orientations and the goodness
of fit. The Bayesian models are applied to the real data with validation. We show that the
double tensors model is more accurate on distinct fibre orientations, more anisotropic or
similar mean diffusivity tensors.
The final contribution of this research is in covariance tensor estimation. We define
the median covariance matrix in terms of Euclidean and various non-Euclidean metrics taking its symmetric semi-positive definiteness into account. We compare with estimation
methods, Euclidean, power Euclidean, square root Euclidean, log-Euclidean, Riemannian
Euclidean and Procrustes median tensors. We provide an analysis of the different metric
between different median covariance tensors. We also provide the weighting functions and
define the weighted non-Euclidean covariance tensors. We finish with manifold-valued
data applications that improve the illustration of DTI images in tensor field processing
with defined non-weighted and weighted median tensors. The validation of non-Euclidean
methods is studied in the tensor field processing. We show that the root square median
estimator is preferable in general, which can effectively exclude outliers and clearly shows
the important structures of the brain. The power Euclidean median estimator is recommended
when producing FA map
Tensor decomposition processes for interpolation of diffusion magnetic resonance imaging
Diffusion magnetic resonance imaging (dMRI) is an established medical technique used for describing water diffusion in an organic tissue. Typically, rank-2 or 2nd-order tensors quantify this diffusion. From this quantification, it is possible to calculate relevant scalar measures (i.e. fractional anisotropy) employed in the clinical diagnosis of neurological diseases. Nonetheless, 2nd-order tensors fail to represent complex tissue structures like crossing fibers. To overcome this limitation, several researchers proposed a diffusion representation with higher order tensors (HOT), specifically 4th and 6th orders. However, the current acquisition protocols of dMRI data allow images with a spatial resolution between 1 mm3 and 2 mm3, and this voxel size is much bigger than tissue structures. Therefore, several clinical procedures derived from dMRI may be inaccurate. Concerning this, interpolation has been used to enhance the resolution of dMRI in a tensorial space. Most interpolation methods are valid only for rank-2 tensors and a generalization for HOT data is missing. In this work, we propose a probabilistic framework for performing HOT data interpolation. In particular, we introduce two novel probabilistic models based on the Tucker and the canonical decompositions. We call our approaches: Tucker decomposition process (TDP) and canonical decomposition process (CDP). We test the TDP and CDP in rank-2, 4 and 6 HOT fields. For rank-2 tensors, we compare against direct interpolation, log-Euclidean approach, and Generalized Wishart processes. For rank-4 and 6 tensors, we compare against direct interpolation and raw dMRI interpolation. Results obtained show that TDP and CDP interpolate accurately the HOT fields in terms of Frobenius distance, anisotropy measurements, and fiber tracts. Besides, CDP and TDP can be generalized to any rank. Also, the proposed framework keeps the mandatory constraint of positive definite tensors, and preserves morphological properties such as fractional anisotropy (FA), generalized anisotropy (GA) and tractography
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