1,472 research outputs found
Improving Fiber Alignment in HARDI by Combining Contextual PDE Flow with Constrained Spherical Deconvolution
We propose two strategies to improve the quality of tractography results
computed from diffusion weighted magnetic resonance imaging (DW-MRI) data. Both
methods are based on the same PDE framework, defined in the coupled space of
positions and orientations, associated with a stochastic process describing the
enhancement of elongated structures while preserving crossing structures. In
the first method we use the enhancement PDE for contextual regularization of a
fiber orientation distribution (FOD) that is obtained on individual voxels from
high angular resolution diffusion imaging (HARDI) data via constrained
spherical deconvolution (CSD). Thereby we improve the FOD as input for
subsequent tractography. Secondly, we introduce the fiber to bundle coherence
(FBC), a measure for quantification of fiber alignment. The FBC is computed
from a tractography result using the same PDE framework and provides a
criterion for removing the spurious fibers. We validate the proposed
combination of CSD and enhancement on phantom data and on human data, acquired
with different scanning protocols. On the phantom data we find that PDE
enhancements improve both local metrics and global metrics of tractography
results, compared to CSD without enhancements. On the human data we show that
the enhancements allow for a better reconstruction of crossing fiber bundles
and they reduce the variability of the tractography output with respect to the
acquisition parameters. Finally, we show that both the enhancement of the FODs
and the use of the FBC measure on the tractography improve the stability with
respect to different stochastic realizations of probabilistic tractography.
This is shown in a clinical application: the reconstruction of the optic
radiation for epilepsy surgery planning
Numerical Approaches for Linear Left-invariant Diffusions on SE(2), their Comparison to Exact Solutions, and their Applications in Retinal Imaging
Left-invariant PDE-evolutions on the roto-translation group (and
their resolvent equations) have been widely studied in the fields of cortical
modeling and image analysis. They include hypo-elliptic diffusion (for contour
enhancement) proposed by Citti & Sarti, and Petitot, and they include the
direction process (for contour completion) proposed by Mumford. This paper
presents a thorough study and comparison of the many numerical approaches,
which, remarkably, is missing in the literature. Existing numerical approaches
can be classified into 3 categories: Finite difference methods, Fourier based
methods (equivalent to -Fourier methods), and stochastic methods (Monte
Carlo simulations). There are also 3 types of exact solutions to the
PDE-evolutions that were derived explicitly (in the spatial Fourier domain) in
previous works by Duits and van Almsick in 2005. Here we provide an overview of
these 3 types of exact solutions and explain how they relate to each of the 3
numerical approaches. We compute relative errors of all numerical approaches to
the exact solutions, and the Fourier based methods show us the best performance
with smallest relative errors. We also provide an improvement of Mathematica
algorithms for evaluating Mathieu-functions, crucial in implementations of the
exact solutions. Furthermore, we include an asymptotical analysis of the
singularities within the kernels and we propose a probabilistic extension of
underlying stochastic processes that overcomes the singular behavior in the
origin of time-integrated kernels. Finally, we show retinal imaging
applications of combining left-invariant PDE-evolutions with invertible
orientation scores.Comment: A final and corrected version of the manuscript is Published in
Numerical Mathematics: Theory, Methods and Applications (NM-TMA), vol. (9),
p.1-50, 201
Diffusion, convection and erosion on R3 x S2 and their application to the enhancement of crossing fibers
In this article we study both left-invariant (convection-)diffusions and left-invariant Hamilton-Jacobi equations (erosions) on the space R3 x S2 of 3D-positions and orientations naturally embedded in the group SE(3) of 3D-rigid body movements. The general motivation for these (convection-)diffusions and erosions is to obtain crossing-preserving fiber enhancement on probability densities defined on the space of positions and orientations. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on R3 x S2 and can be solved by R3 x S2-convolution with the corresponding Green’s functions or by a finite difference scheme. The left-invariant Hamilton-Jacobi equations are Bellman equations of cost processes on R3 x S2 and they are solved by a morphological R3 x S2-convolution with the corresponding Green’s functions. We will reveal the remarkable analogy between these erosions/dilations and diffusions. Furthermore, we consider pseudo-linear scale spaces on the space of positions and orientations that combines dilation and diffusion in a single evolution. In our design and analysis for appropriate linear, non-linear, morphological and pseudo-linear scale spaces on R3 x S2 we employ the underlying differential geometry on SE(3), where the frame of left-invariant vector fields serves as a moving frame of reference. Furthermore, we will present new and simpler finite difference schemes for our diffusions, which are clear improvements of our previous finite difference schemes. We apply our theory to the enhancement of fibres in magnetic resonance imaging (MRI) techniques (HARDI and DTI) for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. We provide experiments of our crossing-preserving (non-linear) left-invariant evolutions on neural images of a human brain containing crossing fibers
Total Variation and Mean Curvature PDEs on
Total variation regularization and total variation flows (TVF) have been
widely applied for image enhancement and denoising. To include a generic
preservation of crossing curvilinear structures in TVF we lift images to the
homogeneous space of positions and
orientations as a Lie group quotient in SE(d). For d = 2 this is called 'total
roto-translation variation' by Chambolle & Pock. We extend this to d = 3, by a
PDE-approach with a limiting procedure for which we prove convergence. We also
include a Mean Curvature Flow (MCF) in our PDE model on M. This was first
proposed for d = 2 by Citti et al. and we extend this to d = 3. Furthermore,
for d = 2 we take advantage of locally optimal differential frames in
invertible orientation scores (OS). We apply our TVF and MCF in the
denoising/enhancement of crossing fiber bundles in DW-MRI. In comparison to
data-driven diffusions, we see a better preservation of bundle boundaries and
angular sharpness in fiber orientation densities at crossings. We support this
by error comparisons on a noisy DW-MRI phantom. We also apply our TVF and MCF
in enhancement of crossing elongated structures in 2D images via OS, and
compare the results to nonlinear diffusions (CED-OS) via OS.Comment: Submission to the Seventh International Conference on Scale Space and
Variational Methods in Computer Vision (SSVM 2019). (v2) Typo correction in
lemma 1. (v3) Typo correction last paragraph page
Fast T2 Mapping with Improved Accuracy Using Undersampled Spin-echo MRI and Model-based Reconstructions with a Generating Function
A model-based reconstruction technique for accelerated T2 mapping with
improved accuracy is proposed using undersampled Cartesian spin-echo MRI data.
The technique employs an advanced signal model for T2 relaxation that accounts
for contributions from indirect echoes in a train of multiple spin echoes. An
iterative solution of the nonlinear inverse reconstruction problem directly
estimates spin-density and T2 maps from undersampled raw data. The algorithm is
validated for simulated data as well as phantom and human brain MRI at 3 T. The
performance of the advanced model is compared to conventional pixel-based
fitting of echo-time images from fully sampled data. The proposed method yields
more accurate T2 values than the mono-exponential model and allows for
undersampling factors of at least 6. Although limitations are observed for very
long T2 relaxation times, respective reconstruction problems may be overcome by
a gradient dampening approach. The analytical gradient of the utilized cost
function is included as Appendix.Comment: 10 pages, 7 figure
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