4,221 research outputs found
A Riemannian Framework for Orientation Distribution Function Computing
International audienceCompared with Diffusion Tensor Imaging (DTI), High Angular Resolution Imaging (HARDI) can better explore the complex microstructure of white matter. Orientation Distribution Function (ODF) is used to describe the probability of the fiber direction. Fisher information metric has been constructed for probability density family in Information Geometry theory and it has been successfully applied for tensor computing in DTI. In this paper, we present a state of the art Riemannian framework for ODF computing based on Information Geometry and sparse representation of orthonormal bases. In this Riemannian framework, the exponential map, logarithmic map and geodesic have closed forms. And the weighted Frechet mean exists uniquely on this manifold. We also propose a novel scalar measurement, named Geometric Anisotropy (GA), which is the Riemannian geodesic distance between the ODF and the isotropic ODF. The Renyi entropy H_{1/2} of the ODF can be computed from the GA. Moreover, we present an Affine-Euclidean framework and a Log-Euclidean framework so that we can work in an Euclidean space. As an application, Lagrange interpolation on ODF field is proposed based on weighted Frechet mean. We validate our methods on synthetic and real data experiments. Compared with existing Riemannian frameworks on ODF, our framework is model-free. The estimation of the parameters, i.e. Riemannian coordinates, is robust and linear. Moreover it should be noted that our theoretical results can be used for any probability density function (PDF) under an orthonormal basis representation
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
Bayesian Estimation of White Matter Atlas from High Angular Resolution Diffusion Imaging
We present a Bayesian probabilistic model to estimate the brain white matter
atlas from high angular resolution diffusion imaging (HARDI) data. This model
incorporates a shape prior of the white matter anatomy and the likelihood of
individual observed HARDI datasets. We first assume that the atlas is generated
from a known hyperatlas through a flow of diffeomorphisms and its shape prior
can be constructed based on the framework of large deformation diffeomorphic
metric mapping (LDDMM). LDDMM characterizes a nonlinear diffeomorphic shape
space in a linear space of initial momentum uniquely determining diffeomorphic
geodesic flows from the hyperatlas. Therefore, the shape prior of the HARDI
atlas can be modeled using a centered Gaussian random field (GRF) model of the
initial momentum. In order to construct the likelihood of observed HARDI
datasets, it is necessary to study the diffeomorphic transformation of
individual observations relative to the atlas and the probabilistic
distribution of orientation distribution functions (ODFs). To this end, we
construct the likelihood related to the transformation using the same
construction as discussed for the shape prior of the atlas. The probabilistic
distribution of ODFs is then constructed based on the ODF Riemannian manifold.
We assume that the observed ODFs are generated by an exponential map of random
tangent vectors at the deformed atlas ODF. Hence, the likelihood of the ODFs
can be modeled using a GRF of their tangent vectors in the ODF Riemannian
manifold. We solve for the maximum a posteriori using the
Expectation-Maximization algorithm and derive the corresponding update
equations. Finally, we illustrate the HARDI atlas constructed based on a
Chinese aging cohort of 94 adults and compare it with that generated by
averaging the coefficients of spherical harmonics of the ODF across subjects
From receptive profiles to a metric model of V1
In this work we show how to construct connectivity kernels induced by the
receptive profiles of simple cells of the primary visual cortex (V1). These
kernels are directly defined by the shape of such profiles: this provides a
metric model for the functional architecture of V1, whose global geometry is
determined by the reciprocal interactions between local elements. Our
construction adapts to any bank of filters chosen to represent a set of
receptive profiles, since it does not require any structure on the
parameterization of the family. The connectivity kernel that we define carries
a geometrical structure consistent with the well-known properties of long-range
horizontal connections in V1, and it is compatible with the perceptual rules
synthesized by the concept of association field. These characteristics are
still present when the kernel is constructed from a bank of filters arising
from an unsupervised learning algorithm.Comment: 25 pages, 18 figures. Added acknowledgement
Finsler geometry on higher order tensor fields and applications to high angular resolution diffusion imaging.
We study 3D-multidirectional images, using Finsler geometry. The application considered here is in medical image analysis, specifically in High Angular Resolution Diffusion Imaging (HARDI) (Tuch et al. in Magn. Reson. Med. 48(6):1358–1372, 2004) of the brain. The goal is to reveal the architecture of the neural fibers in brain white matter. To the variety of existing techniques, we wish to add novel approaches that exploit differential geometry and tensor calculus. In Diffusion Tensor Imaging (DTI), the diffusion of water is modeled by a symmetric positive definite second order tensor, leading naturally to a Riemannian geometric framework. A limitation is that it is based on the assumption that there exists a single dominant direction of fibers restricting the thermal motion of water molecules. Using HARDI data and higher order tensor models, we can extract multiple relevant directions, and Finsler geometry provides the natural geometric generalization appropriate for multi-fiber analysis. In this paper we provide an exact criterion to determine whether a spherical function satisfies the strong convexity criterion essential for a Finsler norm. We also show a novel fiber tracking method in Finsler setting. Our model incorporates a scale parameter, which can be beneficial in view of the noisy nature of the data. We demonstrate our methods on analytic as well as simulated and real HARDI data
The Square Root Velocity Framework for Curves in a Homogeneous Space
In this paper we study the shape space of curves with values in a homogeneous
space , where is a Lie group and is a compact Lie subgroup. We
generalize the square root velocity framework to obtain a reparametrization
invariant metric on the space of curves in . By identifying curves in
with their horizontal lifts in , geodesics then can be computed. We can also
mod out by reparametrizations and by rigid motions of . In each of these
quotient spaces, we can compute Karcher means, geodesics, and perform principal
component analysis. We present numerical examples including the analysis of a
set of hurricane paths.Comment: To appear in 3rd International Workshop on Diff-CVML Workshop, CVPR
201
Total variation regularization for manifold-valued data
We consider total variation minimization for manifold valued data. We propose
a cyclic proximal point algorithm and a parallel proximal point algorithm to
minimize TV functionals with -type data terms in the manifold case.
These algorithms are based on iterative geodesic averaging which makes them
easily applicable to a large class of data manifolds. As an application, we
consider denoising images which take their values in a manifold. We apply our
algorithms to diffusion tensor images, interferometric SAR images as well as
sphere and cylinder valued images. For the class of Cartan-Hadamard manifolds
(which includes the data space in diffusion tensor imaging) we show the
convergence of the proposed TV minimizing algorithms to a global minimizer
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