Symmetric Positive 4 th Order Tensors &amp; Their Estimation from Diffusion Weighted MRI

Barmpoutis, Angelos; Jian, Bing; Vemuri, Baba C.; Shepherd, Timothy M.

doi:10.1007/978-3-540-73273-0_26

Cited by 60 publications

(76 citation statements)

References 16 publications

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“…Given these signals, several reconstruction techniques can be used to characterize diffusion. Higher-order tensors leverage the work done in diffusion tensor imaging (DTI) [2] by using higher-order polynomials to model diffusivity [3,4]. [5] fits the HARDI signals with a mixture of tensors model whose weights are specified by a probability function defined on the space of symmetric positive definite matrices.…”

Section: Introductionmentioning

confidence: 99%

Estimating Orientation Distribution Functions with Probability Density Constraints and Spatial Regularity

Goh

Lenglet

Thompson

et al. 2009

Medical Image Computing and Computer-Assisted Intervention – MICCAI 2009

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Abstract. High angular resolution diffusion imaging (HARDI) has become an important magnetic resonance technique for in vivo imaging. Current techniques for estimating the diffusion orientation distribution function (ODF), i.e., the probability density function of water diffusion along any direction, do not enforce the estimated ODF to be nonnegative or to sum up to one. Very often this leads to an estimated ODF which is not a proper probability density function. In addition, current methods do not enforce any spatial regularity of the data. In this paper, we propose an estimation method that naturally constrains the estimated ODF to be a proper probability density function and regularizes this estimate using spatial information. By making use of the spherical harmonic representation, we pose the ODF estimation problem as a convex optimization problem and propose a coordinate descent method that converges to the minimizer of the proposed cost function. We illustrate our approach with experiments on synthetic and real data.

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Section: Introductionmentioning

confidence: 99%

Estimating Orientation Distribution Functions with Probability Density Constraints and Spatial Regularity

Goh

Lenglet

Thompson

et al. 2009

Medical Image Computing and Computer-Assisted Intervention – MICCAI 2009

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“…This is in part due to an increased demand on the application side (cf. the sample applications in numerical linear algebra [50,26,28], material sciences [57], quantum physics [9,18], and signal processing [16,4,53]), and in part due to its own strong theoretical appeal. Indeed, polynomial optimization is a challenging task; at the same time it is rich enough to be fruitful.…”

mentioning

confidence: 99%

Maximum Block Improvement and Polynomial Optimization

Chen¹,

He²,

Li³

et al. 2012

SIAM J. Optim.

159

177

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Abstract. In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear optimization, with the novelty being that we only accept a block update that achieves the maximum improvement, hence the name of our new search method: Maximum Block Improvement (MBI). Convergence of the sequence produced by the MBI method to a stationary point is proven. Second, we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem, thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the MBI approach. Third, we propose a scheme to reach a KKT point of the polynomial optimization, provided that a stationary solution for the relaxed tensor problem is available. Numerical experiments have shown that our new method works very efficiently: for a majority of the test instances that we have experimented with, the method finds the global optimal solution at a low computational cost.Key words. block coordinate descent, polynomial optimization problem, tensor form AMS subject classifications. 90C26, 90C30, 15A69, 49M271. Introduction. The optimization models whose objective and constraints are polynomial functions have recently attracted much research attention. This is in part due to an increased demand on the application side (cf. the sample applications in numerical linear algebra [50,26,28], material sciences [57], quantum physics [9,18], and signal processing [16,4,53]), and in part due to its own strong theoretical appeal. Indeed, polynomial optimization is a challenging task; at the same time it is rich enough to be fruitful. For instance, even the simplest instances of polynomial optimization, such as maximizing a cubic polynomial over a sphere, is NP-hard (Nesterov [42]). However, the problem is so elementary that it can be even attempted in an undergraduate calculus class. For readers interested in polynomial optimization with simple constraints, we refer to De Klerk [12] for a survey on the computational complexity of optimizing various classes of polynomial functions over a simplex, hypercube or sphere. In particular, De Klerk et al.[13] designed a polynomial-time approximation scheme (PTAS) for minimizing polynomials of fixed degree over the simplex.So far, a few results have been obtained for approximation algorithms with guaranteed worst-case performance ratios for higher degree polynomial optimization problems. Luo and Zhang [39] derived a polynomial-time approximation algorithm to optimize a multivariate quartic polynomial over a region defined by quadratic inequalities.

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“…4 is expressed as (5) where c is a constant whose value is not dependent on r. Note that the left side of Eq. 5 is a Cartesian tensor basis function.…”

Section: Higher-order Basis For Hardi Approximationmentioning

confidence: 99%

“…However, 2 nd -order tensors are incapable of modeling complex geometry of the diffusivity function in practice for many cases (see [2,3], such as in the presence of fiber-crossings, and a higher-order approximation must be employed instead. Higher-order tensors have been used to model either the local diffusivity function [4,5], or the Kurtosis component of it [6]. However, in all cases the peaks of the estimated higher-order tensor do not necessarily yield the distinct orientations of the underlying distinct fiber bundles [2].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we propose a novel representation of the displacement probability profile by using the 4 th -order Cartesian tensor basis. 4 th -order tensors have been studied recently in [5] showing their capability to model multi-lobed diffusivity functions. In our model, the lobes of the probability profile, which is expressed in the form of a 4 th -order tensor, correspond directly to orientations of distinct fiber distributions and thus there is no need to evaluate the integral of Eq.1.…”

Section: Introductionmentioning

confidence: 99%

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Fast displacement probability profile approximation from HARDI using 4th-order tensors

Barmpoutis

Vemuri

Forder

2008

2008 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro

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Cartesian tensor basis have been widely used to approximate spherical functions. In Medical Imaging, tensors of various orders have been used to model the diffusivity function in Diffusion-weighted MRI data sets. However, it is known that the peaks of the diffusivity do not correspond to orientations of the underlying fibers and hence the displacement probability profiles should be employed instead. In this paper, we present a novel representation of the probability profile by a 4 th order tensor, which is a smooth spherical function that can approximate single-fibers as well as multiple-fiber structures. We also present a method for efficiently estimating the unknown tensor coefficients of the probability profile directly from a given high-angular resolution diffusion-weighted (HARDI) data set. The accuracy of our model is validated by experiments on synthetic and real HARDI datasets from a fixed rat spinal cord.

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Symmetric Positive 4 th Order Tensors & Their Estimation from Diffusion Weighted MRI

Cited by 60 publications

References 16 publications

Estimating Orientation Distribution Functions with Probability Density Constraints and Spatial Regularity

Estimating Orientation Distribution Functions with Probability Density Constraints and Spatial Regularity

Maximum Block Improvement and Polynomial Optimization

Fast displacement probability profile approximation from HARDI using 4th-order tensors

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