Brain Connectivity Using Geodesics in HARDI

Péchaud, Mickaël; Descoteaux, Maxime; Keriven, Renaud

doi:10.1007/978-3-642-04271-3_59

Cited by 19 publications

(19 citation statements)

References 18 publications

(26 reference statements)

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“…always provides the optimal solution. Compared with wavefront propagation methods on the extended domain of positions and orientations in image analysis [28,29], we consider a SR metric instead of a Riemannian metric. Results in retinal vessel tracking are promising.…”

Section: Discussionmentioning

confidence: 99%

A PDE Approach to Data-Driven Sub-Riemannian Geodesics in $SE$(2)

Bekkers¹,

Duits²,

Mashtakov³

et al. 2015

SIAM J. Imaging Sci.

109

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Abstract. We present a new flexible wavefront propagation algorithm for the boundary value problem for subRiemannian (SR) geodesics in the roto-translation group SE(2) = R 2 S 1 with a metric tensor depending on a smooth external cost C : SE(2) → [δ, 1], δ > 0, computed from image data. The method consists of a first step where an SR-distance map is computed as a viscosity solution of a Hamilton-Jacobi-Bellman system derived via Pontryagin's maximum principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. For C = 1 we show that our method produces the global minimizers. Comparison with exact solutions shows a remarkable accuracy of the SR-spheres and the SR-geodesics. We present numerical computations of Maxwell points and cusp points, which we again verify for the uniform cost case C = 1. Regarding image analysis applications, trackings of elongated structures in retinal and synthetic images show that our line tracking generically deals with crossings. We show the benefits of including the SR-geometry.

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Section: Discussionmentioning

confidence: 99%

A PDE Approach to Data-Driven Sub-Riemannian Geodesics in $SE$(2)

Bekkers¹,

Duits²,

Mashtakov³

et al. 2015

SIAM J. Imaging Sci.

109

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show abstract

“…: geodesics in such a Riemannian space will tend to align themselves with the eigenvector corresponding with the smallest eigenvalues of T x [231,215,139]. They will thus follow high-diffusion paths, which are likely to correspond to white matter fibers (Figure 3.16).…”

Section: Metric W (X)mentioning

confidence: 99%

“…16 Geodesics corresponding to major white matter fibers bundles in left hemisphere, obtained with the geodesic method of[215].…”

mentioning

confidence: 99%

Geodesic Methods in Computer Vision and Graphics

Peyré¹

2009

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This paper reviews both the theory and practice of the numerical computation of geodesic distances on Riemannian manifolds. The notion of Riemannian manifold allows one to define a local metric (a symmetric positive tensor field) that encodes the information about the problem one wishes to solve. This takes into account a local isotropic cost (whether some point should be avoided or not) and a local anisotropy (which direction should be preferred). Using this local tensor field, the geodesic distance is used to solve many problems of practical interest such as segmentation using geodesic balls and Voronoi regions, sampling points at regular geodesic distance or meshing a domain with geodesic Delaunay triangles. The shortest paths for this Riemannian distance, the so-called geodesics, are also important because they follow salient curvilinear structures in the domain. We show several applications of the numerical computation of geodesic distances and shortest paths to problems in surface and shape processing, in particular segmentation, sampling, meshing and comparison of shapes.

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“…Despite their simplicity, these methods are prone to cumulative errors caused by noise, partial volume effects, and discrete integration, and have difficulty in distinguishing fiber crossing and kissing mostly due to the fact that the entire diffusion information is not globally used and integrated. This led to the development of other successful approaches, including probabilistic techniques (Behrens et al, 2007; Björnemo et al, 2002; Descoteaux et al, 2009; Friman et al, 2006; Jones, 2008; Lazar & Alexander, 2005; Parker et al, 2003), global techniques based on front propagation (Campbell et al, 2005; Jackowski et al, 2005; Parker et al, 2002; Pichon et al, 2005; Prados et al, 2006; Tournier et al, 2003), simulation of the diffusion process or fluid flow (Batchelor et al, 2001; Hageman et al, 2009; Hagmann et al, 2003; Kang et al, 2005; O'Donnell et al, 2002; Yörük et al, 2005), DWI geodesic computations (Jbabdi et al, 2008; Lenglet et al, 2009a; Melonakos et al, 2007; Pechaud et al, 2009), graph theoretical techniques (Iturria-Medina et al, 2007; Sotiropoulos et al, 2010; Zalesky, 2008), spin glass models (Fillard et al, 2009; Mangin et al, 2002), and Gibbs tracking (Kreher et al, 2008). Generally speaking, for virtually every tractography method, a particular putative subset of all possible curves is implicitly considered from which the resulting tracts are chosen according to some criteria, which are different depending on the particular selection strategy.…”

Section: Introductionmentioning

confidence: 99%

A Hough transform global probabilistic approach to multiple-subject diffusion MRI tractography

Aganj

Lenglet

Jahanshad

et al. 2011

Medical Image Analysis

135

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A global probabilistic fiber tracking approach based on the voting process provided by the Hough transform is introduced in this work. The proposed framework tests candidate 3D curves in the volume, assigning to each one a score computed from the diffusion images, and then selects the curves with the highest scores as the potential anatomical connections. The algorithm avoids local minima by performing an exhaustive search at the desired resolution. The technique is easily extended to multiple subjects, considering a single representative volume where the registered high-angular resolution diffusion images (HARDI) from all the subjects are non-linearly combined, thereby obtaining population-representative tracts. The tractography algorithm is run only once for the multiple subjects, and no tract alignment is necessary. We present experimental results on HARDI volumes, ranging from simulated and 1.5T physical phantoms to 7T and 4T human brain and 7T monkey brain datasets.

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Brain Connectivity Using Geodesics in HARDI

Cited by 19 publications

References 18 publications

A PDE Approach to Data-Driven Sub-Riemannian Geodesics in $SE$(2)

A PDE Approach to Data-Driven Sub-Riemannian Geodesics in $SE$(2)

Geodesic Methods in Computer Vision and Graphics

A Hough transform global probabilistic approach to multiple-subject diffusion MRI tractography

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