Riemannian geometry for the statistical analysis of diffusion tensor data

By P. T. Fletcher, Sarang C. Joshi. Signal Processing, 2007

418 citations recorded by the source corpus.

Subfields: Cardiology and Cardiovascular Medicine, Computational Mathematics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/22f281ac734b5cf032a091818c395c5ed7109900

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In Cardiology and Cardiovascular Medicine

In Computational Mathematics

In Radiology, Nuclear Medicine and Imaging

Citations in the corpus

Computing the Karcher mean of symmetric positive definite matrices (2013, methodology): “In principle, one could choose θν in iteration (4) using a line search strategy, but the computation of the optimal θν seems to be too expensive, while heuristic strategies for the step-length [11] may lead to slow convergence in a large number of cases.”
INTRINSIC SHAPE ANALYSIS: GEODESIC PCA FOR RIEMANNIAN MANIFOLDS MODULO ISOMETRIC LIE GROUP ACTIONS (2007, background): “Higher dimensional components would then be sub-manifolds spanned by geodesics, totally geodesic at a point, as proposed by Fletcher and Joshi (2007) and computed in approximation: eigenspaces of the respective covariance matrix in the tangent space at the IM are mapped to geodesics and submanifolds”
Pedestrian Detection via Classification on Riemannian Manifolds (2008, background): “Similarly, the logarithm is given by logð Þ ¼ X1 k¼1 ð 1Þk 1 k ð IÞk ¼ U logðDÞUT : ð17Þ The exponential operator is always defined, whereas the logarithms only exist for symmetric matrices with positive eigenvalues, Symþd .”
Monotonic properties of the least squares mean (2010, background): “λi (X ) denotes the ith eigenvalue of X in non-decreasing order.) Recent contributions that have advocated the use of this metric in applications include [ 12 ,24,28] for tensor computation in medical imaging and [4] for radar processing.”
First-order Methods for Geodesically Convex Optimization (2016, methodology): “Averaging PSD matrices have applications in averaging data of anisotropic symmetric positive-deﬁnite tensors, such as in diffusion tensor imaging (Pennec et al., 2006; Fletcher and Joshi, 2007) and elasticity theory (Cowin and Yang, 1997).”
Differences in White Matter Fiber Tract Development Present from 6 to 24 Months in Infants with Autism (2012, methodology): “The transformed DTI tensor images are averaged using the Riemannian framework (45), resulting in the final 3D average tensor atlas to be used as the reference frame for tractography and tract parameterization.”