Riemannian Analysis of Probability Density Functions with Applications in Vision

Srivastava, Anuj; Jermyn, Ian H.; Joshi, Shantanu H.

doi:10.1109/cvpr.2007.383188

Cited by 163 publications

(161 citation statements)

References 14 publications

(18 reference statements)

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“…We addressed this by taking the square root of the ODF: the Riemannian manifold for the square root of a PDF is isomorphic to a unit sphere and there are closed form expressions defining the geodesic distance, exponential and inverse exponential mappings [9]. The interpolated square-rooted ODF (sqrt-ODF)ϕ at point (x, y, z) was then constructed by finding the weighted Karcher mean of its 8 diagonal neighbors ϕ i in 3D at lattice points (x i , y i , z i ), which minimizes the square sum of the geodesic distance d: (2) Here w i is the trilinear interpolation weight defined as…”

Section: Hardi Processing and Registrationmentioning

confidence: 99%

Brain Fiber Architecture, Genetics, and Intelligence: A High Angular Resolution Diffusion Imaging (HARDI) Study

Chiang

Barysheva

Lee

et al. 2008

Medical Image Computing and Computer-Assisted Intervention – MICCAI 2008

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We developed an analysis pipeline enabling population studies of HARDI data, and applied it to map genetic influences on fiber architecture in 90 twin subjects. We applied tensor-driven 3D fluid registration to HARDI, re-sampling the spherical fiber orientation distribution functions (ODFs) in appropriate Riemannian manifolds, after ODF regularization and sharpening. Fitting structural equation models (SEM) from quantitative genetics, we evaluated genetic influences on the Jensen-Shannon divergence (JSD), a novel measure of fiber spatial coherence, and on the generalized fiber anisotropy (GFA; [1]) a measure of fiber integrity. With random-effects regression, we mapped regions where diffusion profiles were highly correlated with subjects' intelligence quotient (IQ). Fiber complexity was predominantly under genetic control, and higher in more highly anisotropic regions; the proportion of genetic versus environmental control varied spatially. Our methods show promise for discovering genes affecting fiber connectivity in the brain.

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Section: Hardi Processing and Registrationmentioning

confidence: 99%

Brain Fiber Architecture, Genetics, and Intelligence: A High Angular Resolution Diffusion Imaging (HARDI) Study

Chiang

Barysheva

Lee

et al. 2008

Medical Image Computing and Computer-Assisted Intervention – MICCAI 2008

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“…Methods that map a histogram/density to a sphere (e.g., [35]) have such geometry but suffer from two issues. First, they do not respect the ordering of the bins or the real line.…”

Section: Related Workmentioning

confidence: 99%

Highly-Expressive Spaces of Well-Behaved Transformations: Keeping it Simple

Freifeld

Hauberg

Batmanghelich

et al. 2015

2015 IEEE International Conference on Computer Vision (ICCV)

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“…Observe that the Cauchy-Schwarz escort distributions are the square root density representations [26] of distributions.…”

Section: Fact 5 (Hed As a Skew Bhattacharyya Divergence) The Hölder mentioning

confidence: 99%

On Hölder Projective Divergences

Nielsen

Sun

Marchand-Maillet

2017

Entropy

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Abstract:We describe a framework to build distances by measuring the tightness of inequalities and introduce the notion of proper statistical divergences and improper pseudo-divergences. We then consider the Hölder ordinary and reverse inequalities and present two novel classes of Hölder divergences and pseudo-divergences that both encapsulate the special case of the Cauchy-Schwarz divergence. We report closed-form formulas for those statistical dissimilarities when considering distributions belonging to the same exponential family provided that the natural parameter space is a cone (e.g., multivariate Gaussians) or affine (e.g., categorical distributions). Those new classes of Hölder distances are invariant to rescaling and thus do not require distributions to be normalized. Finally, we show how to compute statistical Hölder centroids with respect to those divergences and carry out center-based clustering toy experiments on a set of Gaussian distributions which demonstrate empirically that symmetrized Hölder divergences outperform the symmetric Cauchy-Schwarz divergence.

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Riemannian Analysis of Probability Density Functions with Applications in Vision

Cited by 163 publications

References 14 publications

Brain Fiber Architecture, Genetics, and Intelligence: A High Angular Resolution Diffusion Imaging (HARDI) Study

Brain Fiber Architecture, Genetics, and Intelligence: A High Angular Resolution Diffusion Imaging (HARDI) Study

Highly-Expressive Spaces of Well-Behaved Transformations: Keeping it Simple

On Hölder Projective Divergences

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