Optimization over geodesics for exact principal geodesic analysis

Sommer, Stefan; Lauze, François; Nielsen, Mads

doi:10.1007/s10444-013-9308-1

Cited by 36 publications

(28 citation statements)

References 40 publications

(102 reference statements)

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“…The Exp X Ô¤Õ function is a diffeomorphism from any open 0-neighborhood s X c X onto an open X-neighborhood H X G. Therefore, the inverse function Exp ¡1 X Ô¤Õ is well defined on H X . In computer vision, Exp ¡1 X Ô¤Õ is usually called the Riemannian logarithm function [133,56,57,162,160] and is denoted by Log X Ô¤Õ.…”

Section: Remarksmentioning

confidence: 99%

“…Taking derivatives with respect to on both sides of (6.1) and evaluating at 0, we 20 For abstract groups it can be shown that the corresponding ODE is Integrating (6.2) with initial condition J Ô0Õ 0 results in a vector field along the geodesic which is known as the Jacobi field [50, Chapter 5], [160]. In these works the Jacobi field evolution equation is expressed by a second-order equation which takes into account the curvature of the manifold.…”

Section: Reducing To a First-order Equationmentioning

confidence: 99%

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Left-Invariant Riemannian Geodesics on Spatial Transformation Groups

Zacur¹,

Bossa²,

Olmos³

2014

SIAM J. Imaging Sci.

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Spatial transformations are mappings between locations of a d-dimensional space and are commonly used in computer vision and image analysis. Many of the spatial transformation sets have a group structure and can be represented by matrix groups. In the computer vision and image analysis fields there is a recent and growing interest in performing analyses on spatial transformations data. Differential and Riemannian geometry have been used as a framework to endow the set of spatial transformations with a metric space structure, allowing the extension of the standard analysis techniques defined on vector spaces. This paper presents a review of the concepts and an overview of approaches to computing Riemannian geodesics on spatial transformation groups. The paper is aimed at providing a bridge for researchers from computer vision and image analysis fields to fill in the gap between differential geometry and computer vision and imaging disciplines. Some application examples are shown to illustrate the use of invariant Riemannian geodesics, such as interpolation of spatial transformations and filtering of matrix-valued images.

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Section: Remarksmentioning

confidence: 99%

Section: Reducing To a First-order Equationmentioning

confidence: 99%

Left-Invariant Riemannian Geodesics on Spatial Transformation Groups

Zacur¹,

Bossa²,

Olmos³

2014

SIAM J. Imaging Sci.

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“…The literature also includes ideas related to projective dimensionality reduction methods. For instance, the generalization of Principal Components analysis (PCA) via the so-called PGA [28], Geodesic PCA [29], Exact PGA [30], CCA on manifolds [31], Horizontal Dimension Reduction [32] with frame bundles, and an extension of PGA to the product space of Riemannian manifolds, namely, tensor fields [24]. We should note that an important earlier work dealing with a univariate linear model on manifolds (related to geodesic regression) was studied for the group of diffeomorphisms using metrics which were not strictly Riemmanian [33, 34].…”

Section: Introductionmentioning

confidence: 99%

Riemannian Nonlinear Mixed Effects Models: Analyzing Longitudinal Deformations in Neuroimaging

Kim

Adluru

Suri

et al. 2017

2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)

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Statistical machine learning models that operate on manifold-valued data are being extensively studied in vision, motivated by applications in activity recognition, feature tracking and medical imaging. While non-parametric methods have been relatively well studied in the literature, efficient formulations for parametric models (which may offer benefits in small sample size regimes) have only emerged recently. So far, manifold-valued regression models (such as geodesic regression) are restricted to the analysis of cross-sectional data, i.e., the so-called “fixed effects” in statistics. But in most “longitudinal analysis” (e.g., when a participant provides multiple measurements, over time) the application of fixed effects models is problematic. In an effort to answer this need, this paper generalizes non-linear mixed effects model to the regime where the response variable is manifold-valued, i.e., f : Rd → ℳ. We derive the underlying model and estimation schemes and demonstrate the immediate benefits such a model can provide — both for group level and individual level analysis — on longitudinal brain imaging data. The direct consequence of our results is that longitudinal analysis of manifold-valued measurements (especially, the symmetric positive definite manifold) can be conducted in a computationally tractable manner.

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“…Among the most closely related are ideas related to projective dimensionality reduction methods. For instance, the generalization of Principal Components analysis (PCA) via the so-called Principal Geodesic Analysis (PGA) [9], Geodesic PCA [20], Exact PGA [33], Horizontal Dimension Reduction [32] with frame bundles, and an extension of PGA to the product space of Riemannian manifolds, namely, tensor fields [36]. It is important to note that except the non-parametric method of [34], most of these strategies focus on one rather than two sets of random variables (as is the case in CCA).…”

Section: Introductionmentioning

confidence: 99%

Canonical Correlation Analysis on Riemannian Manifolds and Its Applications

Kim

Adluru

Bendlin

et al. 2014

Computer Vision – ECCV 2014

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Canonical correlation analysis (CCA) is a widely used statistical technique to capture correlations between two sets of multi-variate random variables and has found a multitude of applications in computer vision, medical imaging and machine learning. The classical formulation assumes that the data live in a pair of vector spaces which makes its use in certain important scientific domains problematic. For instance, the set of symmetric positive definite matrices (SPD), rotations and probability distributions, all belong to certain curved Riemannian manifolds where vector-space operations are in general not applicable. Analyzing the space of such data via the classical versions of inference models is rather sub-optimal. But perhaps more importantly, since the algorithms do not respect the underlying geometry of the data space, it is hard to provide statistical guarantees (if any) on the results. Using the space of SPD matrices as a concrete example, this paper gives a principled generalization of the well known CCA to the Riemannian setting. Our CCA algorithm operates on the product Riemannian manifold representing SPD matrix-valued fields to identify meaningful statistical relationships on the product Riemannian manifold. As a proof of principle, we present results on an Alzheimer’s disease (AD) study where the analysis task involves identifying correlations across diffusion tensor images (DTI) and Cauchy deformation tensor fields derived from T1-weighted magnetic resonance (MR) images.

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Optimization over geodesics for exact principal geodesic analysis

Cited by 36 publications

References 40 publications

Left-Invariant Riemannian Geodesics on Spatial Transformation Groups

Left-Invariant Riemannian Geodesics on Spatial Transformation Groups

Riemannian Nonlinear Mixed Effects Models: Analyzing Longitudinal Deformations in Neuroimaging

Canonical Correlation Analysis on Riemannian Manifolds and Its Applications

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