# Statistics on diffeomorphisms via tangent space representations

@article{Vaillant2004StatisticsOD, title={Statistics on diffeomorphisms via tangent space representations}, author={Marc Vaillant and M. I. Miller and Laurent Younes and Alain Trouv{\'e}}, journal={NeuroImage}, year={2004}, volume={23}, pages={S161-S169} }

In this paper, we present a linear setting for statistical analysis of shape and an optimization approach based on a recent derivation of a conservation of momentum law for the geodesics of diffeomorphic flow. Once a template is fixed, the space of initial momentum becomes an appropriate space for studying shape via geodesic flow since the flow at any point along the geodesic is completely determined by the momentum at the origin through geodesic shooting equations. The space of initial…

## 236 Citations

A Log-Euclidean Framework for Statistics on Diffeomorphisms

- Mathematics, Computer ScienceMICCAI
- 2006

This article focuses on the computation of statistics of invertible geometrical deformations (i.e., diffeomorphisms), based on the generalization to this type of data of the notion of principal logarithm, which is a simple 3D vector field and well-defined for diffe morphisms close enough to the identity.

Low-dimensional shape analysis in the space of diffeomorphisms

- Mathematics
- 2020

Abstract Statistical shape analysis via diffeomorphic image registration is challenging due to the high-dimensional and nonlinear nature of the space of diffeomorphisms. In this chapter, we first…

Robust Diffeomorphic Mapping via Geodesically Controlled Active Shapes

- Computer Science, MedicineInt. J. Biomed. Imaging
- 2013

The robustness of this algorithm as applied to the workflow of a large neuroanatomical study is demonstrated by comparing to an existing diffeomorphic landmark matching algorithm.

Evolution Equations with Anisotropic Distributions and Diffusion PCA

- Mathematics, Computer ScienceGSI
- 2015

This paper presents derivations of evolution equations for the family of paths that in the Diffusion PCA framework are used for approximating data likelihood and shows how rank-deficient metrics can be mixed with an underlying Riemannian metric.

Locally Linear Diffeomorphic Metric Embedding (LLDME) for surface-based anatomical shape modeling

- Medicine, MathematicsNeuroImage
- 2011

Compared with Principal Component Analysis and ISOMAP, LLDME provides the most compact and efficient representation of the age-related hippocampal shapes and reveals the nonlinear relationship of the hippocampal morphometry with age.

Diffeomorphic 3D Image Registration via Geodesic Shooting Using an Efficient Adjoint Calculation

- Mathematics, Computer ScienceInternational Journal of Computer Vision
- 2011

The key contribution of this work is to provide an accurate estimation of the so-called initial momentum, which is a scalar function encoding the optimal deformation between two images through the Hamiltonian equations of geodesics.

Multi-manifold diffeomorphic metric mapping for aligning cortical hemispheric surfaces

- Mathematics, Computer ScienceNeuroImage
- 2010

A multi-manifold large deformation diffeomorphic metric mapping (MM-LDDMM) algorithm that allows simultaneously carrying the cortical hemispheric surface and its sulcal curves from one to the other through a flow of diffeomorphisms.

Principal Component Based Diffeomorphic Surface Mapping

- Mathematics, Computer ScienceIEEE Transactions on Medical Imaging
- 2012

This new algorithm reduces the complexity of the estimation of diffeomorphic transformations by incorporating a shape prior in which a nonlinear diffeomorph shape space is represented by a linear space of initial momenta of Diffeomorphic geodesic flows from a fixed template.

A Sub-Riemannian Modular Framework for Diffeomorphism-Based Analysis of Shape Ensembles

- Mathematics, Computer ScienceSIAM J. Imaging Sci.
- 2018

This work proposes a new mathematical and computational framework, in which velocity fields are constrained to be built through a combination of local deformation modules with few degrees of freedom, and derives a method to estimate a Frechet mean from a series of observat...

Transport of Relational Structures in Groups of Diffeomorphisms

- Medicine, Computer ScienceJournal of Mathematical Imaging and Vision
- 2008

This paper discusses two main options for translating the relative variation of one shape with respect to another in a template centered representation, based on the Riemannian metric and the coadjoint transport.

## References

SHOWING 1-10 OF 25 REFERENCES

Geodesic Shooting for Computational Anatomy

- Computer Science, MedicineJournal of Mathematical Imaging and Vision
- 2005

It is shown that this momentum can be also used for describing a deformation of given visual structures, like points, contours or images, and that, it has the same dimension as the described object, as a consequence of the normal momentum constraint the authors introduce.

Soliton dynamics in computational anatomy

- Mathematics, MedicineNeuroImage
- 2004

A recently discovered momentum map for singular solutions of EPDiff yields their canonical Hamiltonian formulation, which provides a complete parameterization of the landmarks by their canonical positions and momenta, suggesting a new dynamical paradigm for CA, as well as new data representation.

Statistics of shape via principal geodesic analysis on Lie groups

- Mathematics, Computer Science2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings.
- 2003

This paper shows that medial descriptions are in fact elements of a Lie group, and develops methodology based on Lie groups for the statistical analysis of medially-defined anatomical objects.

On the metrics and euler-lagrange equations of computational anatomy.

- Mathematics, MedicineAnnual review of biomedical engineering
- 2002

Current experimental results from the Toga & Thompson group in growth, the Van Essen group in macaque and human cortex mapping, and the Csernansky group in hippocampus mapping for neuropsychiatric studies in aging and schizophrenia are shown.

Landmark matching via large deformation diffeomorphisms

- Mathematics, Computer ScienceIEEE Trans. Image Process.
- 2000

Conditions for the existence of solutions in the space of diffeomorphisms are established, with a gradient algorithm provided for generating the optimal flow solving the minimum problem.

Efficient algorithms for inferences on Grassmann manifolds

- Mathematics, Computer ScienceIEEE Workshop on Statistical Signal Processing, 2003
- 2003

This work defines and seeks an optimal linear representation using a Metropolis-Hastings type, stochastic search algorithm, and illustrates computation of sample statistics, such as mean and variances, on a Grassmann manifold for statistical inferences.

The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories

- Mathematics, Physics
- 1998

We study Euler–Poincare systems (i.e., the Lagrangian analogue of Lie–Poisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincare…

Analysis and segmentation of face images using point annotations and linear subspace techniques, in

- Computer Science
- 2002

An analysis of 37 annotated frontal face images is provided and the problem of AAM model truncation is addressed using parallel analysis along with a comparable study of the two prevalent AAM learning methods; principal component regression and estimation of fixed Jacobian matrices.

The Fréchet mean shape and the shape of the means

- MathematicsAdvances in Applied Probability
- 2000

We identify the Fréchet mean shape with respect to the Riemannian metric of a class of probability measures on Bookstein's shape space of labelled triangles and show, in contrast to the case of…

Nonparametic estimation of location and dispersion on Riemannian manifolds

- Mathematics
- 2002

A central limit theorem for intrinsic means on a complete flat manifold and some asymptotic properties of the intrinsic total sample variance on an arbitrary complete manifold are given. A…