Shape Splines and Stochastic Shape Evolutions: A Second Order Point of View

@article{Trouve2010ShapeSA,
  title={Shape Splines and Stochastic Shape Evolutions: A Second Order Point of View},
  author={Alain Trouv'e and Franccois-Xavier Vialard},
  journal={arXiv: Optimization and Control},
  year={2010}
}
This article presents a new mathematical framework to perform statistical analysis on time-indexed sequences of 2D or 3D shapes. At the core of this statistical analysis is the task of time interpolation of such data. Current models in use can be compared to linear interpolation for one dimensional data. We develop a spline interpolation method which is directly related to cubic splines on a Riemannian manifold. Our strategy consists of introducing a control variable on the Hamiltonian… 
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References

SHOWING 1-10 OF 39 REFERENCES
A Pattern-Theoretic Characterization of Biological Growth
TLDR
A structured model, called Growth by Random Iterated Diffeomorphisms (GRID), that treats a cumulative growth deformation as a composition of several elementary deformations, and is demonstrated using an MRI image data of a rat's brain growth.
Representation of time-varying shapes in the large deformation diffeomorphic framework
  • Ali R. Khan, M. Beg
  • Mathematics
    2008 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro
  • 2008
TLDR
A longitudinal growth model is proposed which estimates the diffeomorphic flow of a baseline image passing through a series of time-points that are the observed evolution of the template over time, providing a linear space representation of the shape-change via a time-dependent velocity vector field.
Local Geometry of Deformable Templates
TLDR
This paper provides a rigorous and general construction of this infinite dimensional "shape manifold" on which a Riemannian metric is placed and uses this to provide a geometrically founded linear approximation of the deformations of shapes in the neighborhood of a given template.
The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces
We consider the dynamic interpolation problem for nonlinear control systems modeled by second-order differential equations whose configuration space is a Riemannian manifoldM. In this problem we are
On the metrics and euler-lagrange equations of computational anatomy.
TLDR
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.
On the Diffusion of Shape
The main objective of this thesis is to study the global geometric properties of a manifold embedded in Euclidean space, as it evolves under a stochastic flow of diffeomorphisms. The processes
Modeling Planar Shape Variation via Hamiltonian Flows of Curves
TLDR
The geodesic equations that govern the time evolution of an optimal matching in the case of the action on 2D curves with various driving matching terms are derived, and a Hamiltonian formulation is provided in which the initial momentum is represented by an L 2 vector field on the boundary of the template.
Geodesic Shooting for Computational Anatomy
TLDR
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.
Maximum-Likelihood Estimation of Biological Growth Variables
TLDR
This work first estimates optimal deformation from magnetic resonance image data, and uses an iterative solution to reach maximum-likelihood estimates of seed placements and RDFS, and demonstrates this approach using MRI images of human brain growth.
Geodesic Shooting and Diffeomorphic Matching Via Textured Meshes
TLDR
Two optimization methods formulated in terms of the initial momentum are analyzed and compared: direct optimization by gradient descent, or root-finding for the transversality equation, enhanced by a preconditioning of the Jacobian.
...
1
2
3
4
...