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

@article{Miller2002OnTM,
  title={On the metrics and euler-lagrange equations of computational anatomy.},
  author={Michael I. Miller and Alain Trouv{\'e} and Laurent Younes},
  journal={Annual review of biomedical engineering},
  year={2002},
  volume={4},
  pages={
          375-405
        }
}
This paper reviews literature, current concepts and approaches in computational anatomy (CA). The model of CA is a Grenander deformable template, an orbit generated from a template under groups of diffeomorphisms. The metric space of all anatomical images is constructed from the geodesic connecting one anatomical structure to another in the orbit. The variational problems specifying these metrics are reviewed along with their associated Euler-Lagrange equations. The Euler equations of motion… 

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References

SHOWING 1-10 OF 163 REFERENCES
Computational anatomy: an emerging discipline
This paper studies mathematical methods in the emerging new discipline of Computational Anatomy. Herein we formalize the Brown/Washington University model of anatomy following the global pattern
Statistical methods in computational anatomy
TLDR
This paper reviews recent developments by the Washington/Brown groups for the study of anatomical shape in the emerging new discipline of computational anatomy and presents methods for estimating covariances of vector fields from a family of empirically generated maps.
Volumetric transformation of brain anatomy
TLDR
It is shown that transformations constrained by quadratic regularization methods such as the Laplacian, biharmonic, and linear elasticity models, do not ensure that the transformation maintains topology and, therefore, must only be used for coarse global registration.
Deformable templates using large deformation kinematics
TLDR
Application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability.
An adaptive-focus statistical shape model for segmentation and shape modeling of 3-D brain structures
TLDR
The proposed model is adaptive in that it initially focuses on the most reliable structures of interest, and gradually shifts focus to other structures as those become closer to their respective targets and, therefore, more reliable.
Computing the Differential Characteristics of Isointensity Surfaces
TLDR
A new method to compute the differential characteristics of isointensity surfaces from three-dimensional images, based on the implicit representation of the surface, which allows for entirely new formulas, which make use of only the differentials of the 3D image, and which allow to get rid of the problem of parametrizing the surfaces.
A framework for predictive modeling of anatomical deformations
TLDR
A framework for modeling and predicting anatomical deformations is presented, and results are shown that systematic deformations, such as those resulting from change in position or from tumor growth, can be estimated very well using these models.
Integrated approaches to non-rigid registration in medical images
  • Y. Wang, L. Staib
  • Physics
    Proceedings Fourth IEEE Workshop on Applications of Computer Vision. WACV'98 (Cat. No.98EX201)
  • 1998
TLDR
Two new atlas-based methods of 2D single modality non-rigid registration using the combined power of physical and statistical shape models are described and it is shown that statistical boundary shape information significantly augments and improves physical model based non- rigged registration.
Group Actions, Homeomorphisms, and Matching: A General Framework
TLDR
Left-invariant metrics are defined on the product G × I thus allowing the generation of transformations of the background geometry as well as the image values, and structural generation in which image values are changed supporting notions such as tissue creation in carrying one image to another.
...
1
2
3
4
5
...