A Log-Euclidean Framework for Statistics on Diffeomorphisms

In this article, we focus 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. Remarkably, this logarithm is a simple 3D vector field, and is well-defined for diffeomorphisms close enough to the identity. This allows to perform vectorial statistics on diffeomorphisms, while preserving the invertibility constraint, contrary to Euclidean statistics on displacement fields. We also present here two efficient algorithms to compute logarithms of diffeomorphisms and exponentials of vector fields, whose accuracy is studied on synthetic data. Finally, we apply these tools to compute the mean of a set of diffeomorphisms, in the context of a registration experiment between an atlas an a database of 9 T1 MR images of the human brain.

[1]  Alain Trouvé,et al.  Diffeomorphisms Groups and Pattern Matching in Image Analysis , 1998, International Journal of Computer Vision.

[2]  Erik Johnsen.  David K. Lloyd and Myron Lipow, Reliability: management, methods, and mathematics, Prentice-Hall Space Technology Series, C. W. Besserer and I Floyd E. Nixon, Editors, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, ■ 1962. , 1964 .

[3]  Olivier D. Faugeras,et al.  Flows of diffeomorphisms for multimodal image registration , 2002, Proceedings IEEE International Symposium on Biomedical Imaging.

[4]  Stephen R. Marsland,et al.  Constructing diffeomorphic representations for the groupwise analysis of nonrigid registrations of medical images , 2004, IEEE Transactions on Medical Imaging.

[5]  P. Hartman.  Ordinary Differential Equations , 1965 .

[6]  Laurent Younes,et al.  Geodesic Interpolating Splines , 2001, EMMCVPR.

[7]  Alejandro F. Frangi,et al.  Automatic Construction of 3D Statistical Deformation Models of the Brain using Non-Rigid Registration , 2003, IEEE Trans. Medical Imaging.

[8]  Michael Golomb,et al.  Ordinary Differential Equations.@@@Differential Equations. , 1952 .

[9]  Nicholas Ayache,et al.  Incorporating Statistical Measures of Anatomical Variability in Atlas-to-Subject Registration for Conformal Brain Radiotherapy , 2005, MICCAI.

[10]  S. Sternberg.  Lectures on Differential Geometry , 1964 .

[11]  Nicholas Ayache,et al.  Fast and Simple Calculus on Tensors in the Log-Euclidean Framework , 2005, MICCAI.

[12]  Nicholas Ayache,et al.  Grid powered nonlinear image registration with locally adaptive regularization , 2004, Medical Image Anal..

[13]  Nicholas Ayache,et al.  Riemannian Elasticity: A Statistical Regularization Framework for Non-linear Registration , 2005, MICCAI.

[14]  Alain Trouvé,et al.  Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms , 2005, International Journal of Computer Vision.

[15]  Alejandro F Frangi,et al.  Automatic construction of 3-D statistical deformation models of the brain using nonrigid registration , 2003, IEEE Transactions on Medical Imaging.

[16]  L. Younes,et al.  Statistics on diffeomorphisms via tangent space representations , 2004, NeuroImage.

[17]  Nicholas J. Higham,et al.  The Scaling and Squaring Method for the Matrix Exponential Revisited , 2005, SIAM J. Matrix Anal. Appl..

[18]  Nicholas Ayache,et al.  A Log-Euclidean Polyaffine Framework for Locally Rigid or Affine Registration , 2006, WBIR.

[19]  Guido Gerig,et al.  Medical Image Computing and Computer-Assisted Intervention - MICCAI 2005, 8th International Conference, Palm Springs, CA, USA, October 26-29, 2005, Proceedings, Part II , 2005, MICCAI.