## Abstract

We address the problem of the segmentation of cerebral white matter structures from diffusion tensor images (DTI). A DTI produces, from a set of diffusion-weighted MR images, tensor-valued images where each voxel is assigned with a 3 × 3 symmetric, positive-definite matrix. This second order tensor is simply the covariance matrix of a local Gaussian process, with zero-mean, modeling the average motion of water molecules. As we will show in this paper, the definition of a dissimilarity measure and statistics between such quantities is a nontrivial task which must be tackled carefully. We claim and demonstrate that, by using the theoretically well-founded differential geometrical properties of the manifold of multivariate normal distributions, it is possible to improve the quality of the segmentation results obtained with other dissimilarity measures such as the Euclidean distance or the Kullback-Leibler divergence. The main goal of this paper is to prove that the choice of the probability metric, i.e., the dissimilarity measure, has a deep impact on the tensor statistics and, hence, on the achieved results. We introduce a variational formulation, in the level-set framework, to estimate the optimal segmentation of a DTI according to the following hypothesis: Diffusion tensors exhibit a Gaussian distribution in the different partitions. We must also respect the geometric constraints imposed by the interfaces existing among the cerebral structures and detected by the gradient of the DTI. We show how to express all the statistical quantities for the different probability metrics. We validate and compare the results obtained on various synthetic data-sets, a biological rat spinal cord phantom and human brain DTIs.

Original language | English (US) |
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Article number | 1637527 |

Pages (from-to) | 685-700 |

Number of pages | 16 |

Journal | IEEE Transactions on Medical Imaging |

Volume | 25 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2006 |

### Bibliographical note

Funding Information:Manuscript received September 26, 2005; revised February 22, 2006. This work was supported in part by the US-France (INRIA) Cooperative Research under Grant NSF-0404617, Grant NIH R21-RR019771, Grant NIH-RR008079, the MIND Institute and the Keck foundation, the INRIA-FQNRT program, the French National Project ACI Obs-Cerv and the Région Provence-Alpes-Côte d’Azur. Asterisk indicates corresponding author. *C. Lenglet is with INRIA, 2004 route des lucioles, 06902 Sophia-Antipolis, France (e-mail: clenglet@sophia.inria.fr). M. Rousson is with Siemens Corporate Research, Princeton, NJ 08540 USA. R. Deriche is with INRIA, 06902 Sophia-Antipolis, France. Digital Object Identifier 10.1109/TMI.2006.873299

## Keywords

- Diffusion tensor MRI
- Fisher information matrix
- Information geometry
- Kullback-Leibler divergence
- Level-set
- Probability metric
- Riemannian geometry
- Segmentation