Brain connectivity mapping using riemannian geometry, control theory, and PDEs

Christophe Lenglet, Emmanuel Prados, Jean Philippe Pons, Rachid Deriche, Olivier Faugeras

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


We introduce an original approach for the cerebral white matter connectivity mapping from diffusion tensor imaging (DTI). Our method relies on a global modeling of the acquired magnetic resonance imaging volume as a Riemannian manifold whose metric directly derives from the diffusion tensor. These tensors will be used to measure physical three-dimensional distances between different locations of a brain diffusion tensor image. The key concept is the notion of geodesic distance that will allow us to find optimal paths in the white matter. We claim that such optimal paths are reasonable approximations of neural fiber bundles. The geodesic distance function can be seen as the solution of two theoretically equivalent but, in practice, significantly different problems in the partial differential equation framework: an initial value problem which is intrinsically dynamic, and a boundary value problem which is, on the contrary, intrinsically stationary. The two approaches have very different properties which make them more or less adequate for our problem and more or less computationally efficient. The dynamic formulation is quite easy to implement but has several practical drawbacks. On the contrary, the stationary formulation is much more tedious to implement; we will show, however, that it has many virtues which make it more suitable for our connectivity mapping problem. Finally, we will present different possible measures of connectivity, reflecting the degree of connectivity between different regions of the brain. We will illustrate these notions on synthetic and real DTI datasets.

Original languageEnglish (US)
Pages (from-to)285-322
Number of pages38
JournalSIAM Journal on Imaging Sciences
Issue number2
StatePublished - Jan 1 2009

Bibliographical note

Funding Information:
∗Received by the editors December 17, 2007; accepted for publication (in revised form) October 23, 2008; published electronically April 1, 2009. This work was partially supported by NIH (P41 RR008079, P30 NS057091, R01 EB007813, CON000000004051-3014) and INRIA/NSF (0404671) under the U.S.–France Cooperative Research Program. †Center for Magnetic Resonance Research, Department of Radiology and Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 ( This author was affiliated with INRIA Sophia Antipolis-Méditerranée when the work described here was done and with Siemens Corporate Research, Princeton, NJ, when this manuscript was prepared and submitted. ‡Perception Team, INRIA Rhône-Alpes, 38330 Montbonnot, France ( §MOD-EVE Team, Centre Scientifique et Technique du Bâtiment, 06904 Sophia Antipolis, France (jean-philippe. This author was affiliated with the CERTIS Laboratory, École Nationale des Ponts et Chaussées, when the work described here was done. ¶Odyssée Team, INRIA Sophia Antipolis-Méditerranée Research Center, 06902 Sophia Antipolis, France (Rachid. ‖NeuroMathComp Team, INRIA Sophia Antipolis-Méditerranée Research Center, 06902 Sophia Antipolis, France (

Publisher Copyright:
© 2009 Society for Industrial and Applied Mathematics.


  • Anisotropic Eikonal equation
  • Brain connectivity mapping
  • Brownian motion
  • Control theory
  • Diffusion process
  • Diffusion tensor imaging
  • Fast marching methods
  • Hamilton-jacobi-bellman equations
  • Intrinsic distance function
  • Level set
  • Partial differential equations
  • Riemannian manifolds


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