Finsler Geometry in Diffusion MRI

Tom Dela Haije

Supervisors: Luc Florack Andrea Fuster

Connectomics• Mapping out the structure and function of

the human brain

Multi-modality, Multi-scale

Denk (2004)Feusner (2007)

Palm (2010)

Diffusion MRI

Wedeen (2012)

Diffusion MRI - Basics• Measure diffusion locally• Correlated with fiber orientation

Free diffusionRestricted diffusion

Diffusion MRI - Basics

Stejskal (1965)

Diffusion Tensor Imaging• Diffusion modeled with second order

positive-definite symmetric tensors

Basser (1994)

Diffusion Tensor Imaging

Bangera (2007)

• Diffusion modeled with second order positive-definite symmetric tensors

• Introducing a Riemannian metric

White Matter as a Riemannian Manifold

O’Donnel (2002)

White Matter as a Riemannian Manifold• Elegant perspective:

• Interpolation

• Affine transformations

• Tractography

• Downsides:• Incompatible with complex fiber architecture

High Angular Resolution Diffusion Imaging

Prčkovska (2009)

Diffusion MRI - Basics

White Matter as a Finsler Manifold• Diffusion modeled with a function,

homogeneous of degree 2

White Matter as a Finsler Manifold• Diffusion modeled with a function

• Interpret as a Finsler manifold

Riemann-Finsler Geometry• Advantages:

• Same advantages as Riemannian

• Compatible with complex tissue structure

• Downsides:• More difficult to measure and post-process

Project• Motivation for the metric• Validity of the DTI • Extending the Riemannian case to the

Finsler case• Relating the Finsler interpretation to

existing viewpoints• Operational tools for tractography and

connectivity analysis