Adjugate Diffusion Tensors for Geodesic Tractography in White Matter

Abstract

One of the approaches in diffusion tensor imaging is to consider a Riemannian metric given by the inverse diffusion tensor. Such a metric is used for geodesic tractography and connectivity analysis in white matter. We propose a metric tensor given by the adjugate rather than the previously proposed inverse diffusion tensor. The adjugate metric can also be employed in the sharpening framework. Tractography experiments on synthetic and real brain diffusion data show improvement for high-curvature tracts and in the vicinity of isotropic diffusion regions relative to most results for inverse (sharpened) diffusion tensors, and especially on real data. In addition, adjugate tensors are shown to be more robust to noise.

Adaptive Enhancement in Diffusion MRI Through Propagator Sharpening

“…Since in this case tractography relies on the entire diffusion profile instead of just the main directions of diffusion, enhancement has a more pronounced effect. Though there are alternatives [12,13,17,25], we use the inverse of the diffusion tensor as the metric as it is the most well-known definition.…”

Hamiltonian Fast Marching: A Numerical Solver for Anisotropic and Non-Holonomic Eikonal PDEs

“…In differential geometry, the natural distance on a sub-manifold of R n can be described by a Riemannian metric on a parametrization domain. In image segmentation, the Riemannian tensors may stem naturally from the data [25], or be creatively designed based on some local image analysis [6]. In order to discretize Riemannian eikonal equations, we introduce an adequate decomposition of positive definite tensors.…”

“…From this point, the HFM library applies the previous Riemannian discretization strategy (25), with a positive relaxation parameter. In practice, choosing ε := 0.1 yields good results.…”

Is Affine-Invariance Well Defined on SPD Matrices? A Principled Continuum of Metrics

“…For example, the family A r : diag(λ 1 , λ 2 , λ 3 ) −→ diag(a 1 (r)λ 1 , a 2 (r)λ 2 , a 3 (r)λ 3 ) where λ 1 λ 2 λ 3 > 0 was proposed to map the anisotropy of water measured by diffusion tensors to the one of the diffusion of tumor cells in tumor growth modeling [15]. The inverse function inv = pow −1 : Σ −→ Σ −1 or the adjugate function adj : Σ −→ det(Σ)Σ −1 were also proposed in the context of DTI [16,17]. Let us find some properties satisfied by some of these examples.…”