Riemann-Finsler Geometry for Diffusion Weighted Magnetic Resonance Imaging

Abstract

We consider Riemann-Finsler geometry as a potentially powerful mathematical framework in the context of diffusion weighted magnetic resonance imaging. We explain its basic features in heuristic terms, but also provide mathematical details that are essential for practical applications, such as tractography and voxel-based classification. We stipulate a connection between the (dual) Finsler function and signal attenuation observed in the MRI scanner, which directly generalizes Stejskal-Tanner’s solution of the Bloch-Torrey equations and the diffusion tensor imaging (DTI) model inspired by this. The proposed model can therefore be regarded as an extension of DTI. Technically, reconstruction of the (dual) Finsler function from diffusion weighted measurements is a fairly straightforward generalization of the DTI case. The extension of the Riemann differential geometric paradigm for DTI analysis is, however, nontrivial.

Appendix: Horizontal and Vertical Splitting

We may consider the partial derivatives with respect to x ⁱ and $\dot{x}^{i}$ as coordinate vector fields on the tangent bundle TM, and consider the effect induced by a change of coordinates of the base manifold M, x = x(ξ) say. Since $\dot{x}$ is a vector, this induces the following vector transformation law for its components $\dot{x}^{i}$ expressed in terms of its new components, $\dot{\xi }^{p}$, say:

$$\displaystyle\begin{array}{rcl} \dot{x}^{i} = \frac{\partial x^{i}} {\partial \xi ^{p}} \dot{\xi }^{p}\,,& &{}\end{array}$$

(63)

or, equivalently,

$$\displaystyle\begin{array}{rcl} \frac{\partial } {\partial \dot{\xi }^{p}} = \frac{\partial x^{i}} {\partial \xi ^{p}} \frac{\partial } {\partial \dot{x}^{i}}\,,& &{}\end{array}$$

(64)

so that, by construction,

$$\displaystyle\begin{array}{rcl} \dot{x}^{i} \frac{\partial } {\partial x^{i}} = \dot{\xi }^{p} \frac{\partial } {\partial \xi ^{p}}\,.& &{}\end{array}$$

(65)

As a result,

$$\displaystyle\begin{array}{rcl} \frac{\partial } {\partial \xi ^{p}} = \frac{\partial x^{i}} {\partial \xi ^{p}} \frac{\partial } {\partial x^{i}} + \frac{\partial ^{2}x^{i}} {\partial \xi ^{p}\partial \xi ^{q}}\dot{\xi }^{q} \frac{\partial } {\partial \dot{x}^{i}}\,.& &{}\end{array}$$

(66)

Given the definition of the horizontal vectors, Eq. (22), and of the nonlinear connection, Eq. (23), it is then a tedious but straightforward exercise to deduce that

$$\displaystyle{ \frac{\delta } {\delta \xi ^{p}} = \frac{\partial x^{i}} {\partial \xi ^{p}} \frac{\delta } {\delta x^{i}}\,, }$$

(67)

similar to the vector transformation law for the vertical components, recall Eq. (64).

Likewise one has the covector transformation law for the components of the horizontal and vertical one-forms, recall Eq. (33):

$$\displaystyle{ \mathit{dx}^{i} = \frac{\partial x^{i}} {\partial \xi ^{p}} d\xi ^{p}\,, }$$

(68)

respectively

$$\displaystyle{ \delta \dot{x}^{i} = \frac{\partial x^{i}} {\partial \xi ^{p}} \delta \dot{\xi }^{i}\,. }$$

(69)

The “natural” transformation behavior expressed by Eqs. (64) and (67)–(69) motivates the definitions of horizontal and vertical vectors and covectors in Sect. 2.6.