A Closed-Form Solution of Rotation Invariant Spherical Harmonic Features in Diffusion MRI

Abstract

Rotation invariant features are an indispensable tool for characterizing diffusion Magnetic Resonance Imaging (MRI) and in particular for brain tissue microstructure estimation. In this work, we propose a new mathematical framework for efficiently calculating a complete set of such invariants from any spherical function. Specifically, our method is based on the spherical harmonics series expansion of a given function of any order and can be applied directly to the resulting coefficients by performing a simple integral operation analytically. This enable us to derive a general closed-form equation for the invariants. We test our invariants on the diffusion MRI fiber orientation distribution function obtained from the diffusion signal both in-vivo and in synthetic data. Results show how it is possible to use these invariants for characterizing the white matter using a small but complete set of features.

6 Relation Between Signal SH Coefficients and fODF SH Coefficients

$I^d_\mathbf{l } [f]$ is rotation invariant iff $I^d_\mathbf{l } [f] =I^d_\mathbf{l } [R f] =I^d_\mathbf{l } [h]$. Given the rotated SH expansion $h(\mathbf u )$ we can calculate $I^d_\mathbf{l } [h]$ as

$$\begin{aligned} \begin{aligned} I^d_{l_1 \dots l_d}[h]&= \int _{S^2} \prod _{i=1}^d \left[ \sum _{m_i'=-l_i}^{l_i} g_{l_i m_i'} Y_{l_i}^{m_i'}(\mathbf u )\right] d \mathbf u \\&= \int _{S^2} \prod _{i=1}^d \left[ \sum _{m_i'=-l_i}^{l_i} \sum _{m_i=-l_i}^{l_i} c_{l_i m_i} D_{m_i',m_i}^{l_i}(R)Y_{l_i}^{m_i'}(\mathbf u ) \right] d \mathbf u \\&= \int _{S^2} \prod _{i=1}^d \left[ \sum _{m_i=-l_i}^{l_i} c_{l_i m_i} \sum _{m_i'=-l_i}^{l_i} D_{m_i',m_i}^{l_i}(R)Y_{l_i}^{m_i'}(\mathbf u ) \right] d \mathbf u \\&= \int _{S^2} \prod _{i=1}^d \left[ \sum _{m_i=-l_i}^{l_i} c_{l_i m_i} Y_l^{m}(R^{-1}{} \mathbf u ) \right] d \mathbf u \\&=\int _{S^2} \sum _{m_1=-l_1}^{l_1}\cdots \sum _{m_d=-l_d}^{l_d} c_{l_1 m_1} \cdots c_{l_d m_d} Y_{l_1}^{m_1}(R^{-1}{} \mathbf u ) \cdots Y_{l_d}^{m_d}(R^{-1}{} \mathbf u ) d \mathbf u \\&=\sum _{m_1=-l_1}^{l_1}\cdots \sum _{m_d=-l_d}^{l_d} c_{l_1 m_1} \cdots c_{l_d m_d} \int _{S^2} Y_{l_1}^{m_1}(R^{-1}{} \mathbf u ) \cdots Y_{l_d}^{m_d}(R^{-1}{} \mathbf u ) d \mathbf u \\&=\sum _{m_1=-l_1}^{l_1}\cdots \sum _{m_d=-l_d}^{l_d} c_{l_1 m_1} \cdots c_{l_d m_d} G(l_1,m_1 | \cdots |l_d,m_d)\\&= I^d_{l_1 \dots l_d}[f] \end{aligned} \end{aligned}$$

(17)

which proves the invariance property $I^d_\mathbf{l } [f] =I^d_\mathbf{l } [R f]$.