Spherical Harmonic bases

Spherical Harmonics (SH) are functions defined on the sphere. A collection of SH can used as a basis function to represent and reconstruct any function on the surface of a unit sphere.

Spherical harmonics are ortho-normal functions defined by:

\[Y_l^m(\theta, \phi) = \sqrt{\frac{2l + 1}{4 \pi} \frac{(l - m)!}{(l + m)!}} P_l^m( cos \theta) e^{i m \phi}\]

where \(l\) is the band index, \(m\) is the order, \(P_l^m\) is an associated \(l\)-th degree, \(m\)-th order Legendre polynomial, and \((\theta, \phi)\) is the representation of the direction vector in the spherical coordinate.

A function \(f(\theta, \phi)\) can be represented using a spherical harmonics basis using the spherical harmonics coefficients \(a_l^m\), which can be computed using the expression:

\[a_l^m = \int_S f(\theta, \phi) Y_l^m(\theta, \phi) ds\]

Once the coefficients are computed, the function \(f(\theta, \phi)\) can be approximately computed as:

\[f(\theta, \phi) = \sum_{l = 0}^{\inf} \sum_{m = -l}^{l} a^m_l Y_l^m(\theta, \phi)\]

In HARDI, the Orientation Distribution Function (ODF) is a function on the sphere.

Several Spherical Harmonics bases have been proposed in the diffusion imaging literature for the computation of the ODF. DIPY implements two of these in the shm module tool set:

  • The basis proposed by Descoteaux et al. 1:

\[\begin{split}Y_i(\theta, \phi) = \begin{cases} \sqrt{2} \Re(Y_l^{|m|}(\theta, \phi)) & -l \leq m < 0, \\ Y_l^0(\theta, \phi) & m = 0, \\ \sqrt{2} \Im(Y_l^m(\theta, \phi)) & 0 < m \leq l \end{cases}\end{split}\]
  • The basis proposed by Tournier et al. 2:

\[\begin{split}Y_i(\theta, \phi) = \begin{cases} \Im(Y_l^{|m|}(\theta, \phi)) & -l \leq m < 0, \\ Y_l^0(\theta, \phi) & m = 0, \\ \Re(Y_{l}^m(\theta, \phi)) & 0 < m \leq l \end{cases}\end{split}\]

In both cases, \(\Re\) denotes the real part of the spherical harmonic basis, and \(\Im\) denotes the imaginary part.

In practice, a maximum even order \(k\) is chosen such that \(k \leq l\). By only taking into account even order SH functions, the above bases can be used to reconstruct symmetric spherical functions. The choice of an even order is motivated by the symmetry of the diffusion process around the origin.

Both bases are also available as full SH bases, where odd order SH functions are also taken into account when reconstructing a spherical function. These full bases can successfully reconstruct asymmetric signals as well as symmetric signals.

Descoteaux et al. 1 use the Q-Ball Imaging (QBI) formalization to recover the ODF, while Tournier et al. 2 use the Spherical Deconvolution (SD) framework to recover the ODF.

References

1(1,2)

Descoteaux, M., Angelino, E., Fitzgibbons, S. and Deriche, R. Regularized, Fast, and Robust Analytical Q‐ball Imaging. Magn. Reson. Med. 2007;58:497-510.

2(1,2)

Tournier J.D., Calamante F. and Connelly A. Robust determination of the fibre orientation distribution in diffusion MRI: Non-negativity constrained super-resolved spherical deconvolution. NeuroImage. 2007;35(4):1459–1472.