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Pull in extra stuff from my doc PR #30

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merged 1 commit into from May 3, 2018
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Pull in extra stuff from my doc PR #30

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50 changes: 26 additions & 24 deletions dipy/reconst/dki.py
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Expand Up @@ -668,34 +668,36 @@ def mean_kurtosis(dki_params, min_kurtosis=-3./7, max_kurtosis=3):

.. math::

MK=F_1(\lambda_1,\lambda_2,\lambda_3)\hat{W}_{1111}+
F_1(\lambda_2,\lambda_1,\lambda_3)\hat{W}_{2222}+
F_1(\lambda_3,\lambda_2,\lambda_1)\hat{W}_{3333}+ \\
F_2(\lambda_1,\lambda_2,\lambda_3)\hat{W}_{2233}+
F_2(\lambda_2,\lambda_1,\lambda_3)\hat{W}_{1133}+
F_2(\lambda_3,\lambda_2,\lambda_1)\hat{W}_{1122}
MK=F_1(\lambda_1,\lambda_2,\lambda_3)\hat{W}_{1111}+
F_1(\lambda_2,\lambda_1,\lambda_3)\hat{W}_{2222}+
F_1(\lambda_3,\lambda_2,\lambda_1)\hat{W}_{3333}+ \\
F_2(\lambda_1,\lambda_2,\lambda_3)\hat{W}_{2233}+
F_2(\lambda_2,\lambda_1,\lambda_3)\hat{W}_{1133}+
F_2(\lambda_3,\lambda_2,\lambda_1)\hat{W}_{1122}

where $\hat{W}_{ijkl}$ are the components of the $W$ tensor in the
coordinates system defined by the eigenvectors of the diffusion tensor
$\mathbf{D}$ and

F_1(\lambda_1,\lambda_2,\lambda_3)=
\frac{(\lambda_1+\lambda_2+\lambda_3)^2}
{18(\lambda_1-\lambda_2)(\lambda_1-\lambda_3)}
[\frac{\sqrt{\lambda_2\lambda_3}}{\lambda_1}
R_F(\frac{\lambda_1}{\lambda_2},\frac{\lambda_1}{\lambda_3},1)+\\
\frac{3\lambda_1^2-\lambda_1\lambda_2-\lambda_2\lambda_3-
\lambda_1\lambda_3}
{3\lambda_1 \sqrt{\lambda_2 \lambda_3}}
R_D(\frac{\lambda_1}{\lambda_2},\frac{\lambda_1}{\lambda_3},1)-1 ]

F_2(\lambda_1,\lambda_2,\lambda_3)=
\frac{(\lambda_1+\lambda_2+\lambda_3)^2}
{3(\lambda_2-\lambda_3)^2}
[\frac{\lambda_2+\lambda_3}{\sqrt{\lambda_2\lambda_3}}
R_F(\frac{\lambda_1}{\lambda_2},\frac{\lambda_1}{\lambda_3},1)+\\
\frac{2\lambda_1-\lambda_2-\lambda_3}{3\sqrt{\lambda_2 \lambda_3}}
R_D(\frac{\lambda_1}{\lambda_2},\frac{\lambda_1}{\lambda_3},1)-2]
.. math::

F_1(\lambda_1,\lambda_2,\lambda_3)=
\frac{(\lambda_1+\lambda_2+\lambda_3)^2}
{18(\lambda_1-\lambda_2)(\lambda_1-\lambda_3)}
[\frac{\sqrt{\lambda_2\lambda_3}}{\lambda_1}
R_F(\frac{\lambda_1}{\lambda_2},\frac{\lambda_1}{\lambda_3},1)+\\
\frac{3\lambda_1^2-\lambda_1\lambda_2-\lambda_2\lambda_3-
\lambda_1\lambda_3}
{3\lambda_1 \sqrt{\lambda_2 \lambda_3}}
R_D(\frac{\lambda_1}{\lambda_2},\frac{\lambda_1}{\lambda_3},1)-1 ]

F_2(\lambda_1,\lambda_2,\lambda_3)=
\frac{(\lambda_1+\lambda_2+\lambda_3)^2}
{3(\lambda_2-\lambda_3)^2}
[\frac{\lambda_2+\lambda_3}{\sqrt{\lambda_2\lambda_3}}
R_F(\frac{\lambda_1}{\lambda_2},\frac{\lambda_1}{\lambda_3},1)+\\
\frac{2\lambda_1-\lambda_2-\lambda_3}{3\sqrt{\lambda_2 \lambda_3}}
R_D(\frac{\lambda_1}{\lambda_2},\frac{\lambda_1}{\lambda_3},1)-2]

where $R_f$ and $R_d$ are the Carlson's elliptic integrals.

Expand Down Expand Up @@ -906,7 +908,7 @@ def radial_kurtosis(dki_params, min_kurtosis=-3./7, max_kurtosis=10):

Notes
--------
RK is calculated with the following equation [1]_::
RK is calculated with the following equation [1]_:

.. math::

Expand Down