BEP016: Revise MRtrix3 SH description

Revising based on MRtrix3:mrtrix3#1635.

src/05-derivatives/05-diffusion-derivatives.md

@@ -449,31 +449,47 @@ another.
 
 #### Spherical Harmonics bases
 
--  `MRtrix3`
+Basis functions:
 
-   -  Antipodally symmetric: all basis functions with odd degree are
-      assumed zero; `AntipodalSymmetry` MUST NOT be set to True.
+![SH basis functions][https://latex.codecogs.com/gif.latex?Y_l^m(\theta,\phi)&space;=&space;\sqrt{\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}}&space;P_l^m(\cos&space;\theta)&space;e^{im\phi}"]
+
+for integer *order* *l*, *phase* *m*, associated Legendre polynomials *P*.
+
+(Truncated) basis coefficients:
+
+![SH basis coefficients][https://latex.codecogs.com/gif.latex?f(\theta,\phi)&space;=&space;\sum_{l=0}^{l_\text{max}}&space;\sum_{m=-l}^{l}&space;c_l^m&space;Y_l^m(\theta,\phi)"]
+
+for *maximum* spherical harmonic order *l<sub>max<\sub>*.
+
+-  `MRtrix3`
 
    -  Functions assumed to be real: conjugate symmetry is assumed, i.e.
       *Y*(*l*,-*m*) = *Y*(*l*,*m*)\*, where \* denotes the complex
       conjugate.
 
-   -  Mapping of image volumes to spherical harmonic basis function
-      coefficients:
+   -  Antipodally symmetric: all basis functions with odd degree are
+      assumed zero; `AntipodalSymmetry` MUST NOT be set to True.
+
+   -  Utilised basis functions:
+
+      ![MRtrix3 SH basis functions][https://latex.codecogs.com/gif.latex?Y_{lm}(\theta,\phi)&space;=&space;\begin{Bmatrix}&space;0&space;&&space;\text{if&space;}l\text{&space;is&space;odd},&space;\\&space;\sqrt{2}&space;\:&space;\text{Im}&space;\left[&space;Y_l^{-m}(\theta,\phi)&space;\right]&space;&&space;\text{if&space;}m&space;<&space;0,&space;\\&space;Y_l^0(\theta,\phi)&space;&&space;\text{if&space;}m&space;=&space;0,&space;\\&space;\sqrt{2}&space;\:&space;\text{Re}&space;\left[&space;Y_l^m(\theta,\phi)&space;\right]&space;&&space;\text{if&space;}m&space;>&space;0,&space;\\&space;\end{Bmatrix}]
 
-      | **Volume** | **Coefficient**                   |
-      | ---------- | --------------------------------- |
-      | 0          | *l* = 0, *m* = 0                  |
-      | 1          | *l* = 2, *m* = 2 (imaginary part) |
-      | 2          | *l* = 2, *m* = 1 (imaginary part) |
-      | 3          | *l* = 2, *m* = 0                  |
-      | 4          | *l* = 2, *m* = 1 (real part)      |
-      | 5          | *l* = 2, *m* = 2 (real part)      |
-      | 6          | *l* = 4, *m* = 4 (imaginary part) |
-      | 7          | *l* = 4, *m* = 3 (imaginary part) |
-      | ...        | etc.                              |
+   -  Mapping between image volume *V<sub>lm</sub>* and spherical harmonic basis
+      function coefficient *c<sub>lm</sub>*:
 
-   -  Normalisation: ***TODO***
+      *V<sub>lm</sub>* = (*l*(*l*+1)) / 2 + *m*
+
+      | ***V<sub>lm</sub>** | ***c<sub>lm<\sub>*** |
+      | ------------------- | -------------------- |
+      | 0                   | *l* = 0, *m* = 0     |
+      | 1                   | *l* = 2, *m* = -2    |
+      | 2                   | *l* = 2, *m* = -1    |
+      | 3                   | *l* = 2, *m* = 0     |
+      | 4                   | *l* = 2, *m* = 1     |
+      | 5                   | *l* = 2, *m* = 2     |
+      | 6                   | *l* = 4, *m* = -4    |
+      | 7                   | *l* = 4, *m* = -3    |
+      | ...                 | etc.                 |
 
    -  Relationship between maximal spherical harmonic degree *l<sub>max</sub>*
       and number of image volumes *N*:
@@ -495,10 +511,6 @@ another.
       | --------------------- | --: | --: | --: | --: | --: | :--: |
       | ***N***               |   1 |   2 |  3  |  4  |  5  | etc. |
 
--  `Descoteaux`
-
-   ***TODO***
-
 ## Demonstrative examples
 
 -  A basic Diffusion Tensor fit: