This example shows how you can use a spherical harmonics (SH) function to reconstruct any spherical function using DIPY. In order to generate a signal, we will need to generate an evenly distributed sphere. Let’s import some standard packages.

```
import numpy as np
from dipy.core.sphere import disperse_charges, Sphere, HemiSphere
```

We can first create some random points on a `HemiSphere`

using spherical
polar coordinates.

```
n_pts = 64
theta = np.pi * np.random.rand(n_pts)
phi = 2 * np.pi * np.random.rand(n_pts)
hsph_initial = HemiSphere(theta=theta, phi=phi)
```

Next, we call `disperse_charges`

which will iteratively move the points so
that the electrostatic potential energy is minimized. In `hsph_updated`

we
have the updated `HemiSphere`

with the points nicely distributed on the
hemisphere.

```
hsph_updated, potential = disperse_charges(hsph_initial, 5000)
sphere = Sphere(xyz=np.vstack((hsph_updated.vertices, -hsph_updated.vertices)))
```

We now need to create our initial signal. To do so, we will use our sphere’s
vertices as the sampled points of our spherical function (SF). We will
use `multi_tensor_odf`

to simulate an ODF. For more information on how to use
DIPY to simulate a signal and ODF, see MultiTensor Simulation.

```
from dipy.sims.voxel import multi_tensor_odf
mevals = np.array([[0.0015, 0.00015, 0.00015],
[0.0015, 0.00015, 0.00015]])
angles = [(0, 0), (60, 0)]
odf = multi_tensor_odf(sphere.vertices, mevals, angles, [50, 50])
from dipy.viz import window, actor
# Enables/disables interactive visualization
interactive = False
scene = window.Scene()
scene.SetBackground(1, 1, 1)
odf_actor = actor.odf_slicer(odf[None, None, None, :], sphere=sphere)
odf_actor.RotateX(90)
scene.add(odf_actor)
print('Saving illustration as symm_signal.png')
window.record(scene, out_path='symm_signal.png', size=(300, 300))
if interactive:
window.show(scene)
```

We can now express this signal as a series of SH coefficients using
`sf_to_sh`

. This function converts a series of SF coefficients in a series of
SH coefficients. For more information on SH basis, see Spherical Harmonic bases. For
this example, we will use the `descoteaux07`

basis up to a maximum SH order
of 8.

```
from dipy.reconst.shm import sf_to_sh
# Change this value to try out other bases
sh_basis = 'descoteaux07'
# Change this value to try other maximum orders
sh_order = 8
sh_coeffs = sf_to_sh(odf, sphere, sh_order, sh_basis)
```

`sh_coeffs`

is an array containing the SH coefficients multiplying the SH
functions of the chosen basis. We can use it as input of `sh_to_sf`

to
reconstruct our original signal. We will now reproject our signal on a high
resolution sphere using `sh_to_sf`

.

```
from dipy.data import get_sphere
from dipy.reconst.shm import sh_to_sf
high_res_sph = get_sphere('symmetric724').subdivide(2)
reconst = sh_to_sf(sh_coeffs, high_res_sph, sh_order, sh_basis)
scene.clear()
odf_actor = actor.odf_slicer(reconst[None, None, None, :], sphere=high_res_sph)
odf_actor.RotateX(90)
scene.add(odf_actor)
print('Saving output as symm_reconst.png')
window.record(scene, out_path='symm_reconst.png', size=(300, 300))
if interactive:
window.show(scene)
```

While a symmetric SH basis works well for reconstructing symmetric SF, it fails to do so on asymmetric signals. We will now create such a signal by using a different ODF for each hemisphere of our sphere.

```
mevals = np.array([[0.0015, 0.0003, 0.0003]])
angles = [(0, 0)]
odf2 = multi_tensor_odf(sphere.vertices, mevals, angles, [100])
n_pts_hemisphere = int(sphere.vertices.shape[0] / 2)
asym_odf = np.append(odf[:n_pts_hemisphere], odf2[n_pts_hemisphere:])
scene.clear()
odf_actor = actor.odf_slicer(asym_odf[None, None, None, :], sphere=sphere)
odf_actor.RotateX(90)
scene.add(odf_actor)
print('Saving output as asym_signal.png')
window.record(scene, out_path='asym_signal.png', size=(300, 300))
if interactive:
window.show(scene)
```

Let’s try to reconstruct this SF using a symmetric SH basis.

```
sh_coeffs = sf_to_sh(asym_odf, sphere, sh_order, sh_basis)
reconst = sh_to_sf(sh_coeffs, high_res_sph, sh_order, sh_basis)
scene.clear()
odf_actor = actor.odf_slicer(reconst[None, None, None, :], sphere=high_res_sph)
odf_actor.RotateX(90)
scene.add(odf_actor)
print('Saving output as asym_reconst.png')
window.record(scene, out_path='asym_reconst.png', size=(300, 300))
if interactive:
window.show(scene)
```

As we can see, a symmetric basis fails to properly represent asymmetric SF.
Fortunately, DIPY also implements full SH bases, which can deal with symmetric
as well as asymmetric signals. For this tutorial, we will demonstrate it using
the `descoteaux07`

full SH basis by setting `full_basis=true`

.

```
sh_coeffs = sf_to_sh(asym_odf, sphere, sh_order,
sh_basis, full_basis=True)
reconst = sh_to_sf(sh_coeffs, high_res_sph, sh_order,
sh_basis, full_basis=True)
scene.clear()
odf_actor = actor.odf_slicer(reconst[None, None, None, :], sphere=high_res_sph)
odf_actor.RotateX(90)
scene.add(odf_actor)
print('Saving output as asym_reconst_full.png')
window.record(scene, out_path='asym_reconst_full.png', size=(300, 300))
if interactive:
window.show(scene)
```

As we can see, a full SH basis properly reconstruct asymmetric signal.

Example source code

You can download `the full source code of this example`

. This same script is also included in the dipy source distribution under the `doc/examples/`

directory.