Python: How to unwrap circular data to remove discontinuities?

Question

I have a grid of circular data, e.g. the data is given by its angle from 0 to π. Within this data, I have another smaller grid.

This might look like this:

What I want to do is to interpolate the black data on the red dots. Therefore I'm using scipy.interpolate.griddata. This will give me the following result:

As you see, there is a discontinuity when the angle changes from 'almost 0' to 'almost π'.

To remove that, I tried to unwrap the data before interpolation. According to this answer (here). And I get this (better) result, but surprisingly, there is a new disconituity on the right that I don't understand.

So my question is: How to use np.unwrap to get a continous interpolation? Or is there a better way to do so?

Here is the code to reproduce:

import numpy as np
from matplotlib import pyplot as plt
from scipy.interpolate import griddata

ax = plt.subplot()
ax.set_aspect(1)

# Simulate some given data.
x, y = np.meshgrid(np.linspace(-10, 10, 20), np.linspace(-10, 10, 20))
phi = np.arctan2(y, x) % (2 * np.pi)
data = np.arctan2(np.cos(phi), np.sin(phi)) % np.pi

# Plot data.
u = np.cos(data)
v = np.sin(data)
ax.quiver(x, y, u, v, headlength=0.01, headaxislength=0, pivot='middle', units='xy')

# Create a smaller grid within.
x1, y1 = np.meshgrid(np.linspace(-6, 5, 20), np.linspace(-4, 8, 25))
# ax.plot(x1, y1, '.', color='red', markersize=2)

# Prepare data.
data = np.unwrap(2 * data) / 2

# Interpolate data on grid.
interpolation = griddata((x.flatten(), y.flatten()), data.flatten(), (x1.flatten(), y1.flatten()))

# Plot interpolated data.
u1 = np.cos(interpolation)
v1 = np.sin(interpolation)
ax.quiver(x1, y1, u1, v1, headlength=0.01, headaxislength=0, pivot='middle', units='xy',
          scale=3, width=0.03, color='red')

plt.show()

Answer 1

To operate correctly on circular quantities, convert the angles to complex numbers before calling griddata and back to angles afterwards:

c=np.exp(2j*data)  # 0,pi -> 1
# …
a=np.angle(interpolation)/2

The factors of 2 spread your [0,π) out to the whole circle and back again. Note that the normalization implicit in the call to angle will be very sensitive to input that changes too much within one “grid cell” of the input data.