Fast interpolation of regularly sampled 3D data with different intervals in x,y, and z

Question

I have some volumetric imaging data consisting of values sampled on a regular grid in x,y,z, but with a non-cubic voxel shape (the space between adjacent points in z is grea

Answer 1

You can use map_coordinates with a little bit of algebra. Lets say the spacings of your grid are dx, dy and dz. We need to map these real world coordinates to array index coordinates, so lets define three new variables:

xx = x / dx
yy = y / dy
zz = z / dz

The array index input to map_coordinates is an array of shape (d, ...) where d is the number of dimensions of your original data. If you define an array such as:

scaling = np.array([dx, dy, dz])

you can transform your real world coordinates to array index coordinates by dividing by scaling with a little broadcasting magic:

idx = coords / scaling[(slice(None),) + (None,)*(coords.ndim-1)]

To put it all together in an example:

dx, dy, dz = 1, 1, 2
scaling = np.array([dx, dy, dz])
data = np.random.rand(10, 15, 5)

Lets say we want to interpolate values along the plane 2*y - z = 0. We take two vectors perpendicular to the planes normal vector:

u = np.array([1, 0 ,0])
v = np.array([0, 1, 2])

And get the coordinates at which we want to interpolate as:

coords = (u[:, None, None] * np.linspace(0, 9, 10)[None, :, None] +
          v[:, None, None] * np.linspace(0, 2.5, 10)[None, None, :])

We convert them to array index coordinates and interpoalte using map_coordinates:

idx = coords / scaling[(slice(None),) + (None,)*(coords.ndim-1)]
new_data = ndi.map_coordinates(data, idx)

This last array is of shape (10, 10) and has in position [u_idx, v_idx] the value corresponding to the coordinate coords[:, u_idx, v_idx].

You could build on this idea to handle interpolation where your coordinates don't start at zero, by adding an offset before the scaling.