Fast interpolation of regularly sampled 3D data with different intervals in x,y, and z
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 greater than in x,y). I would eventually like to be able to interpolate the values on some arbitrary 2D plane that passes through the volume, like this:
I'm aware of scipy.ndimage.map_coordinates, but in my case using it is less straightforward because it implicitly assumes that the spacing of the elements in the input array are equal across dimensions. I could first resample my input array according to the smallest voxel dimension (so that all of my voxels would then be cubes), then use map_coordinates to interpolate over my plane, but it doesn't seem like a great idea to interpolate my data twice.
I'm also aware that scipy has various interpolators for irregularly-spaced ND data (LinearNDInterpolator, NearestNDInterpolator etc.), but these are very slow and memory-intensive for my purposes. What is the best way of interpolating my data given that I know that the values are regularly spaced within each dimension?
Answer
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.