Fast interpolation of regularly sampled 3D data with different intervals in x,y, and z
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.
Here's a simple class Intergrid
that maps / scales non-uniform to uniform grids,
then does map_coordinates.
On a 4d test case it runs at about 1 μsec per query point.
pip install [--user] intergrid should work (February 2020), in python2 or python3; see intergrid on PyPi.
""" interpolate data given on an Nd rectangular grid, uniform or non-uniform.
Purpose: extend the fast N-dimensional interpolator
`scipy.ndimage.map_coordinates` to non-uniform grids, using `np.interp`.
Background: please look at
http://en.wikipedia.org/wiki/Bilinear_interpolation
https://stackoverflow.com/questions/6238250/multivariate-spline-interpolation-in-python-scipy
http://docs.scipy.org/doc/scipy-dev/reference/generated/scipy.ndimage.interpolation.map_coordinates.html
Example
-------
Say we have rainfall on a 4 x 5 grid of rectangles, lat 52 .. 55 x lon -10 .. -6,
and want to interpolate (estimate) rainfall at 1000 query points
in between the grid points.
# define the grid --
griddata = np.loadtxt(...) # griddata.shape == (4, 5)
lo = np.array([ 52, -10 ]) # lowest lat, lowest lon
hi = np.array([ 55, -6 ]) # highest lat, highest lon
# set up an interpolator function "interfunc()" with class Intergrid --
interfunc = Intergrid( griddata, lo=lo, hi=hi )
# generate 1000 random query points, lo <= [lat, lon] <= hi --
query_points = lo + np.random.uniform( size=(1000, 2) ) * (hi - lo)
# get rainfall at the 1000 query points --
query_values = interfunc( query_points ) # -> 1000 values
What this does:
for each [lat, lon] in query_points:
1) find the square of griddata it's in,
e.g. [52.5, -8.1] -> [0, 3] [0, 4] [1, 4] [1, 3]
2) do bilinear (multilinear) interpolation in that square,
using `scipy.ndimage.map_coordinates` .
Check:
interfunc( lo ) -> griddata[0, 0],
interfunc( hi ) -> griddata[-1, -1] i.e. griddata[3, 4]
Parameters
----------
griddata: numpy array_like, 2d 3d 4d ...
lo, hi: user coordinates of the corners of griddata, 1d array-like, lo < hi
maps: a list of `dim` descriptors of piecewise-linear or nonlinear maps,
e.g. [[50, 52, 62, 63], None] # uniformize lat, linear lon
copy: make a copy of query_points, default True;
copy=False overwrites query_points, runs in less memory
verbose: default 1: print a 1-line summary for each call, with run time
order=1: see `map_coordinates`
prefilter: 0 or False, the default: smoothing B-spline
1 or True: exact-fit interpolating spline (IIR, not C-R)
1/3: Mitchell-Netravali spline, 1/3 B + 2/3 fit
(prefilter is only for order > 1, since order = 1 interpolates)
Non-uniform rectangular grids
-----------------------------
What if our griddata above is at non-uniformly-spaced latitudes,
say [50, 52, 62, 63] ? `Intergrid` can "uniformize" these
before interpolation, like this:
lo = np.array([ 50, -10 ])
hi = np.array([ 63, -6 ])
maps = [[50, 52, 62, 63], None] # uniformize lat, linear lon
interfunc = Intergrid( griddata, lo=lo, hi=hi, maps=maps )
This will map (transform, stretch, warp) the lats in query_points column 0
to array coordinates in the range 0 .. 3, using `np.interp` to do
piecewise-linear (PWL) mapping:
50 51 52 53 54 55 56 57 58 59 60 61 62 63 # lo[0] .. hi[0]
0 .5 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 3
`maps[1] None` says to map the lons in query_points column 1 linearly:
-10 -9 -8 -7 -6 # lo[1] .. hi[1]
0 1 2 3 4
More doc: https://denis-bz.github.com/docs/intergrid.html
"""
# split class Gridmap ?
from __future__ import division
from time import time
# warnings
import numpy as np
from scipy.ndimage import map_coordinates, spline_filter
__version__ = "2014-01-15 jan denis" # 15jan: fix bug in linear scaling
__author_email__ = "[email protected]" # comments welcome, testcases most welcome
#...............................................................................
class Intergrid:
__doc__ = globals()["__doc__"]
def __init__( self, griddata, lo, hi, maps=[], copy=True, verbose=1,
order=1, prefilter=False ):
griddata = np.asanyarray( griddata )
dim = griddata.ndim # - (griddata.shape[-1] == 1) # ??
assert dim >= 2, griddata.shape
self.dim = dim
if np.isscalar(lo):
lo *= np.ones(dim)
if np.isscalar(hi):
hi *= np.ones(dim)
self.loclip = lo = np.asarray_chkfinite( lo ).copy()
self.hiclip = hi = np.asarray_chkfinite( hi ).copy()
assert lo.shape == (dim,), lo.shape
assert hi.shape == (dim,), hi.shape
self.copy = copy
self.verbose = verbose
self.order = order
if order > 1 and 0 < prefilter < 1: # 1/3: Mitchell-Netravali = 1/3 B + 2/3 fit
exactfit = spline_filter( griddata ) # see Unser
griddata += prefilter * (exactfit - griddata)
prefilter = False
self.griddata = griddata
self.prefilter = (prefilter == True)
self.maps = maps
self.nmap = 0
if len(maps) > 0:
assert len(maps) == dim, "maps must have len %d, not %d" % (
dim, len(maps))
# linear maps (map None): Xcol -= lo *= scale -> [0, n-1]
# nonlinear: np.interp e.g. [50 52 62 63] -> [0 1 2 3]
self._lo = np.zeros(dim)
self._scale = np.ones(dim)
for j, (map, n, l, h) in enumerate( zip( maps, griddata.shape, lo, hi )):
## print "test: j map n l h:", j, map, n, l, h
if map is None or callable(map):
self._lo[j] = l
if h > l:
self._scale[j] = (n - 1) / (h - l) # _map lo -> 0, hi -> n - 1
else:
self._scale[j] = 0 # h <= l: X[:,j] -> 0
continue
self.maps[j] = map = np.asanyarray(map)
self.nmap += 1
assert len(map) == n, "maps[%d] must have len %d, not %d" % (
j, n, len(map) )
mlo, mhi = map.min(), map.max()
if not (l <= mlo <= mhi <= h):
print "Warning: Intergrid maps[%d] min %.3g max %.3g " \
"are outside lo %.3g hi %.3g" % (
j, mlo, mhi, l, h )
#...............................................................................
def _map_to_uniform_grid( self, X ):
""" clip, map X linear / nonlinear inplace """
np.clip( X, self.loclip, self.hiclip, out=X )
# X nonlinear maps inplace --
for j, map in enumerate(self.maps):
if map is None:
continue
if callable(map):
X[:,j] = map( X[:,j] ) # clip again ?
else:
# PWL e.g. [50 52 62 63] -> [0 1 2 3] --
X[:,j] = np.interp( X[:,j], map, np.arange(len(map)) )
# linear map the rest, inplace (nonlinear _lo 0, _scale 1: noop)
if self.nmap < self.dim:
X -= self._lo
X *= self._scale # (griddata.shape - 1) / (hi - lo)
## print "test: _map_to_uniform_grid", X.T
#...............................................................................
def __call__( self, X, out=None ):
""" query_values = Intergrid(...) ( query_points npt x dim )
"""
X = np.asanyarray(X)
assert X.shape[-1] == self.dim, ("the query array must have %d columns, "
"but its shape is %s" % (self.dim, X.shape) )
Xdim = X.ndim
if Xdim == 1:
X = np.asarray([X]) # in a single point -> out scalar
if self.copy:
X = X.copy()
assert X.ndim == 2, X.shape
npt = X.shape[0]
if out is None:
out = np.empty( npt, dtype=self.griddata.dtype )
t0 = time()
self._map_to_uniform_grid( X ) # X inplace
#...............................................................................
map_coordinates( self.griddata, X.T,
order=self.order, prefilter=self.prefilter,
mode="nearest", # outside -> edge
# test: mode="constant", cval=np.NaN,
output=out )
if self.verbose:
print "Intergrid: %.3g msec %d points in a %s grid %d maps order %d" % (
(time() - t0) * 1000, npt, self.griddata.shape, self.nmap, self.order )
return out if Xdim == 2 else out[0]
at = __call__
# end intergrid.py
I created the regulargrid package (https://pypi.python.org/pypi/regulargrid/, source at https://github.com/JohannesBuchner/regulargrid)
It provides support for n-dimensional Cartesian grids (as needed here) via the very fast scipy.ndimage.map_coordinates for arbitrary coordinate scales.
Also see this answer: Fast interpolation of grid data