我有一些体积成像数据，这些数据由在 x、y、z 中的规则网格上采样的值组成，但具有非立方体素形状（z 中相邻点之间的空间大于 x、y 中的空间）。我最终希望能够在穿过体积的任意 2D 平面上插值，如下所示：
我知道scipy.ndimage.map_coordinates，但在我的情况下使用它并不那么简单，因为它隐含地假设输入数组中元素的间距在各个维度上是相等的。我可以首先根据最小的体素维度对输入数组进行重新采样（这样我的所有体素都将成为立方体），然后用于map_coordinates在我的平面上进行插值，但对我的数据进行两次插值似乎不是一个好主意。
我也知道scipy对于不规则间隔的 ND 数据（等）有各种插值器LinearNDInterpolator， NearestNDInterpolator但对于我的目的而言，这些插值器非常缓慢且占用大量内存。鉴于我知道值在每个维度内有规律地间隔，插值数据的最佳方法是什么？
你可以使用map_coordinates一点代数。假设您的网格间距是dx,dy和dz。我们需要将这些真实世界坐标映射到数组索引坐标，所以让我们定义三个新变量：
xx = x / dx
yy = y / dy
zz = z / dz
数组索引输入map_coordinates是一个形状数组，其中(d, ...)是d原始数据的维数。如果您定义一个数组，例如：
scaling = np.array([dx, dy, dz])
您可以通过使用一点广播魔术除以将您的真实世界坐标转换为数组索引坐标：scaling
idx = coords / scaling[(slice(None),) + (None,)*(coords.ndim-1)]
举个例子：
dx, dy, dz = 1, 1, 2
scaling = np.array([dx, dy, dz])
data = np.random.rand(10, 15, 5)
假设我们要沿平面插入值2*y - z = 0。我们取两个垂直于平面法线向量的向量：
u = np.array([1, 0 ,0])
v = np.array([0, 1, 2])
并获得我们想要插值的坐标：
coords = (u[:, None, None] * np.linspace(0, 9, 10)[None, :, None] +
v[:, None, None] * np.linspace(0, 2.5, 10)[None, None, :])
我们将它们转换为数组索引坐标并使用map_coordinates：
idx = coords / scaling[(slice(None),) + (None,)*(coords.ndim-1)]
new_data = ndi.map_coordinates(data, idx)
最后一个数组是有形状的(10, 10)，并且在位置[u_idx, v_idx]上具有对应于坐标的值coords[:, u_idx, v_idx]。
您可以基于这个想法来处理坐标不从零开始的插值，方法是在缩放之前添加一个偏移量。
这是一个简单的类Intergrid
，将非均匀网格映射/缩放到均匀网格，然后map_coordinates.
在4d 测试用例中，它以每个查询点大约 1 微秒的速度运行。
pip install [--user] intergrid应该在 python2 或 python3 中工作（2020 年 2 月）；请参阅PyPi 上的 intergrid。
""" 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__ = "denis-bz-py@t-online.de" # 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
我创建了regulargrid包（https://pypi.python.org/pypi/regulargrid/，来源https://github.com/JohannesBuchner/regulargrid）
它通过非常快速的 scipy.ndimage.map_coordinates 为任意坐标比例提供对 n 维笛卡尔网格（根据需要）的支持。
另请参阅此答案：网格数据的快速插值