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?
Linear interpolation is the most straightforward and commonly used interpolation method.
Overview. Linear interpolation is a method of calculating intermediate data between known values by conceptually drawing a straight line between two adjacent known values. An interpolated value is any point along that line.
Linear Interpolation is the technique of determining the values of the functions of any intermediate points when the values of two adjacent points are known. Linear interpolation is basically the estimation of an unknown value that falls within two known values.
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
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