Purpose: interpolate data given on an N-d rectangular grid, uniform or non-uniform, using the fast scipy.ndimage.map_coordinates. Non-uniform grids are first uniformized with numpy.interp.
Keywords, tags: interpolation, rectangular grid, box grid, python, numpy, scipy
Background:
the reader should know some Python and NumPy
(IPython is invaluable for learning both).
For basics of interpolation, see
Bilinear interpolation
on Wikipedia. For map_coordinates, see the example under
multivariate-spline-interpolation-in-python-scipy
on stackoverflow.
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.
from intergrid.intergrid import Intergrid # .../intergrid/intergrid.py
# 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.at( query_points ) # -> 1000 values
What this does: for each [lat, lon] in query_points,
- find the square of
griddatait's in, e.g. [52.5, -8.1] -> [0, 3] [0, 4] [1, 4] [1, 3]\ - 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]
griddata: numpy array_like, 2d 3d 4d ...
lo, hi: user coordinates of the corners of griddata, 1d array-like, lo < hi
maps: an optional list of dim descriptors of piecewise-linear or nonlinear maps,
e.g. [[50, 52, 62, 63], None] \ \ # uniformize lat, linear lon; see below
copy: make a copy of query_points, default True;
copy=False overwrites query_points, runs in less memory
verbose: the default 1 prints a summary of each call, with run time
order: interpolation order:
default 1: bilinear, trilinear ... interpolation using all 2^dim corners
0: each query point -> the nearest grid point -> the value there
2 to 5: spline, see below
prefilter: the kind of spline:
default False: smoothing B-spline
True: exact-fit C-R spline
1/3: Mitchell-Netravali spline, 1/3 B + 2/3 fit
After setting up interfunc = Intergrid(...), either
query_values = interfunc.at( query_points ) # or
query_values = interfunc( query_points )
do the interpolation. (The latter is __call__ in python.)
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
Mapping details
The query_points are first clipped, then columns mapped linearly or non-linearly,
then fed to map_coordinates .
In dim dimensions (i.e. axes or columns), lo and hi are each dim numbers,
the low and high corners of the data grid.
maps is an optional list of dim map descriptors, which can be
None: linear-transform that column,query_points[:,j], togriddata:
lo[j] -> 0
hi[j] -> griddata.shape[j] - 1- a callable function: e.g.
np.logdoes
query_points[:,j] = np.log( query_points[:,j] ) - a sorted array describing a non-uniform grid:
query_points[:,j] =
np.interp( query_points[:,j], [50, 52, 62, 63], [0, 1, 2, 3] )
git clone https://github.com/denis-bz/intergrid.git
# ? pip install --user git+https://github.com/denis-bz/intergrid.git
# ? pip install --user intergrid
# tell python where the intergrid directory is, e.g. in your ~/.bashrc:
# export PYTHONPATH=$PYTHONPATH:.../intergrid/
# test in python or IPython:
from intergrid.intergrid import Intergrid # i.e. .../intergrid/intergrid.py
Intergrid( ... order = 0 to 5 ) gives the spline order:
order=1, the default, does bilinear, trilinear ...
interpolation, which looks at the grid data at all 4 8 16 .. 2^dim corners of
the box around each query point.
order=0 looks at only the one gridpoint
nearest each query point — crude but fast.
order = 2 to 5 does spline interpolation on a uniform
or uniformized grid, looking at (order+1)^dim neighbors of each query point.
Intergrid( ... prefilter = False | True | 1/3 )
specifies the kind of spline, for order >= 2:
prefilter=0 or False, the default: B-spline
prefilter=1 or True: exact-fit spline
prefilter=1/3: M-N spline.
A B-spline goes through smoothed data points,
with [1 4 1] smoothing, [0 0 1 0 0] -> [0 1 4 1 0] / 6.
An exact-fit a.k.a interpolating spline
goes through the data points exactly.
This is not what you want for noisy data,
and may also wiggle or overshoot more than B-splines do.
An M-N spline blends 1/3 B-spline and 2/3 exact-fit; see
Mitchell and Netravali,
Reconstruction filters in computer-graphics ,
1988, and the plots from intergrid/test/MNspline.py.
Prefiltering is a clever transformation
such that Bspline( transform( data )) = exactfitspline( data ).
It is described in a paper by M. Unser,
Splines: A perfect fit for signal and image processing ,
1999.
Uniformizing a grid with PWL, then uniform-splining, is fast and simple, but not as smooth as true splining on the original non-uniform grid. The differences will of course depend on the grid spacings and on how rough the function is.
Run any interpolator on your data with orders 0, 1 ... to get an idea of how the results get smoother, and take longer. Check a few query points by hand; plot some cross-sections.
griddata values can be of any numpy integer or floating type: int8 uint8
.. int32 int64 float32 float64.
Beware of overflow: interpolating uint8 s can give values outside the range 0 .. 255.
(Interpolation in d dimensions can overshoot by (9/8)^d .)
np.float32 will use less memory than np.float64,
but beware of functions in the flow that silently convert everything
to float64. The values must be numbers, not vectors.
Coordinate scaling doesn't matter to Intergrid;
corner weights are calculated in unit cubes of griddata,
after scaling and mapping. If for example griddata column 3
is multiplied by 1000, and lo[3] hi[3] too, the weights are unchanged.
Box grids get big and slow above 5d. A cube with steps 0 .1 .2 .. 1.0 in all dimensions has 11^6 ~ 1.8M points in 6d, 11^8 ~ 200M in 8d. One can reduce that only with a coarser grid like 0 .5 1 in some dimensions (those that vary the least). But time ~ 2^d per query point grows pretty fast.
map_coordinates in 5d with order=1 looks at 32 corner values, with average weight 3 %.
If the weights are roughly equal
(which they will tend to be, by the central limit theorem ?),
sharp edges or gradients will be blurred, and colors mixed to a grey fog.
To see how different interpolators affect images, run matplotlib
plt.imshow( interpolation = "nearest" / "bilinear" / ... ) .
Terminology varies, so the basic kinds of box grids a.k.a. rectangular grids are defined here.
An integer or Cartesian grid has integer coordinates,
e.g. 2 x 3 x 5 points in a numpy array:
A = np.array((2,3,5)); A[0,0,0], A[0,0,1] .. A[1,2,4].
A uniform box grid has nx x ny x nz ... points uniformly spaced, linspace x linspace x linspace ... so all boxes have the same size and are axis-aligned. Examples: 1024 x 768 pixels on a screen, or 4 x 5 points at latitudes [10 20 30 40] x longitudes [-10 -9 -8 -7 -6].
A non-uniform box grid also has nx x ny x nz ... points, but allows non-uniform spacings, e.g. latitudes [-10 0 60 70] x longitudes [-10 -9 0 20 40]; the boxes have different sizes but are still axis-aligned.
(Scattered data, as the name says, has points anywhere,
not only on grid lines.
To interpolate scattered data in scipy, see
scipy.interpolate.griddata
and
scipy.spatial.cKDTree
.)
There are countless varieties of grids: grids with holes, grids warped to various map projections, multiscale / multiresolution grids ...
See intergrid/test/test-4d.py: a 4d grid with 1M scattered query points, uniform / non-uniform box grid, on a 2.5Gz i5 iMac:
shape (361, 720, 47, 8) 98M * 8
Intergrid: 617 msec 1000000 points in a (361, 720, 47, 8) grid 0 maps order 1
Intergrid: 788 msec 1000000 points in a (361, 720, 47, 8) grid 4 maps order 1
scipy.ndimage.interpolation.map_coordinates
scipy reference ndimage
stackoverflow.com/questions/tagged/scipy+interpolation
interpol 2014 -- intergrid + barypol
Google "regrid | resample"
pip search interpol (also gets string interpolation)
and testcases most welcome
— denis-bz-py at t-online dot de