Interpolate data given on an Nd box grid, uniform or non-uniform, using numpy and scipy

## Intergrid: interpolate data given on an N-d rectangular grid

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.

### 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.

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,

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: 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

### Methods

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.)

### 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 .. hi
0  .5   1  1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9  2   3


maps None says to map the lons in query_points column 1 linearly:

-10  -9  -8  -7  -6   # lo .. hi
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], to griddata:
lo[j] -> 0
hi[j] -> griddata.shape[j] - 1
• a callable function: e.g. np.log does
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


### Splines

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.

<small> Fine print: Exact-fit or interpolating splines can be local or global. Catmull-Rom splines and the original M-N splines are local: they look at 4 neighbors of each query point in 1d, 16 in 2d, 64 in 3d. Prefiltering is global, with IIR falloff ~ 1 / 4^distance. (I don't know of test images that show a visible difference to local C-R). Confusingly, the term "Cardinal spline" is sometimes used for local (C-R, FIR), and sometimes for global (IIR prefilter, then B-spline).

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. </small>

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.

### Notes

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 hi 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" / ... ) .

### Kinds of grids

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 ...

### Run times

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
pip search interpol (also gets string interpolation)

and testcases most welcome
— denis-bz-py at t-online dot de

## Project details

This version 2020.2.20

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