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

Purpose: interpolate data given on an N-dimensional rectangular grid, uniform or non-uniform, with the fast `scipy.ndimage.map_coordinates` . Non-uniform grids are first uniformized with `numpy.interp` .

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 .

## 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 import Intergrid  # intergrid.py in \$PYTHONPATH

# 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 "iinterfunc()" 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([ 60, -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]`, 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 http://github.com/denis-bz/interpol.git
# rm -rf interpol/barypol
# add .../interpol/intergrid to PYTHONPATH in ~/.bashrc or ~/.cshrc``````

## 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`: 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`.

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 looks at all neighbors, so is a bit smoother — although I don't know of test images that show a visible difference. 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.

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. `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 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.

## 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 ... Google "regridding" or "resampling".

## 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``````