# Overlapping Computations¶

Some array operations require communication of borders between neighboring blocks. Example operations include the following:

• Convolve a filter across an image
• Sliding sum/mean/max, …
• Search for image motifs like a Gaussian blob that might span the border of a block
• Evaluate a partial derivative
• Play the game of Life

Dask Array supports these operations by creating a new array where each block is slightly expanded by the borders of its neighbors. This costs an excess copy and the communication of many small chunks, but allows localized functions to evaluate in an embarrassingly parallel manner.

The main API for these computations is the map_overlap method defined below:

 map_overlap(x, func, depth[, boundary, trim]) Map a function over blocks of the array with some overlap
dask.array.map_overlap(x, func, depth, boundary=None, trim=True, **kwargs)

Map a function over blocks of the array with some overlap

We share neighboring zones between blocks of the array, then map a function, then trim away the neighboring strips.

Parameters: func: function The function to apply to each extended block depth: int, tuple, or dict The number of elements that each block should share with its neighbors If a tuple or dict then this can be different per axis. Asymmetric depths may be specified using a dict value of (-/+) tuples. Note that asymmetric depths are currently only supported when boundary is ‘none’. boundary: str, tuple, dict How to handle the boundaries. Values include ‘reflect’, ‘periodic’, ‘nearest’, ‘none’, or any constant value like 0 or np.nan trim: bool Whether or not to trim depth elements from each block after calling the map function. Set this to False if your mapping function already does this for you **kwargs: Other keyword arguments valid in map_blocks

Examples

>>> import numpy as np

>>> x = np.array([1, 1, 2, 3, 3, 3, 2, 1, 1])
>>> x = da.from_array(x, chunks=5)
>>> def derivative(x):
...     return x - np.roll(x, 1)

>>> y = x.map_overlap(derivative, depth=1, boundary=0)
>>> y.compute()
array([ 1,  0,  1,  1,  0,  0, -1, -1,  0])

>>> x = np.arange(16).reshape((4, 4))
>>> d = da.from_array(x, chunks=(2, 2))
>>> d.map_overlap(lambda x: x + x.size, depth=1).compute()
array([[16, 17, 18, 19],
[20, 21, 22, 23],
[24, 25, 26, 27],
[28, 29, 30, 31]])

>>> func = lambda x: x + x.size
>>> depth = {0: 1, 1: 1}
>>> boundary = {0: 'reflect', 1: 'none'}
>>> d.map_overlap(func, depth, boundary).compute()  # doctest: +NORMALIZE_WHITESPACE
array([[12,  13,  14,  15],
[16,  17,  18,  19],
[20,  21,  22,  23],
[24,  25,  26,  27]])


## Explanation¶

Consider two neighboring blocks in a Dask array:

We extend each block by trading thin nearby slices between arrays:

We do this in all directions, including also diagonal interactions with the overlap function:

>>> import dask.array as da
>>> import numpy as np

>>> x = np.arange(64).reshape((8, 8))
>>> d = da.from_array(x, chunks=(4, 4))
>>> d.chunks
((4, 4), (4, 4))

>>> g = da.overlap.overlap(d, depth={0: 2, 1: 1},
...                       boundary={0: 100, 1: 'reflect'})
>>> g.chunks
((8, 8), (6, 6))

>>> np.array(g)
array([[100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100],
[100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100],
[  0,   0,   1,   2,   3,   4,   3,   4,   5,   6,   7,   7],
[  8,   8,   9,  10,  11,  12,  11,  12,  13,  14,  15,  15],
[ 16,  16,  17,  18,  19,  20,  19,  20,  21,  22,  23,  23],
[ 24,  24,  25,  26,  27,  28,  27,  28,  29,  30,  31,  31],
[ 32,  32,  33,  34,  35,  36,  35,  36,  37,  38,  39,  39],
[ 40,  40,  41,  42,  43,  44,  43,  44,  45,  46,  47,  47],
[ 16,  16,  17,  18,  19,  20,  19,  20,  21,  22,  23,  23],
[ 24,  24,  25,  26,  27,  28,  27,  28,  29,  30,  31,  31],
[ 32,  32,  33,  34,  35,  36,  35,  36,  37,  38,  39,  39],
[ 40,  40,  41,  42,  43,  44,  43,  44,  45,  46,  47,  47],
[ 48,  48,  49,  50,  51,  52,  51,  52,  53,  54,  55,  55],
[ 56,  56,  57,  58,  59,  60,  59,  60,  61,  62,  63,  63],
[100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100],
[100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100]])


## Boundaries¶

With respect to overlaping, you can specify how to handle the boundaries. Current policies include the following:

• periodic - wrap borders around to the other side
• reflect - reflect each border outwards
• any-constant - pad the border with this value

An example boundary kind argument might look like the following:

{0: 'periodic',
1: 'reflect',
2: np.nan}


Alternatively, you can use dask.array.pad() for other types of paddings.

## Map a function across blocks¶

Overlapping goes hand-in-hand with mapping a function across blocks. This function can now use the additional information copied over from the neighbors that is not stored locally in each block:

>>> from scipy.ndimage.filters import gaussian_filter
>>> def func(block):
...    return gaussian_filter(block, sigma=1)

>>> filt = g.map_blocks(func)


While in this case we used a SciPy function, any arbitrary function could have been used instead. This is a good interaction point with Numba.

If your function does not preserve the shape of the block, then you will need to provide a chunks keyword argument. If your block size is regular, then this argument can take a block shape of, for example, (1000, 1000). In case of irregular block sizes, it must be a tuple with the full chunks shape like ((1000, 700, 1000), (200, 300)).

>>> g.map_blocks(myfunc, chunks=(5, 5))


If your function needs to know the location of the block on which it operates, you can give your function a keyword argument block_id:

def func(block, block_id=None):
...


This extra keyword argument will be given a tuple that provides the block location like (0, 0) for the upper-left block or (0, 1) for the block just to the right of that block.

## Trim Excess¶

After mapping a blocked function, you may want to trim off the borders from each block by the same amount by which they were expanded. The function trim_internal is useful here and takes the same depth argument given to overlap:

>>> x.chunks
((10, 10, 10, 10), (10, 10, 10, 10))

>>> y = da.overlap.trim_internal(x, {0: 2, 1: 1})
>>> y.chunks
((6, 6, 6, 6), (8, 8, 8, 8))


## Full Workflow¶

And so, a pretty typical overlaping workflow includes overlap, map_blocks and trim_internal:

>>> x = ...
>>> g = da.overlap.overlap(x, depth={0: 2, 1: 2},
...                       boundary={0: 'periodic', 1: 'periodic'})
>>> g2 = g.map_blocks(myfunc)
>>> result = da.overlap.trim_internal(g2, {0: 2, 1: 2})