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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Einops tutorial, part 1
Step1:  Load a batch of images to play with
Step2:  Composition of axes
transposition is very common and useful, but let's move to other capabilities provided by einops
Step3:  Decomposition of axis
Step4:  Order of axes matters
Step5:  Meet einops.reduce
In einops-land you don't need to guess what happened
python
x.mean(-1)
Because you write what the operation does
python
reduce(x, 'b h w c -&gt; b h w', 'mean')
if axis is not present in the output — you guessed it — axis was reduced.
Step6:  Stack and concatenate
Step7:  Addition or removal of axes
You can write 1 to create a new axis of length 1. Similarly you can remove such axis.
There is also a synonym () that you can use. That's a composition of zero axes and it also has a unit length.
Step8:  Repeating elements
Third operation we introduce is repeat
Step9:  Note
Step10:  Fancy examples in random order
(a.k.a. mad designer gallery)
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Python Code:
# Examples are given for numpy. This code also setups ipython/jupyter
# so that numpy arrays in the output are displayed as images
import numpy
from utils import display_np_arrays_as_images
display_np_arrays_as_images()
Explanation: Einops tutorial, part 1: basics
<!-- <img src='http://arogozhnikov.github.io/images/einops/einops_logo_350x350.png' height="80" /> -->
Welcome to einops-land!
We don't write 
python
y = x.transpose(0, 2, 3, 1)
We write comprehensible code
python
y = rearrange(x, 'b c h w -&gt; b h w c')
einops supports widely used tensor packages (such as numpy, pytorch, chainer, gluon, tensorflow), and extends them.
What's in this tutorial?
fundamentals: reordering, composition and decomposition of axes
operations: rearrange, reduce, repeat
how much you can do with a single operation!
Preparations
End of explanation
ims = numpy.load('./resources/test_images.npy', allow_pickle=False)
# There are 6 images of shape 96x96 with 3 color channels packed into tensor
print(ims.shape, ims.dtype)
# display the first image (whole 4d tensor can't be rendered)
ims[0]
# second image in a batch
ims[1]
# we'll use three operations
from einops import rearrange, reduce, repeat
# rearrange, as its name suggests, rearranges elements
# below we swapped height and width.
# In other words, transposed first two axes (dimensions)
rearrange(ims[0], 'h w c -> w h c')
Explanation: Load a batch of images to play with
End of explanation
# einops allows seamlessly composing batch and height to a new height dimension
# We just rendered all images by collapsing to 3d tensor!
rearrange(ims, 'b h w c -> (b h) w c')
# or compose a new dimension of batch and width
rearrange(ims, 'b h w c -> h (b w) c')
# resulting dimensions are computed very simply
# length of newly composed axis is a product of components
# [6, 96, 96, 3] -> [96, (6 * 96), 3]
rearrange(ims, 'b h w c -> h (b w) c').shape
# we can compose more than two axes. 
# let's flatten 4d array into 1d, resulting array has as many elements as the original
rearrange(ims, 'b h w c -> (b h w c)').shape
Explanation: Composition of axes
transposition is very common and useful, but let's move to other capabilities provided by einops
End of explanation
# decomposition is the inverse process - represent an axis as a combination of new axes
# several decompositions possible, so b1=2 is to decompose 6 to b1=2 and b2=3
rearrange(ims, '(b1 b2) h w c -> b1 b2 h w c ', b1=2).shape
# finally, combine composition and decomposition:
rearrange(ims, '(b1 b2) h w c -> (b1 h) (b2 w) c ', b1=2)
# slightly different composition: b1 is merged with width, b2 with height
# ... so letters are ordered by w then by h
rearrange(ims, '(b1 b2) h w c -> (b2 h) (b1 w) c ', b1=2)
# move part of width dimension to height. 
# we should call this width-to-height as image width shrunk by 2 and height doubled. 
# but all pixels are the same!
# Can you write reverse operation (height-to-width)?
rearrange(ims, 'b h (w w2) c -> (h w2) (b w) c', w2=2)
Explanation: Decomposition of axis
End of explanation
# compare with the next example
rearrange(ims, 'b h w c -> h (b w) c')
# order of axes in composition is different
# rule is just as for digits in the number: leftmost digit is the most significant, 
# while neighboring numbers differ in the rightmost axis.
# you can also think of this as lexicographic sort
rearrange(ims, 'b h w c -> h (w b) c')
# what if b1 and b2 are reordered before composing to width?
rearrange(ims, '(b1 b2) h w c -> h (b1 b2 w) c ', b1=2) # produces 'einops'
rearrange(ims, '(b1 b2) h w c -> h (b2 b1 w) c ', b1=2) # produces 'eoipns'
Explanation: Order of axes matters
End of explanation
# average over batch
reduce(ims, 'b h w c -> h w c', 'mean')
# the previous is identical to familiar:
ims.mean(axis=0)
# but is so much more readable
# Example of reducing of several axes 
# besides mean, there are also min, max, sum, prod
reduce(ims, 'b h w c -> h w', 'min')
# this is mean-pooling with 2x2 kernel
# image is split into 2x2 patches, each patch is averaged
reduce(ims, 'b (h h2) (w w2) c -> h (b w) c', 'mean', h2=2, w2=2)
# max-pooling is similar
# result is not as smooth as for mean-pooling
reduce(ims, 'b (h h2) (w w2) c -> h (b w) c', 'max', h2=2, w2=2)
# yet another example. Can you compute result shape?
reduce(ims, '(b1 b2) h w c -> (b2 h) (b1 w)', 'mean', b1=2)
Explanation: Meet einops.reduce
In einops-land you don't need to guess what happened
python
x.mean(-1)
Because you write what the operation does
python
reduce(x, 'b h w c -&gt; b h w', 'mean')
if axis is not present in the output — you guessed it — axis was reduced.
End of explanation
# rearrange can also take care of lists of arrays with the same shape
x = list(ims)
print(type(x), 'with', len(x), 'tensors of shape', x[0].shape)
# that's how we can stack inputs
# "list axis" becomes first ("b" in this case), and we left it there
rearrange(x, 'b h w c -> b h w c').shape
# but new axis can appear in the other place:
rearrange(x, 'b h w c -> h w c b').shape
# that's equivalent to numpy stacking, but written more explicitly
numpy.array_equal(rearrange(x, 'b h w c -> h w c b'), numpy.stack(x, axis=3))
# ... or we can concatenate along axes
rearrange(x, 'b h w c -> h (b w) c').shape
# which is equivalent to concatenation
numpy.array_equal(rearrange(x, 'b h w c -> h (b w) c'), numpy.concatenate(x, axis=1))
Explanation: Stack and concatenate
End of explanation
x = rearrange(ims, 'b h w c -> b 1 h w 1 c') # functionality of numpy.expand_dims
print(x.shape)
print(rearrange(x, 'b 1 h w 1 c -> b h w c').shape) # functionality of numpy.squeeze
# compute max in each image individually, then show a difference 
x = reduce(ims, 'b h w c -> b () () c', 'max') - ims
rearrange(x, 'b h w c -> h (b w) c')
Explanation: Addition or removal of axes
You can write 1 to create a new axis of length 1. Similarly you can remove such axis.
There is also a synonym () that you can use. That's a composition of zero axes and it also has a unit length.
End of explanation
# repeat along a new axis. New axis can be placed anywhere
repeat(ims[0], 'h w c -> h new_axis w c', new_axis=5).shape
# shortcut
repeat(ims[0], 'h w c -> h 5 w c').shape
# repeat along w (existing axis)
repeat(ims[0], 'h w c -> h (repeat w) c', repeat=3)
# repeat along two existing axes
repeat(ims[0], 'h w c -> (2 h) (2 w) c')
# order of axes matters as usual - you can repeat each element (pixel) 3 times 
# by changing order in parenthesis
repeat(ims[0], 'h w c -> h (w repeat) c', repeat=3)
Explanation: Repeating elements
Third operation we introduce is repeat
End of explanation
repeated = repeat(ims, 'b h w c -> b h new_axis w c', new_axis=2)
reduced = reduce(repeated, 'b h new_axis w c -> b h w c', 'min')
assert numpy.array_equal(ims, reduced)
Explanation: Note: repeat operation covers functionality identical to numpy.repeat, numpy.tile and actually more than that.
Reduce ⇆ repeat
reduce and repeat are like opposite of each other: first one reduces amount of elements, second one increases.
In the following example each image is repeated first, then we reduce over new axis to get back original tensor. Notice that operation patterns are "reverse" of each other
End of explanation
# interweaving pixels of different pictures
# all letters are observable
rearrange(ims, '(b1 b2) h w c -> (h b1) (w b2) c ', b1=2)
# interweaving along vertical for couples of images
rearrange(ims, '(b1 b2) h w c -> (h b1) (b2 w) c', b1=2)
# interweaving lines for couples of images
# exercise: achieve the same result without einops in your favourite framework
reduce(ims, '(b1 b2) h w c -> h (b2 w) c', 'max', b1=2)
# color can be also composed into dimension
# ... while image is downsampled
reduce(ims, 'b (h 2) (w 2) c -> (c h) (b w)', 'mean')
# disproportionate resize
reduce(ims, 'b (h 4) (w 3) c -> (h) (b w)', 'mean')
# spilt each image in two halves, compute mean of the two
reduce(ims, 'b (h1 h2) w c -> h2 (b w)', 'mean', h1=2)
# split in small patches and transpose each patch
rearrange(ims, 'b (h1 h2) (w1 w2) c -> (h1 w2) (b w1 h2) c', h2=8, w2=8)
# stop me someone!
rearrange(ims, 'b (h1 h2 h3) (w1 w2 w3) c -> (h1 w2 h3) (b w1 h2 w3) c', h2=2, w2=2, w3=2, h3=2)
rearrange(ims, '(b1 b2) (h1 h2) (w1 w2) c -> (h1 b1 h2) (w1 b2 w2) c', h1=3, w1=3, b2=3)
# patterns can be arbitrarily complicated
reduce(ims, '(b1 b2) (h1 h2 h3) (w1 w2 w3) c -> (h1 w1 h3) (b1 w2 h2 w3 b2) c', 'mean', 
       h2=2, w1=2, w3=2, h3=2, b2=2)
# subtract background in each image individually and normalize
# pay attention to () - this is composition of 0 axis, a dummy axis with 1 element.
im2 = reduce(ims, 'b h w c -> b () () c', 'max') - ims
im2 /= reduce(im2, 'b h w c -> b () () c', 'max')
rearrange(im2, 'b h w c -> h (b w) c')
# pixelate: first downscale by averaging, then upscale back using the same pattern
averaged = reduce(ims, 'b (h h2) (w w2) c -> b h w c', 'mean', h2=6, w2=8)
repeat(averaged, 'b h w c -> (h h2) (b w w2) c', h2=6, w2=8)
rearrange(ims, 'b h w c -> w (b h) c')
# let's bring color dimension as part of horizontal axis
# at the same time horizontal axis is downsampled by 2x
reduce(ims, 'b (h h2) (w w2) c -> (h w2) (b w c)', 'mean', h2=3, w2=3)
Explanation: Fancy examples in random order
(a.k.a. mad designer gallery)
End of explanation
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