.. _sec_padding:
パディングとストライド
======================
:numref:`fig_correlation` の畳み込みの例を思い出してほしい。
入力の高さと幅はともに 3 で、 畳み込みカーネルの高さと幅はともに 2
であった。 その結果、出力表現の次元は :math:`2\times2` になった。
入力の形状が :math:`n_\textrm{h}\times n_\textrm{w}` で、
畳み込みカーネルの形状が :math:`k_\textrm{h}\times k_\textrm{w}`
だとすると、 出力の形状は
:math:`(n_\textrm{h}-k_\textrm{h}+1) \times (n_\textrm{w}-k_\textrm{w}+1)`
になる。 つまり、畳み込みを適用できる画素が尽きるまでしか、
畳み込みカーネルを移動できないのである。
以下では、パディングやストライド付き畳み込みを含むいくつかの手法を見ていく。
これらは出力サイズをより柔軟に制御するためのものである。
その動機として、カーネルの幅と高さは通常 :math:`1` より大きいため、
多くの畳み込みを連続して適用すると、
出力は入力よりかなり小さくなりがちであることに注意してほしい。
:math:`240 \times 240` ピクセルの画像から始めると、 :math:`5 \times 5`
の畳み込みを 10 層重ねることで 画像は :math:`200 \times 200`
ピクセルまで縮小され、 画像の :math:`30 \%` が切り落とされるうえに、
元画像の境界にある興味深い情報も すべて失われてしまう。 *パディング*
は、この問題に対処するための最も一般的な手段である。
別の状況では、たとえば元の入力解像度が扱いにくいと感じる場合などに、
次元を大幅に削減したいこともある。 そのような場合に役立つ一般的な手法が
*ストライド付き畳み込み* である。
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import torch
from torch import nn
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from mxnet import np, npx
from mxnet.gluon import nn
npx.set_np()
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\diilbookstyleinputcell
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from d2l import jax as d2l
from flax import linen as nn
import jax
from jax import numpy as jnp
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No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)
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import tensorflow as tf
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# We define a helper function to calculate convolutions. It initializes the
# convolutional layer weights and performs corresponding dimensionality
# elevations and reductions on the input and output
def comp_conv2d(conv2d, X):
# (1, 1) indicates that batch size and the number of channels are both 1
X = X.reshape((1, 1) + X.shape)
Y = conv2d(X)
# Strip the first two dimensions: examples and channels
return Y.reshape(Y.shape[2:])
# 1 row and column is padded on either side, so a total of 2 rows or columns
# are added
conv2d = nn.LazyConv2d(1, kernel_size=3, padding=1)
X = torch.rand(size=(8, 8))
comp_conv2d(conv2d, X).shape
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torch.Size([8, 8])
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\diilbookstyleinputcell
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# We define a helper function to calculate convolutions. It initializes
# the convolutional layer weights and performs corresponding dimensionality
# elevations and reductions on the input and output
def comp_conv2d(conv2d, X):
conv2d.initialize()
# (1, 1) indicates that batch size and the number of channels are both 1
X = X.reshape((1, 1) + X.shape)
Y = conv2d(X)
# Strip the first two dimensions: examples and channels
return Y.reshape(Y.shape[2:])
# 1 row and column is padded on either side, so a total of 2 rows or columns are added
conv2d = nn.Conv2D(1, kernel_size=3, padding=1)
X = np.random.uniform(size=(8, 8))
comp_conv2d(conv2d, X).shape
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\diilbookstyleoutputcell
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[07:21:49] ../src/storage/storage.cc:196: Using Pooled (Naive) StorageManager for CPU
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(8, 8)
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\diilbookstyleinputcell
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# We define a helper function to calculate convolutions. It initializes
# the convolutional layer weights and performs corresponding dimensionality
# elevations and reductions on the input and output
def comp_conv2d(conv2d, X):
# (1, X.shape, 1) indicates that batch size and the number of channels are both 1
key = jax.random.PRNGKey(d2l.get_seed())
X = X.reshape((1,) + X.shape + (1,))
Y, _ = conv2d.init_with_output(key, X)
# Strip the dimensions: examples and channels
return Y.reshape(Y.shape[1:3])
# 1 row and column is padded on either side, so a total of 2 rows or columns are added
conv2d = nn.Conv(1, kernel_size=(3, 3), padding='SAME')
X = jax.random.uniform(jax.random.PRNGKey(d2l.get_seed()), shape=(8, 8))
comp_conv2d(conv2d, X).shape
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(8, 8)
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\diilbookstyleinputcell
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# We define a helper function to calculate convolutions. It initializes
# the convolutional layer weights and performs corresponding dimensionality
# elevations and reductions on the input and output
def comp_conv2d(conv2d, X):
# (1, 1) indicates that batch size and the number of channels are both 1
X = tf.reshape(X, (1, ) + X.shape + (1, ))
Y = conv2d(X)
# Strip the first two dimensions: examples and channels
return tf.reshape(Y, Y.shape[1:3])
# 1 row and column is padded on either side, so a total of 2 rows or columns
# are added
conv2d = tf.keras.layers.Conv2D(1, kernel_size=3, padding='same')
X = tf.random.uniform(shape=(8, 8))
comp_conv2d(conv2d, X).shape
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TensorShape([8, 8])
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\diilbookstyleinputcell
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# We use a convolution kernel with height 5 and width 3. The padding on either
# side of the height and width are 2 and 1, respectively
conv2d = nn.LazyConv2d(1, kernel_size=(5, 3), padding=(2, 1))
comp_conv2d(conv2d, X).shape
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\diilbookstyleoutputcell
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:class: output
torch.Size([8, 8])
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\diilbookstyleinputcell
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# We use a convolution kernel with height 5 and width 3. The padding on
# either side of the height and width are 2 and 1, respectively
conv2d = nn.Conv2D(1, kernel_size=(5, 3), padding=(2, 1))
comp_conv2d(conv2d, X).shape
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\diilbookstyleoutputcell
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(8, 8)
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\diilbookstyleinputcell
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# We use a convolution kernel with height 5 and width 3. The padding on
# either side of the height and width are 2 and 1, respectively
conv2d = nn.Conv(1, kernel_size=(5, 3), padding=(2, 1))
comp_conv2d(conv2d, X).shape
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\diilbookstyleoutputcell
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:class: output
(8, 8)
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\diilbookstyleinputcell
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# We use a convolution kernel with height 5 and width 3. The padding on
# either side of the height and width are 2 and 1, respectively
conv2d = tf.keras.layers.Conv2D(1, kernel_size=(5, 3), padding='same')
comp_conv2d(conv2d, X).shape
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\diilbookstyleoutputcell
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:class: output
TensorShape([8, 8])
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\diilbookstyleinputcell
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conv2d = nn.LazyConv2d(1, kernel_size=3, padding=1, stride=2)
comp_conv2d(conv2d, X).shape
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\diilbookstyleoutputcell
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torch.Size([4, 4])
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conv2d = nn.Conv2D(1, kernel_size=3, padding=1, strides=2)
comp_conv2d(conv2d, X).shape
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\diilbookstyleoutputcell
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:class: output
(4, 4)
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conv2d = nn.Conv(1, kernel_size=(3, 3), padding=1, strides=2)
comp_conv2d(conv2d, X).shape
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(4, 4)
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conv2d = tf.keras.layers.Conv2D(1, kernel_size=3, padding='same', strides=2)
comp_conv2d(conv2d, X).shape
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TensorShape([4, 4])
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conv2d = nn.LazyConv2d(1, kernel_size=(3, 5), padding=(0, 1), stride=(3, 4))
comp_conv2d(conv2d, X).shape
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torch.Size([2, 2])
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conv2d = nn.Conv2D(1, kernel_size=(3, 5), padding=(0, 1), strides=(3, 4))
comp_conv2d(conv2d, X).shape
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(2, 2)
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conv2d = nn.Conv(1, kernel_size=(3, 5), padding=(0, 1), strides=(3, 4))
comp_conv2d(conv2d, X).shape
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(2, 2)
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conv2d = tf.keras.layers.Conv2D(1, kernel_size=(3,5), padding='valid',
strides=(3, 4))
comp_conv2d(conv2d, X).shape
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TensorShape([2, 1])
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