.. _sec_conv_layer:
画像のための畳み込み
====================
ここまでで、畳み込み層が理論上どのように動作するかを理解した。
いよいよ、実際にどのように機能するかを見ていこう。
画像データの構造を効率よく探索するためのアーキテクチャとして畳み込みニューラルネットワークを考えるという動機に基づき、
ここでは引き続き画像を例として用いる。
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\diilbookstyleinputcell
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from d2l import torch as d2l
import torch
from torch import nn
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\diilbookstyleinputcell
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from d2l import mxnet as d2l
from mxnet import autograd, np, npx
from mxnet.gluon import nn
npx.set_np()
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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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\diilbookstyleoutputcell
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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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from d2l import tensorflow as d2l
import tensorflow as tf
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def corr2d(X, K): #@save
"""Compute 2D cross-correlation."""
h, w = K.shape
Y = d2l.zeros((X.shape[0] - h + 1, X.shape[1] - w + 1))
for i in range(Y.shape[0]):
for j in range(Y.shape[1]):
Y[i, j] = d2l.reduce_sum((X[i: i + h, j: j + w] * K))
return Y
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def corr2d(X, K): #@save
"""Compute 2D cross-correlation."""
h, w = K.shape
Y = d2l.zeros((X.shape[0] - h + 1, X.shape[1] - w + 1))
for i in range(Y.shape[0]):
for j in range(Y.shape[1]):
Y[i, j] = d2l.reduce_sum((X[i: i + h, j: j + w] * K))
return Y
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def corr2d(X, K): #@save
"""Compute 2D cross-correlation."""
h, w = K.shape
Y = jnp.zeros((X.shape[0] - h + 1, X.shape[1] - w + 1))
for i in range(Y.shape[0]):
for j in range(Y.shape[1]):
Y = Y.at[i, j].set((X[i:i + h, j:j + w] * K).sum())
return Y
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def corr2d(X, K): #@save
"""Compute 2D cross-correlation."""
h, w = K.shape
Y = tf.Variable(tf.zeros((X.shape[0] - h + 1, X.shape[1] - w + 1)))
for i in range(Y.shape[0]):
for j in range(Y.shape[1]):
Y[i, j].assign(tf.reduce_sum(
X[i: i + h, j: j + w] * K))
return Y
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class Conv2D(nn.Module):
def __init__(self, kernel_size):
super().__init__()
self.weight = nn.Parameter(torch.rand(kernel_size))
self.bias = nn.Parameter(torch.zeros(1))
def forward(self, x):
return corr2d(x, self.weight) + self.bias
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class Conv2D(nn.Block):
def __init__(self, kernel_size, **kwargs):
super().__init__(**kwargs)
self.weight = self.params.get('weight', shape=kernel_size)
self.bias = self.params.get('bias', shape=(1,))
def forward(self, x):
return corr2d(x, self.weight.data()) + self.bias.data()
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class Conv2D(nn.Module):
kernel_size: int
def setup(self):
self.weight = nn.param('w', nn.initializers.uniform, self.kernel_size)
self.bias = nn.param('b', nn.initializers.zeros, 1)
def forward(self, x):
return corr2d(x, self.weight) + self.bias
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class Conv2D(tf.keras.layers.Layer):
def __init__(self):
super().__init__()
def build(self, kernel_size):
initializer = tf.random_normal_initializer()
self.weight = self.add_weight(name='w', shape=kernel_size,
initializer=initializer)
self.bias = self.add_weight(name='b', shape=(1, ),
initializer=initializer)
def call(self, inputs):
return corr2d(inputs, self.weight) + self.bias
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X = d2l.ones((6, 8))
X[:, 2:6] = 0
X
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\diilbookstyleoutputcell
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:class: output
tensor([[1., 1., 0., 0., 0., 0., 1., 1.],
[1., 1., 0., 0., 0., 0., 1., 1.],
[1., 1., 0., 0., 0., 0., 1., 1.],
[1., 1., 0., 0., 0., 0., 1., 1.],
[1., 1., 0., 0., 0., 0., 1., 1.],
[1., 1., 0., 0., 0., 0., 1., 1.]])
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X = d2l.ones((6, 8))
X[:, 2:6] = 0
X
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\diilbookstyleoutputcell
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:class: output
array([[1., 1., 0., 0., 0., 0., 1., 1.],
[1., 1., 0., 0., 0., 0., 1., 1.],
[1., 1., 0., 0., 0., 0., 1., 1.],
[1., 1., 0., 0., 0., 0., 1., 1.],
[1., 1., 0., 0., 0., 0., 1., 1.],
[1., 1., 0., 0., 0., 0., 1., 1.]])
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X = jnp.ones((6, 8))
X = X.at[:, 2:6].set(0)
X
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\diilbookstyleoutputcell
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:class: output
Array([[1., 1., 0., 0., 0., 0., 1., 1.],
[1., 1., 0., 0., 0., 0., 1., 1.],
[1., 1., 0., 0., 0., 0., 1., 1.],
[1., 1., 0., 0., 0., 0., 1., 1.],
[1., 1., 0., 0., 0., 0., 1., 1.],
[1., 1., 0., 0., 0., 0., 1., 1.]], dtype=float32)
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X = tf.Variable(tf.ones((6, 8)))
X[:, 2:6].assign(tf.zeros(X[:, 2:6].shape))
X
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# Construct a two-dimensional convolutional layer with 1 output channel and a
# kernel of shape (1, 2). For the sake of simplicity, we ignore the bias here
conv2d = nn.LazyConv2d(1, kernel_size=(1, 2), bias=False)
# The two-dimensional convolutional layer uses four-dimensional input and
# output in the format of (example, channel, height, width), where the batch
# size (number of examples in the batch) and the number of channels are both 1
X = X.reshape((1, 1, 6, 8))
Y = Y.reshape((1, 1, 6, 7))
lr = 3e-2 # Learning rate
for i in range(10):
Y_hat = conv2d(X)
l = (Y_hat - Y) ** 2
conv2d.zero_grad()
l.sum().backward()
# Update the kernel
conv2d.weight.data[:] -= lr * conv2d.weight.grad
if (i + 1) % 2 == 0:
print(f'epoch {i + 1}, loss {l.sum():.3f}')
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\diilbookstyleoutputcell
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:class: output
epoch 2, loss 14.442
epoch 4, loss 3.970
epoch 6, loss 1.299
epoch 8, loss 0.477
epoch 10, loss 0.186
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# Construct a two-dimensional convolutional layer with 1 output channel and a
# kernel of shape (1, 2). For the sake of simplicity, we ignore the bias here
conv2d = nn.Conv2D(1, kernel_size=(1, 2), use_bias=False)
conv2d.initialize()
# The two-dimensional convolutional layer uses four-dimensional input and
# output in the format of (example, channel, height, width), where the batch
# size (number of examples in the batch) and the number of channels are both 1
X = X.reshape(1, 1, 6, 8)
Y = Y.reshape(1, 1, 6, 7)
lr = 3e-2 # Learning rate
for i in range(10):
with autograd.record():
Y_hat = conv2d(X)
l = (Y_hat - Y) ** 2
l.backward()
# Update the kernel
conv2d.weight.data()[:] -= lr * conv2d.weight.grad()
if (i + 1) % 2 == 0:
print(f'epoch {i + 1}, loss {float(l.sum()):.3f}')
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\diilbookstyleoutputcell
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:class: output
epoch 2, loss 4.949
epoch 4, loss 0.831
epoch 6, loss 0.140
epoch 8, loss 0.024
epoch 10, loss 0.004
[07:06:32] ../src/base.cc:48: GPU context requested, but no GPUs found.
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# Construct a two-dimensional convolutional layer with 1 output channel and a
# kernel of shape (1, 2). For the sake of simplicity, we ignore the bias here
conv2d = nn.Conv(1, kernel_size=(1, 2), use_bias=False, padding='VALID')
# The two-dimensional convolutional layer uses four-dimensional input and
# output in the format of (example, height, width, channel), where the batch
# size (number of examples in the batch) and the number of channels are both 1
X = X.reshape((1, 6, 8, 1))
Y = Y.reshape((1, 6, 7, 1))
lr = 3e-2 # Learning rate
params = conv2d.init(jax.random.PRNGKey(d2l.get_seed()), X)
def loss(params, X, Y):
Y_hat = conv2d.apply(params, X)
return ((Y_hat - Y) ** 2).sum()
for i in range(10):
l, grads = jax.value_and_grad(loss)(params, X, Y)
# Update the kernel
params = jax.tree_map(lambda p, g: p - lr * g, params, grads)
if (i + 1) % 2 == 0:
print(f'epoch {i + 1}, loss {l:.3f}')
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epoch 2, loss 8.136
epoch 4, loss 1.726
epoch 6, loss 0.437
epoch 8, loss 0.134
epoch 10, loss 0.047
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# Construct a two-dimensional convolutional layer with 1 output channel and a
# kernel of shape (1, 2). For the sake of simplicity, we ignore the bias here
conv2d = tf.keras.layers.Conv2D(1, (1, 2), use_bias=False)
# The two-dimensional convolutional layer uses four-dimensional input and
# output in the format of (example, height, width, channel), where the batch
# size (number of examples in the batch) and the number of channels are both 1
X = tf.reshape(X, (1, 6, 8, 1))
Y = tf.reshape(Y, (1, 6, 7, 1))
lr = 3e-2 # Learning rate
Y_hat = conv2d(X)
for i in range(10):
with tf.GradientTape(watch_accessed_variables=False) as g:
g.watch(conv2d.weights[0])
Y_hat = conv2d(X)
l = (abs(Y_hat - Y)) ** 2
# Update the kernel
update = tf.multiply(lr, g.gradient(l, conv2d.weights[0]))
weights = conv2d.get_weights()
weights[0] = conv2d.weights[0] - update
conv2d.set_weights(weights)
if (i + 1) % 2 == 0:
print(f'epoch {i + 1}, loss {tf.reduce_sum(l):.3f}')
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\diilbookstyleoutputcell
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epoch 2, loss 7.524
epoch 4, loss 1.912
epoch 6, loss 0.587
epoch 8, loss 0.207
epoch 10, loss 0.079
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d2l.reshape(conv2d.weight.data, (1, 2))
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tensor([[ 0.9414, -1.0291]])
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d2l.reshape(conv2d.weight.data(), (1, 2))
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array([[ 0.9895 , -0.9873705]])
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params['params']['kernel'].reshape((1, 2))
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Array([[ 1.0077564, -0.965434 ]], dtype=float32)
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d2l.reshape(conv2d.get_weights()[0], (1, 2))
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