转置卷积Transposed Convolution

我们为卷积神经网络引入的层，包括卷积层和池层，通常会减小输入的宽度和高度，或者保持不变。然而，语义分割和生成对抗网络等应用程序需要预测每个像素的值，因此需要增加输入宽度和高度。转置卷积，也称为分步卷积或反卷积，就是为了达到这一目的。

from mxnet import np, npx, init

from mxnet.gluon import nn

from d2l import mxnet as d2l

npx.set_np()

Basic 2D Transposed Convolution

让我们考虑一个基本情况，输入和输出通道都是1，填充为0，步长为1。图1说明了如何用2×2输入矩阵计算2×2内核的。

Fig. 1. Transposed convolution layer with a 2×22×2 kernel.

可以通过给出矩阵核来实现这个运算
K和矩阵输入X。

def trans_conv(X, K):

h, w = K.shapeY = np.zeros((X.shape[0] + h - 1, X.shape[1] + w - 1))for i in range(X.shape[0]):for j in range(X.shape[1]):Y[i: i + h, j: j + w] += X[i, j] * K

Return

卷积通过Y[i, j] = (X[i: i + h, j: j + w] * K).sum()计算结果，它通过内核汇总输入值。而转置卷积则通过核来传输输入值，从而得到更大的输出。

X = np.array([[0, 1], [2, 3]])

K = np.array([[0, 1], [2, 3]])

trans_conv(X, K)

array([[ 0., 0., 1.],

   [ 0., 4.,  6.],[ 4., 12., 9.]])

或者我们可以用nn.Conv2D转置得到同样的结果。作为nn.Conv2D，输入和核都应该是四维张量。

X, K = X.reshape(1, 1, 2, 2), K.reshape(1, 1, 2, 2)

tconv = nn.Conv2DTranspose(1, kernel_size=2)

tconv.initialize(init.Constant(K))

tconv(X)

array([[[[ 0., 0., 1.],

     [ 0., 4.,  6.],[ 4., 12., 9.])

Padding, Strides, and Channels

在卷积中，我们将填充元素应用于输入，而在转置卷积中将它们应用于输出。A 1×1 padding意味着我们首先正常计算输出，然后删除第一行/最后一列。

tconv = nn.Conv2DTranspose(1, kernel_size=2, padding=1)

tconv.initialize(init.Constant(K))

tconv(X)

array([[4.])

同样，在输出中也应用了这个策略。

tconv = nn.Conv2DTranspose(1, kernel_size=2, strides=2)

tconv.initialize(init.Constant(K))

tconv(X)

array([0., 0., 0., 1.],

     [0., 0., 2., 3.],[0., 2., 0., 3.],[4., 6., 6., 9.])

X = np.random.uniform(size=(1, 10, 16, 16))

conv = nn.Conv2D(20, kernel_size=5, padding=2, strides=3)

tconv = nn.Conv2DTranspose(10, kernel_size=5, padding=2, strides=3)

conv.initialize()

tconv.initialize()

tconv(conv(X)).shape == X.shape

True

Analogy to Matrix Transposition

转置卷积因矩阵转置而得名。实际上，卷积运算也可以通过矩阵乘法来实现。在下面的示例中，我们定义了一个3×3× input XX with a 2×22×2 kernel K，然后使用corr2d计算卷积输出。

X = np.arange(9).reshape(3, 3)

K = np.array([[0, 1], [2, 3]])

Y = d2l.corr2d(X, K)

array([[19., 25.], [37., 43.]])

Next, we rewrite convolution kernel KK as a matrix WW. Its shape will be (4,9)(4,9), where the ithith row present applying the kernel to the input to generate the ithith output element.

def kernel2matrix(K):

k, W = np.zeros(5), np.zeros((4, 9))k[:2], k[3:5] = K[0, :], K[1, :]W[0, :5], W[1, 1:6], W[2, 3:8], W[3, 4:] = k, k, k, kreturn W

W = kernel2matrix(K)

array([[0., 1., 0., 2., 3., 0., 0., 0., 0.],

   [0., 0., 1., 0., 2., 3., 0., 0., 0.],[0., 0., 0., 0., 1., 0., 2., 3., 0.],[0., 0., 0., 0., 0., 1., 0., 2., 3.]])

然后通过适当的整理，用矩阵乘法实现卷积算子。

Y == np.dot(W, X.reshape(-1)).reshape(2, 2)

array([[ True, True],

   [ True, True]])

We can implement transposed convolution as a matrix multiplication as well by reusing kernel2matrix. To reuse the generated WW, we construct a 2×22×2 input, so the corresponding weight matrix will
have a shape (9,4)(9,4), which is W⊤W⊤. Let us verify the results.

X = np.array([0, 1], [2, 3])

Y = trans_conv(X, K)

Y == np.dot(W.T, X.reshape(-1)).reshape(3, 3)

array([[ True, True, True],

   [ True, True,  True],[ True, True,  True]])

Summary

· Compared to convolutions that reduce inputs through kernels, transposed convolutions broadcast inputs.

· If a convolution layer reduces the input width and height by nwnw and hhhh time, respectively. Then a transposed convolution layer with the same kernel sizes, padding and strides will increase the input width and height by nwnw and nhnh, respectively.

· We can implement convolution operations by the matrix multiplication, the corresponding transposed convolutions can be done by transposed matrix multiplication.

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