1. Pix2Pix的生成器

Pix2Pix模型的生成器的输入输出都是图像格式，因此采用UNet网络结构。

class UNetDown(nn.Module):
    def __init__(self, in_size, out_size, normalize=True, dropout=0.0):
        super(UNetDown, self).__init__()
        layers = [nn.Conv2d(in_size, out_size, 4, 2, 1, bias=False)]
        if normalize:
            layers.append(nn.InstanceNorm2d(out_size))
        layers.append(nn.LeakyReLU(0.2))
        if dropout:
            layers.append(nn.Dropout(dropout))
        self.model = nn.Sequential(*layers)

    def forward(self, x):
        return self.model(x)

class UNetUp(nn.Module):
    def __init__(self, in_size, out_size, dropout=0.0):
        super(UNetUp, self).__init__()
        layers = [
            nn.ConvTranspose2d(in_size, out_size, 4, 2, 1, bias=False),
            nn.InstanceNorm2d(out_size),
            nn.ReLU(inplace=True),
        ]
        if dropout:
            layers.append(nn.Dropout(dropout))
        self.model = nn.Sequential(*layers)

    def forward(self, x, skip_input):
        x = self.model(x)
        x = torch.cat((x, skip_input), 1)
        return x

class GeneratorUNet(nn.Module):
    def __init__(self, in_channels=3, out_channels=3):
        super(GeneratorUNet, self).__init__()

        self.down1 = UNetDown(in_channels, 64, normalize=False)
        self.down2 = UNetDown(64, 128)
        self.down3 = UNetDown(128, 256)
        self.down4 = UNetDown(256, 512, dropout=0.5)
        self.down5 = UNetDown(512, 512, dropout=0.5)
        self.down6 = UNetDown(512, 512, dropout=0.5)
        self.down7 = UNetDown(512, 512, dropout=0.5)
        self.down8 = UNetDown(512, 512, normalize=False, dropout=0.5)

        self.up1 = UNetUp(512, 512, dropout=0.5)
        self.up2 = UNetUp(1024, 512, dropout=0.5)
        self.up3 = UNetUp(1024, 512, dropout=0.5)
        self.up4 = UNetUp(1024, 512, dropout=0.5)
        self.up5 = UNetUp(1024, 256)
        self.up6 = UNetUp(512, 128)
        self.up7 = UNetUp(256, 64)

        self.final = nn.Sequential(
            nn.Upsample(scale_factor=2),
            nn.ZeroPad2d((1, 0, 1, 0)),
            nn.Conv2d(128, out_channels, 4, padding=1),
            nn.Tanh(),
        )

    def forward(self, x):
        # U-Net generator with skip connections from encoder to decoder
        d1 = self.down1(x)
        d2 = self.down2(d1)
        d3 = self.down3(d2)
        d4 = self.down4(d3)
        d5 = self.down5(d4)
        d6 = self.down6(d5)
        d7 = self.down7(d6)
        d8 = self.down8(d7)
        u1 = self.up1(d8, d7)
        u2 = self.up2(u1, d6)
        u3 = self.up3(u2, d5)
        u4 = self.up4(u3, d4)
        u5 = self.up5(u4, d3)
        u6 = self.up6(u5, d2)
        u7 = self.up7(u6, d1)
        return self.final(u7)

2. Pix2Pix的判别器

通常GAN的判别器相当于一个二分类器，把输入图像转化成是否属于真实图像的一个标量值。本文作者设计了一种PatchGAN结构，把判别器设计为全卷积网络，输出为一个$N \times N$矩阵，其中的每个元素对应输入图像的一个子区域，用来评估该子区域的真实性。

class Discriminator(nn.Module):
    def __init__(self, in_channels=3):
        super(Discriminator, self).__init__()

        def discriminator_block(in_filters, out_filters, normalization=True):
            """Returns downsampling layers of each discriminator block"""
            layers = [nn.Conv2d(in_filters, out_filters, 4, stride=2, padding=1)]
            if normalization:
                layers.append(nn.InstanceNorm2d(out_filters))
            layers.append(nn.LeakyReLU(0.2, inplace=True))
            return layers

        self.model = nn.Sequential(
            *discriminator_block(in_channels * 2, 64, normalization=False),
            *discriminator_block(64, 128),
            *discriminator_block(128, 256),
            *discriminator_block(256, 512),
            nn.ZeroPad2d((1, 0, 1, 0)),
            nn.Conv2d(512, 1, 4, padding=1, bias=False),
            nn.Sigmoid()
        )

    def forward(self, img_A, img_B):
        # Concatenate image and condition image by channels to produce input
        img_input = torch.cat((img_A, img_B), 1)
        return self.model(img_input)

3. Pix2Pix的目标函数

Pix2Pix的判别器采用标准的交叉熵损失，生成器除交叉熵损失外，还引入了转换图像$\hat{y}=G(x)$和对应风格图像$y$的L1重构损失：

\[\begin{aligned} \mathop{\max}_{D} & \Bbb{E}_{(x,y) \text{~} (P_{data}(x),P_{data}(y))}[\log D(x,y)] + \Bbb{E}_{x \text{~} P_{data}(x)}[\log(1-D(x,G(x)))] \\ \mathop{ \min}_{G} & -\Bbb{E}_{x \text{~} P_{data}(x)}[\log(D(x,G(x))] + \lambda \Bbb{E}_{(x,y) \text{~} (P_{data}(x),P_{data}(y))}[||y-G(x)||_1] \end{aligned}\]

Pix2Pix的完整pytorch实现可参考PyTorch-GAN，下面给出其损失函数的计算和参数更新过程：

# Loss functions
criterion_GAN = torch.nn.BCELoss()
criterion_pixelwise = torch.nn.L1Loss()

# Calculate output of image discriminator (PatchGAN)
patch = (1, opt.img_height // 2 ** 4, opt.img_width // 2 ** 4)

for epoch in range(opt.n_epochs):
    for i, (real_A, real_B) in enumerate(zip(dataloader_A, dataloader_B)):
        # Adversarial ground truths
        valid = torch.ones(real_A.shape[0], *patch).requires_grad_.(False)
        fake = torch.zeros(real_A.shape[0], *patch).requires_grad_.(False)

        # Generate a batch of images
        fake_B = generator(real_A)

        # ---------------------
        #  Train Discriminator
        # ---------------------
        optimizer_D.zero_grad()

        # Loss for real images
        # Real loss
        pred_real = discriminator(real_B, real_A)
        loss_real = criterion_GAN(pred_real, valid)

        # Fake loss
        pred_fake = discriminator(fake_B.detach(), real_A)
        loss_fake = criterion_GAN(pred_fake, fake)

        # Total loss
        d_loss = 0.5 * (loss_real + loss_fake)
        d_loss.backward()
        optimizer_D.step()

        # -----------------
        #  Train Generator
        # -----------------
        optimizer_G.zero_grad()

        # GAN loss
        pred_fake = discriminator(fake_B, real_A)
        loss_GAN = criterion_GAN(pred_fake, valid)
        # Pixel-wise loss
        loss_pixel = criterion_pixelwise(fake_B, real_B)

        # Total loss
        g_loss = loss_GAN + lambda_pixel * loss_pixel
        g_loss.backward()
        optimizer_G.step()