Lesson 10: Diffusion Models - From Theory to Practice
Introduction
Welcome to Lesson 10! Today, we're diving into the fascinating world of Diffusion Models. These models have revolutionized image generation, producing incredibly realistic images from noise. By the end of this lesson, you'll understand how diffusion models work and create your own simple image generator!
We'll cover three main topics: the mathematics behind diffusion models, implementing a basic diffusion model, and scaling up to a Stable Diffusion-like model. Don't worry if some concepts seem challenging - we'll break them down with simple analogies and hands-on examples.
1. Mathematics of Diffusion Models
Diffusion models work by gradually adding noise to an image and then learning to reverse this process. It's like slowly stirring milk into coffee and then learning how to un-stir it!
The process involves two main steps:
- Forward diffusion: Gradually add noise to an image over several steps
- Reverse diffusion: Learn to remove the noise step by step, eventually recreating the original image
The magic lies in the reverse process. Once a model learns how to remove noise, it can start with pure noise and generate entirely new images!
Interactive Visualization: Diffusion Process
Let's visualize how the diffusion process works. Use the slider to add noise to an image, then click "Generate" to see a simulated reverse diffusion process:
2. Implementing a Basic Diffusion Model
Now that we understand the concept, let's implement a basic diffusion model using PyTorch. We'll create a simple neural network that learns to remove noise from images.
Here's a basic implementation of a diffusion model:
import torch
import torch.nn as nn
import torch.nn.functional as F
class SimpleDiffusionModel(nn.Module):
def __init__(self, input_channels=1, hidden_channels=64, output_channels=1):
super(SimpleDiffusionModel, self).__init__()
self.conv1 = nn.Conv2d(input_channels, hidden_channels, kernel_size=3, padding=1)
self.conv2 = nn.Conv2d(hidden_channels, hidden_channels, kernel_size=3, padding=1)
self.conv3 = nn.Conv2d(hidden_channels, output_channels, kernel_size=3, padding=1)
def forward(self, x, t):
# t is the timestep, we'll use it to condition the model
t = t.unsqueeze(-1).unsqueeze(-1)
x = torch.cat([x, t.expand(-1, -1, x.shape[2], x.shape[3])], dim=1)
x = F.relu(self.conv1(x))
x = F.relu(self.conv2(x))
return self.conv3(x)
def diffusion_process(model, x, num_steps=1000, beta_start=1e-4, beta_end=0.02):
betas = torch.linspace(beta_start, beta_end, num_steps)
alphas = 1 - betas
alphas_cumprod = torch.cumprod(alphas, dim=0)
for t in reversed(range(num_steps)):
t_tensor = torch.full((x.shape[0],), t, device=x.device)
noise_pred = model(x, t_tensor)
alpha_t = alphas_cumprod[t]
alpha_t_prev = alphas_cumprod[t-1] if t > 0 else torch.tensor(1.0)
beta_t = 1 - alpha_t / alpha_t_prev
if t > 0:
noise = torch.randn_like(x)
else:
noise = torch.zeros_like(x)
x = (1 / torch.sqrt(alpha_t)) * (x - ((1 - alpha_t) / torch.sqrt(1 - alpha_t)) * noise_pred) + torch.sqrt(beta_t) * noise
return x
# Usage
model = SimpleDiffusionModel()
x = torch.randn(1, 1, 28, 28) # Start with random noise
generated_image = diffusion_process(model, x)
This code defines a simple diffusion model with a few convolutional layers. The `diffusion_process` function simulates the reverse diffusion process, gradually removing noise from the input.
3. Scaling up to a Stable Diffusion-like Model
Stable Diffusion, one of the most popular image generation models, is essentially a scaled-up version of our basic diffusion model with some key improvements:
- U-Net architecture for better image understanding
- Attention mechanisms to capture long-range dependencies
- Conditioning on text descriptions for controlled generation
Here's a simplified version of a Stable Diffusion-like model:
import torch
import torch.nn as nn
import torch.nn.functional as F
class ResidualBlock(nn.Module):
def __init__(self, in_channels, out_channels):
super(ResidualBlock, self).__init__()
self.conv1 = nn.Conv2d(in_channels, out_channels, 3, padding=1)
self.conv2 = nn.Conv2d(out_channels, out_channels, 3, padding=1)
self.bn1 = nn.BatchNorm2d(out_channels)
self.bn2 = nn.BatchNorm2d(out_channels)
self.shortcut = nn.Conv2d(in_channels, out_channels, 1) if in_channels != out_channels else nn.Identity()
def forward(self, x):
residual = self.shortcut(x)
x = F.relu(self.bn1(self.conv1(x)))
x = self.bn2(self.conv2(x))
return F.relu(x + residual)
class AttentionBlock(nn.Module):
def __init__(self, channels):
super(AttentionBlock, self).__init__()
self.attn = nn.MultiheadAttention(channels, 4, batch_first=True)
self.ln = nn.LayerNorm(channels)
def forward(self, x):
x = x.flatten(2).transpose(1, 2)
attn_output, _ = self.attn(x, x, x)
return self.ln(x + attn_output).transpose(1, 2).view(*x.shape)
class UNet(nn.Module):
def __init__(self, in_channels=3, out_channels=3, time_emb_dim=256):
super(UNet, self).__init__()
self.time_mlp = nn.Sequential(
nn.Linear(1, time_emb_dim),
nn.ReLU(),
nn.Linear(time_emb_dim, time_emb_dim),
)
self.conv_in = nn.Conv2d(in_channels, 64, 3, padding=1)
self.down1 = nn.Sequential(ResidualBlock(64, 128), AttentionBlock(128))
self.down2 = nn.Sequential(ResidualBlock(128, 256), AttentionBlock(256))
self.down3 = nn.Sequential(ResidualBlock(256, 256), AttentionBlock(256))
self.bot1 = ResidualBlock(256, 512)
self.bot2 = ResidualBlock(512, 512)
self.bot3 = ResidualBlock(512, 256)
self.up1 = nn.Sequential(ResidualBlock(512, 128), AttentionBlock(128))
self.up2 = nn.Sequential(ResidualBlock(256, 64), AttentionBlock(64))
self.up3 = nn.Sequential(ResidualBlock(128, 64), AttentionBlock(64))
self.conv_out = nn.Conv2d(64, out_channels, 3, padding=1)
def forward(self, x, t):
t = self.time_mlp(t.unsqueeze(-1))
x1 = self.conv_in(x)
x2 = self.down1(x1)
x3 = self.down2(x2)
x4 = self.down3(x3)
x4 = self.bot1(x4)
x4 = self.bot2(x4)
x4 = self.bot3(x4)
x = self.up1(torch.cat([x4, x3], dim=1))
x = self.up2(torch.cat([x, x2], dim=1))
x = self.up3(torch.cat([x, x1], dim=1))
return self.conv_out(x)
# Usage
model = UNet()
x = torch.randn(1, 3, 256, 256) # Start with random noise
t = torch.tensor([500]) # Example timestep
output = model(x, t)
This model uses a U-Net architecture with residual blocks and attention mechanisms. It's much more powerful than our basic model and can generate high-quality images when trained on large datasets.
Challenge: Extend the Diffusion Model
Now it's your turn to experiment with diffusion models! Here are some ideas to extend our basic implementation:
- Implement a training loop for the basic diffusion model using a simple dataset (e.g., MNIST)
- Add text conditioning to the model to generate images based on text descriptions
- Experiment with different noise schedules in the forward and reverse processes
- Implement a simple web interface to interact with your trained diffusion model
- Try to generate images of a specific category (e.g., only faces or only landscapes)
This challenge will help you deepen your understanding of diffusion models and give you hands-on experience with state-of-the-art image generation techniques.
Conclusion
Congratulations! You've just taken your first steps into the world of diffusion models. We've covered the basic mathematics behind these models, implemented a simple version, and explored how to scale up to more powerful architectures like Stable Diffusion.
Remember, the field of AI and machine learning is constantly evolving, and diffusion models are at the cutting edge. Keep experimenting, stay curious, and don't be afraid to dive deeper into the topics that interest you most!