我在上篇博客深入浅出 diffusion（1）：白话 diffusion 原理（无公式）中介绍了 diffusion 的一些基本原理，其中谈到了 diffusion 的加噪过程，本文用pytorch 实现下到底是怎么加噪的。

import torch
import math
import numpy as np
from PIL import Image
import requests
import matplotlib.pyplot as plot
import cv2


def linear_beta_schedule(timesteps):
    """
    linear schedule, proposed in original ddpm paper
    """
    scale = 1000 / timesteps
    beta_start = scale * 0.0001
    beta_end = scale * 0.02
    return torch.linspace(beta_start, beta_end, timesteps, dtype = torch.float64)

def cosine_beta_schedule(timesteps, s = 0.008):
    """
    cosine schedule
    as proposed in https://openreview.net/forum?id=-NEXDKk8gZ
    """
    steps = timesteps + 1
    t = torch.linspace(0, timesteps, steps, dtype = torch.float64) / timesteps
    alphas_cumprod = torch.cos((t + s) / (1 + s) * math.pi * 0.5) ** 2
    alphas_cumprod = alphas_cumprod / alphas_cumprod[0]
    betas = 1 - (alphas_cumprod[1:] / alphas_cumprod[:-1])
    return torch.clip(betas, 0, 0.999)
    
   
# 时间步(timestep)定义为1000
timesteps = 1000

# 定义Beta Schedule, 选择线性版本，同DDPM原文一致，当然也可以换成cosine_beta_schedule
betas = linear_beta_schedule(timesteps=timesteps)

# 根据beta定义alpha 
alphas = 1. - betas
alphas_cumprod = torch.cumprod(alphas, axis=0)
sqrt_recip_alphas = torch.sqrt(1.0 / alphas)

# 计算前向过程 diffusion q(x_t | x_{t-1}) 中所需的
sqrt_alphas_cumprod = torch.sqrt(alphas_cumprod)
sqrt_one_minus_alphas_cumprod = torch.sqrt(1. - alphas_cumprod)


def extract(a, t, x_shape):
    batch_size = t.shape[0]
    out = a.gather(-1, t.cpu())
    return out.reshape(batch_size, *((1,) * (len(x_shape) - 1))).to(t.device)

# 前向加噪过程: forward diffusion process
def q_sample(x_start, t, noise=None):
    if noise is None:
        noise = torch.randn_like(x_start)
        cv2.imwrite('noise.png', noise.numpy()*255)

    sqrt_alphas_cumprod_t = extract(sqrt_alphas_cumprod, t, x_start.shape)
    sqrt_one_minus_alphas_cumprod_t = extract(
        sqrt_one_minus_alphas_cumprod, t, x_start.shape
    )
    
    print('sqrt_alphas_cumprod_t :', sqrt_alphas_cumprod_t)
    print('sqrt_one_minus_alphas_cumprod_t :', sqrt_one_minus_alphas_cumprod_t)
    return sqrt_alphas_cumprod_t * x_start + sqrt_one_minus_alphas_cumprod_t * noise

# 图像后处理
def get_noisy_image(x_start, t):
  # add noise
  x_noisy = q_sample(x_start, t=t)

  # turn back into PIL image
  noisy_image = x_noisy.squeeze().numpy()

  return noisy_image

...

# 展示图像, t=0, 50, 100, 500的效果
x_start = cv2.imread('img.png') / 255.0
x_start = torch.tensor(x_start, dtype=torch.float)
cv2.imwrite('img_0.png', get_noisy_image(x_start, torch.tensor([0])) * 255.0)
cv2.imwrite('img_50.png', get_noisy_image(x_start, torch.tensor([50])) * 255.0)
cv2.imwrite('img_100.png', get_noisy_image(x_start, torch.tensor([100])) * 255.0)
cv2.imwrite('img_500.png', get_noisy_image(x_start, torch.tensor([500])) * 255.0)
cv2.imwrite('img_999.png', get_noisy_image(x_start, torch.tensor([999])) * 255.0)


sqrt_alphas_cumprod_t : tensor([[[0.9999]]], dtype=torch.float64)
sqrt_one_minus_alphas_cumprod_t : tensor([[[0.0100]]], dtype=torch.float64)
sqrt_alphas_cumprod_t : tensor([[[0.9849]]], dtype=torch.float64)
sqrt_one_minus_alphas_cumprod_t : tensor([[[0.1733]]], dtype=torch.float64)
sqrt_alphas_cumprod_t : tensor([[[0.9461]]], dtype=torch.float64)
sqrt_one_minus_alphas_cumprod_t : tensor([[[0.3238]]], dtype=torch.float64)
sqrt_alphas_cumprod_t : tensor([[[0.2789]]], dtype=torch.float64)
sqrt_one_minus_alphas_cumprod_t : tensor([[[0.9603]]], dtype=torch.float64)
sqrt_alphas_cumprod_t : tensor([[[0.0064]]], dtype=torch.float64)
sqrt_one_minus_alphas_cumprod_t : tensor([[[1.0000]]], dtype=torch.float64)

        以下分别为原图，t = 0, 50, 100, 500, 999 的结果。

        可见，随着 t 的加大，原图对应的比例系数减小，噪声的强度系数加大，t = 500的时候，隐约可见人脸轮廓，t = 999 的时候，人脸彻底淹没在噪声里面了。