class_docstr: ## DDPM Sampler
  
  This extends the [`DiffusionSampler` base class](index.html).
  
  DDPM samples images by repeatedly removing noise by sampling step by step from
  $p_ heta(x_{t-1} | x_t)$,
  
  egin{align}
  
  p_  heta(x_{t-1} | x_t) &= \mathcal{N}ig(x_{t-1}; \mu_     heta(x_t, t),   ildeeta_t \mathbf{I} ig) \
  
  \mu_t(x_t, t) &= rac{\sqrt{arlpha_{t-1}}eta_t}{1 - arlpha_t}x_0
                       + rac{\sqrt{lpha_t}(1 - arlpha_{t-1})}{1-arlpha_t}x_t \
  
      ildeeta_t &= rac{1 - arlpha_{t-1}}{1 - arlpha_t} eta_t \
  
  x_0 &= rac{1}{\sqrt{arlpha_t}} x_t -  \Big(\sqrt{rac{1}{arlpha_t} - 1}\Big)\epsilon_  heta \
  
  \end{align}
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docstr: ### Sample from $q(x_t|x_0)$
  
  $$q(x_t|x_0) = \mathcal{N} \Big(x_t; \sqrt{arlpha_t} x_0, (1-arlpha_t) \mathbf{I} \Big)$$
  
  :param x0: is $x_0$ of shape `[batch_size, channels, height, width]`
  :param index: is the time step $t$ index
  :param noise: is the noise, $\epsilon$
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code: def q_sample(
      self,
      x0: torch.Tensor,
      index: int,
      noise: Optional[torch.Tensor] = None,
  ):
      """
      ### Sample from $q(x_t|x_0)$
  
      $$q(x_t|x_0) = \mathcal{N} \Big(x_t; \sqrt{\bar\alpha_t} x_0, (1-\bar\alpha_t) \mathbf{I} \Big)$$
  
      :param x0: is $x_0$ of shape `[batch_size, channels, height, width]`
      :param index: is the time step $t$ index
      :param noise: is the noise, $\epsilon$
      """
  
      # Random noise, if noise is not specified
      if noise is None:
          noise = torch.randn_like(x0)
  
      # Sample from $\mathcal{N} \Big(x_t; \sqrt{\bar\alpha_t} x_0, (1-\bar\alpha_t) \mathbf{I} \Big)$
      return (
          self.sqrt_alpha_bar[index] * x0
          + self.sqrt_1m_alpha_bar[index] * noise
      )
  
file_path: annotated_deep_learning_paper_implementations/labml_nn/diffusion/stable_diffusion/sampler/ddpm.py
imports: from typing import Optional, List
  import numpy
  import torch
  from labml import monit
  from labml_nn.diffusion.stable_diffusion.latent_diffusion import LatentDiffusion
  from labml_nn.diffusion.stable_diffusion.sampler import DiffusionSampler
label_desc: Use image node_id 4 for calling the DDPMSampler obj's underlying member method code with example usage: obj.q_sample(x0, index, noise) and returns: unknown