Answers.Body: <blockquote>
  <p>Figure 1. The proposed modeling framework trained on 2-d swiss roll data</p>
  </blockquote>
  <p>The data is actually 2d coordinates, plotted on a scatter-plot. It doesn't represent an <span class="math-container">$n \times n$</span> image with individual pixels. And the added noise is 2d, well 1d to the x-coordinate and 1d to the y-coordinate, if you consider them separately.</p>
  <p>But when the process is applied to images, as seen in Figure 3, it behaves as you described and expected the result to look like.</p>
  
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Answers.CreationDate: 2023-04-14T05:09:58.823Z
Answers.Text: I did not realize that "2-d swiss roll data" is actually represented as a (x,y) scatter plot, thanks very much!!
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Body: <p>In the 2015 paper &quot;<a href="https://arxiv.org/pdf/1503.03585.pdf" rel="nofollow noreferrer">Deep Unsupervised Learning using Nonequilibrium Thermodynamics</a>&quot; by Sohl-Dickstein et al. on diffusion for generative models, Figure 1 shows the forward trajectory for a 2-d swiss-roll image using Gaussian diffusion. The thin lines are gradually blurred into wider and fuzzier lines, and eventually into an identity-covariance Gaussian. Table App.1 gives the diffusion kernel as:</p>
  <p><span class="math-container">$$
  q(\mathbf{x}^{(t)} \mid \mathbf{x}^{(t-1)}) = \mathcal{N}(\mathbf{x}^{(t)} ; \mathbf{x}^{(t-1)} \sqrt{1 - \beta_t}, \mathbf{I} \beta_t )
  $$</span></p>
  <p>The covariance of the diffusion kernel is diagonal, so each component <span class="math-container">$x_i^{(t)}$</span> (i.e., each pixel in the image at time step <span class="math-container">$t$</span>) is independently sampled from a 1-d Gaussian based on the prior time step's pixel value at the <em>same x-y location in the image</em>. So a given pixel should <em>NOT</em> diffuse into neighboring pixels; instead, the action of the diffusion step is a linear Gaussian 1-d transformation of the number held in the pixel, with the mean slightly reduced and some noise added.</p>
  <p><strong>Question</strong>: This seems inconsistent with Figure 1? Instead of the blurred line (wider and fuzzier line), we should have a line that has the same width, but exhibits more noise? In order to have a pixel diffuse into neighboring pixels, we would need a diffusion kernel with a non-diagonal covariance, so that there is nonzero covariance between components?</p>
  
Tags: <generative-model><image-generation><diffusion-models>
Title: Blurring of image in generative model using diffusion probabilistic method
UserDisplayName: sunfishstanford
ThreadText: # Blurring of image in generative model using diffusion probabilistic method
  
  (Asked by: username_0 on 2023-04-12)
  
  In the 2015 paper "[Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://arxiv.org/pdf/1503.03585.pdf)" by Sohl-Dickstein et al. on diffusion for generative models, Figure 1 shows the forward trajectory for a 2-d swiss-roll image using Gaussian diffusion. The thin lines are gradually blurred into wider and fuzzier lines, and eventually into an identity-covariance Gaussian. Table App.1 gives the diffusion kernel as:
  
  $$
  q(\mathbf{x}^{(t)} \mid \mathbf{x}^{(t-1)}) = \mathcal{N}(\mathbf{x}^{(t)} ; \mathbf{x}^{(t-1)} \sqrt{1 - \beta\_t}, \mathbf{I} \beta\_t )
  $$
  
  The covariance of the diffusion kernel is diagonal, so each component $x\_i^{(t)}$ (i.e., each pixel in the image at time step $t$) is independently sampled from a 1-d Gaussian based on the prior time step's pixel value at the *same x-y location in the image*. So a given pixel should *NOT* diffuse into neighboring pixels; instead, the action of the diffusion step is a linear Gaussian 1-d transformation of the number held in the pixel, with the mean slightly reduced and some noise added.
  
  **Question**: This seems inconsistent with Figure 1? Instead of the blurred line (wider and fuzzier line), we should have a line that has the same width, but exhibits more noise? In order to have a pixel diffuse into neighboring pixels, we would need a diffusion kernel with a non-diagonal covariance, so that there is nonzero covariance between components?
  
  **Tags:** generative-model, image-generation, diffusion-models
  
  
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  ## 1 answers
  
  ### Answer by username_1 on 2023-04-13. Score: 0 | Accepted answer
  
  > Figure 1. The proposed modeling framework trained on 2-d swiss roll data
  
  The data is actually 2d coordinates, plotted on a scatter-plot. It doesn't represent an $n \times n$ image with individual pixels. And the added noise is 2d, well 1d to the x-coordinate and 1d to the y-coordinate, if you consider them separately.
  
  But when the process is applied to images, as seen in Figure 3, it behaves as you described and expected the result to look like.
  
  > **Comments:**
  > **username_0**: I did not realize that "2-d swiss roll data" is actually represented as a (x,y) scatter plot, thanks very much!!  
  
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