Answers.Body: <p>TL,DR Use the Jacobian to get smoothness and the L2 norm to limit the magnitude + they could be complementary.</p>
  <p>I just skimmed at the paper (that you haven't really referenced) and the regularization term using the derivatives makes sense to make <span class="math-container">$\phi$</span> smooth. It's basically penalizing large changes between neighboring values of the registration field. Using the L2 norm would penalize the magnitude of the registration field, not the local variations.</p>
  
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Answers.Text: That means taking the jacobian(derivatives, partial derivatives) measures the smoothness and then applying a penalty to the magnitude of the derivatives using L2 norm? is that what the statement "We use the L2 norm over the deformation field gradients as a regularization term to encourage the deformation field smoothness" means (I've included the links sorry about that)
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Answers.Text: Exactly, if the gradient or changes are small, the field is smooth.
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Body: <p>How do L2 norm and The Jacobian act as a regularisation term to encourage smoothness in a deformation field? from the VoxelMorph original paper (<a href="https://arxiv.org/pdf/1809.05231.pdf" rel="nofollow noreferrer">here</a>) they used Jacobian as a means to smoothen the deformation field, a similar paper (<a href="https://arxiv.org/pdf/2008.03949.pdf" rel="nofollow noreferrer">here</a>) made use of L2 norm to smoothen the deformation field. However, I don't understand how this is achieved using either L2 norm or Jacobian. Thanks in advance.</p>
  
Tags: <computer-vision><image-processing><l2-regularization>
Title: How do L2 norm and Jacobian act as a regularisation term to encourage smoothness in a deformation field?
UserDisplayName: a__ys
ThreadText: # How do L2 norm and Jacobian act as a regularisation term to encourage smoothness in a deformation field?
  
  (Asked by: username_0 on 2022-12-03)
  
  How do L2 norm and The Jacobian act as a regularisation term to encourage smoothness in a deformation field? from the VoxelMorph original paper ([here](https://arxiv.org/pdf/1809.05231.pdf)) they used Jacobian as a means to smoothen the deformation field, a similar paper ([here](https://arxiv.org/pdf/2008.03949.pdf)) made use of L2 norm to smoothen the deformation field. However, I don't understand how this is achieved using either L2 norm or Jacobian. Thanks in advance.
  
  **Tags:** computer-vision, image-processing, l2-regularization
  
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  ## 1 answers
  
  ### Answer by username_1 on 2022-12-03. Score: 1 | Accepted answer
  
  TL,DR Use the Jacobian to get smoothness and the L2 norm to limit the magnitude + they could be complementary.
  
  I just skimmed at the paper (that you haven't really referenced) and the regularization term using the derivatives makes sense to make $\phi$ smooth. It's basically penalizing large changes between neighboring values of the registration field. Using the L2 norm would penalize the magnitude of the registration field, not the local variations.
  
  > **Comments:**
  > **username_0**: That means taking the jacobian(derivatives, partial derivatives) measures the smoothness and then applying a penalty to the magnitude of the derivatives using L2 norm? is that what the statement "We use the L2 norm over the deformation field gradients as a regularization term to encourage the deformation field smoothness" means (I've included the links sorry about that)  
  > **username_1**: Exactly, if the gradient or changes are small, the field is smooth.  
  
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