<p>Kobbelt et al. published a paper some 20 years ago where they specified how solving Poisson equations on the positions of a mesh lead to removing noise. This can be used for reconstructing mesh topology on holes, smooth out noisy meshes, mesh joints...</p>
<p>I am trying to do something similar, but in my case the meshes are piecewise smooth:</p>
<p><a href="https://i.sstatic.net/qHnzr.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/qHnzr.png" alt="enter image description here" /></a></p>
<p>What can I do at the corners so that they better reflect the geometry of the edges?</p>
<p>For the record what I am doing is splititng the mesh into fixed and mobile vertices, computing a cotan laplacian matrix and then solving for <span class="math-container">$Lx=0$</span> where x are the interleaved coordinates of the mobile vertices.</p>

CC BY-SA 4.0
2023-10-09T06:06:54.293Z
It will not work with a Laplace equation. It may be better with $L^q$ where $q$ is large - my guess is that for $q\to \infty$ this becomes a sharp corner, but I am not sure. In your case probably a transport PDE will work better to create a sharp corner. It could also be that some low order and nonlinear operator works well.
lightxbulb
CC BY-SA 4.0
2023-10-09T06:12:17.310Z
By nature the Poisson equation gives you a smooth solution. I don't think you can get what you want with this method. Can you show the original point cloud maybe?
user8469759
CC BY-SA 4.0
2023-10-09T06:16:05.197Z
Also maybe variational methods might be useful in your case (the poisson equation is a specific case of those).
user8469759
CC BY-SA 4.0
2023-10-09T18:49:55.547Z
@user8469759 I am not using poisson reconstruction, there is no point cloud, I am working directly on meshes.
Makogan
CC BY-SA 4.0
2023-10-09T18:50:29.970Z
@lightxbulb Do you have a suggestion on what the operator could be?
Makogan
CC BY-SA 4.0
2023-10-10T05:45:50.830Z
@Makogan I mentioned "Poisson equation" not "Poisson reconstruction". I thought also the use of a mesh for you was a bit of a toy. Another thing you can try is to solve the polyharmonic equation $$\Delta^k x = 0$$ for some $k > 0$. For $k=0$ this reduces to $\Delta x = 0$ which is your current case. Another idea could be trying to solve $$\Delta^k x = g(x)$$ where $g(x)$ gives you some info about the corners of your geometry, but this needs to be designed heuristically.
user8469759
CC BY-SA 4.0
2023-10-10T08:15:36.750Z
@Makogan In 1D the transport equation  $d_t - d_x = 0$ with Dirichlet conditions propagates a front from the Dirichlet conditions. Note that this problem is hyperbloic, so you need special discretisation schemes (upwinding) even with FDM. I am not sure how this needs to be reformulated in your manifold case in order to work, or whether that is even feasible. As for nonlinear methods you can look at edge enhancing diffusion, but I feel like it will yield something round for you too, unless you add Dirichlet nodes at the corners.
lightxbulb
<mathematics><computational-geometry><algorithm><geometry><mesh>
Piecewise smoothing of a mesh?
Makogan
# Piecewise smoothing of a mesh?

(Asked by: username_0 on 2023-10-08)

Kobbelt et al. published a paper some 20 years ago where they specified how solving Poisson equations on the positions of a mesh lead to removing noise. This can be used for reconstructing mesh topology on holes, smooth out noisy meshes, mesh joints...

I am trying to do something similar, but in my case the meshes are piecewise smooth:

[![enter image description here](https://i.sstatic.net/qHnzr.png)](https://i.sstatic.net/qHnzr.png)

What can I do at the corners so that they better reflect the geometry of the edges?

For the record what I am doing is splititng the mesh into fixed and mobile vertices, computing a cotan laplacian matrix and then solving for $Lx=0$ where x are the interleaved coordinates of the mobile vertices.

**Tags:** mathematics, computational-geometry, algorithm, geometry, mesh

> **Comments:**
> **username_1**: It will not work with a Laplace equation. It may be better with $L^q$ where $q$ is large - my guess is that for $q\to \infty$ this becomes a sharp corner, but I am not sure. In your case probably a transport PDE will work better to create a sharp corner. It could also be that some low order and nonlinear operator works well.  
> **username_2**: By nature the Poisson equation gives you a smooth solution. I don't think you can get what you want with this method. Can you show the original point cloud maybe?  
> **username_2**: Also maybe variational methods might be useful in your case (the poisson equation is a specific case of those).  
> **username_0**: @username_2 I am not using poisson reconstruction, there is no point cloud, I am working directly on meshes.  
> **username_0**: @username_1 Do you have a suggestion on what the operator could be?  
> **username_2**: @username_0 I mentioned "Poisson equation" not "Poisson reconstruction". I thought also the use of a mesh for you was a bit of a toy. Another thing you can try is to solve the polyharmonic equation $$\Delta^k x = 0$$ for some $k > 0$. For $k=0$ this reduces to $\Delta x = 0$ which is your current case. Another idea could be trying to solve $$\Delta^k x = g(x)$$ where $g(x)$ gives you some info about the corners of your geometry, but this needs to be designed heuristically.  
> **username_1**: @username_0 In 1D the transport equation  $d_t - d_x = 0$ with Dirichlet conditions propagates a front from the Dirichlet conditions. Note that this problem is hyperbloic, so you need special discretisation schemes (upwinding) even with FDM. I am not sure how this needs to be reformulated in your manifold case in order to work, or whether that is even feasible. As for nonlinear methods you can look at edge enhancing diffusion, but I feel like it will yield something round for you too, unless you add Dirichlet nodes at the corners.  

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## 0 answers

