Mesh Denoising¶

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This tour explores denoising of 3-D meshes using linear filtering, heat diffusion and Sobolev regularization.

3-D Triangulated Meshes¶

The topology of a triangulation is defined via a set of indexes $\Vv = \{1,\ldots,n\}$ that indexes the $n$ vertices, a set of edges $\Ee \subset \Vv \times \Vv$ and a set of $m$ faces $\Ff \subset \Vv \times \Vv \times \Vv$.

We load a mesh. The set of faces $\Ff$ is stored in a matrix $F \in \{1,\ldots,n\}^{3 \times m}$. The positions $x_i \in \RR^3$, for $i \in V$, of the $n$ vertices are stored in a matrix $X_0 = (x_{0,i})_{i=1}^n \in \RR^{3 \times n}$.

Number $n$ of vertices and number $m$ of faces.

Display the mesh in 3-D.

Noisy Mesh¶

We generate artificially a noisy mesh by random normal displacement along the normal. We only perform normal displacements because tangencial displacements do not impact the geometry of the mesh.

The parameter $\rho>0$ controls the amount of noise.

We compute the normals $N = (N_i)_{i=1}^n$ to the mesh. This is obtained by averaging the normal to the faces ajacent to each vertex.

We create a noisy mesh by displacement of the vertices along the normal direction $$ x_i = x_{0,i} + \rho \epsilon_i N_i \in \RR^3 $$ where $\epsilon_i \sim \Nn(0,1)$ is a realization of a Gaussian random variable, and where $N_i \in \RR^3$ is the normal of the mesh for each vertex index $i$.

Display the noisy mesh.