Mesh Denoising¶
Important: Please read the installation page for details about how to install the toolboxes. $\newcommand{\dotp}[2]{\langle #1, #2 \rangle}$ $\newcommand{\enscond}[2]{\lbrace #1, #2 \rbrace}$ $\newcommand{\pd}[2]{ \frac{ \partial #1}{\partial #2} }$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\umax}[1]{\underset{#1}{\max}\;}$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\uargmin}[1]{\underset{#1}{argmin}\;}$ $\newcommand{\norm}[1]{\|#1\|}$ $\newcommand{\abs}[1]{\left|#1\right|}$ $\newcommand{\choice}[1]{ \left\{ \begin{array}{l} #1 \end{array} \right. }$ $\newcommand{\pa}[1]{\left(#1\right)}$ $\newcommand{\diag}[1]{{diag}\left( #1 \right)}$ $\newcommand{\qandq}{\quad\text{and}\quad}$ $\newcommand{\qwhereq}{\quad\text{where}\quad}$ $\newcommand{\qifq}{ \quad \text{if} \quad }$ $\newcommand{\qarrq}{ \quad \Longrightarrow \quad }$ $\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\RR}{\mathbb{R}}$ $\newcommand{\EE}{\mathbb{E}}$ $\newcommand{\Zz}{\mathcal{Z}}$ $\newcommand{\Ww}{\mathcal{W}}$ $\newcommand{\Vv}{\mathcal{V}}$ $\newcommand{\Nn}{\mathcal{N}}$ $\newcommand{\NN}{\mathcal{N}}$ $\newcommand{\Hh}{\mathcal{H}}$ $\newcommand{\Bb}{\mathcal{B}}$ $\newcommand{\Ee}{\mathcal{E}}$ $\newcommand{\Cc}{\mathcal{C}}$ $\newcommand{\Gg}{\mathcal{G}}$ $\newcommand{\Ss}{\mathcal{S}}$ $\newcommand{\Pp}{\mathcal{P}}$ $\newcommand{\Ff}{\mathcal{F}}$ $\newcommand{\Xx}{\mathcal{X}}$ $\newcommand{\Mm}{\mathcal{M}}$ $\newcommand{\Ii}{\mathcal{I}}$ $\newcommand{\Dd}{\mathcal{D}}$ $\newcommand{\Ll}{\mathcal{L}}$ $\newcommand{\Tt}{\mathcal{T}}$ $\newcommand{\si}{\sigma}$ $\newcommand{\al}{\alpha}$ $\newcommand{\la}{\lambda}$ $\newcommand{\ga}{\gamma}$ $\newcommand{\Ga}{\Gamma}$ $\newcommand{\La}{\Lambda}$ $\newcommand{\si}{\sigma}$ $\newcommand{\Si}{\Sigma}$ $\newcommand{\be}{\beta}$ $\newcommand{\de}{\delta}$ $\newcommand{\De}{\Delta}$ $\newcommand{\phi}{\varphi}$ $\newcommand{\th}{\theta}$ $\newcommand{\om}{\omega}$ $\newcommand{\Om}{\Omega}$
This tour explores denoising of 3-D meshes using linear filtering, heat diffusion and Sobolev regularization.
addpath('toolbox_signal')
addpath('toolbox_general')
addpath('toolbox_graph')
addpath('solutions/meshproc_3_denoising')
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 $X_0 = (x_{0,i})_{i=1}^n \in \RR^{3 \times n}$.
clear options;
name = 'nefertiti';
name = 'elephant-50kv';
options.name = name; % useful for displaying
[X0,F] = read_mesh(name);
Number $n$ of vertices and number $m$ of faces.
n = size(X0,2);
m = size(F,2);
Display the mesh in 3-D.
options.lighting = 1;
% clf;
plot_mesh(X0,F, options);
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.
rho = 0.015;
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.
N = compute_normal(X0,F);
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 $i$.
X = X0 + repmat(rho*randn(1,n),[3,1]).*N;
Display the noisy mesh.
clf;
plot_mesh(X,F,options); axis('tight');