Laplacian coordinates

When computing the Laplacian coordinates of a grid, the following formula can be used:

For a person with A grid of $N$ $(x_i, y_i, z_i)$ , where $\ldots, N$ 。

Laplacian coordinates $L_i$ Represents the $The weighted average of the difference between the coordinates of i$ vertices and the coordinates of their neighbors.

Commonly used calculation formulas are as follows:

$L_i = \frac{1}{m} \sum_{j=1}^{m} w_{ij} (v_i - v_j)$

in:

$L_i$ This is number $Laplacian coordinates of i$ vertices.
$m$ is theThe number of adjacent vertices for $i vertices.$
$v_i = (x_i, y_i, z_i)$ is no. $The coordinates of i$ vertices.
$v_j = (x_j, y_j, z_j)$ is no.vertex of $i$ $The coordinates of j$ adjacent vertices.
$w_{ij}$ is the weight, indicating the $i$ vertex andThe connection strength between $j adjacent vertices.$ Common weight calculation methods include uniform weight, distance weight and so on.

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More general formula

Using uniform weights
$L_i = \frac{1}{m} \sum_{j=1}^{m} (v_i - v_j)$
Write it another way:

$L_i =v_i- \frac{1}{m} \sum_{j=1}^{m} v_j$

That is to say, the Laplace coordinates of a vertex are the difference between the coordinates of the vertex and the mean of its neighbors.
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It should be noted that Laplacian coordinates are a feature describing the grid structure and shape, and are used to analyze the smoothness, deformation and other properties of the grid. It has a wide range of applications in computer graphics, computer vision and other fields. The specific weight calculation methods and application scenarios may vary, and you can choose a suitable method according to your specific needs.

Application scenarios

In physics and computer graphics, the Laplace coordinates of vertices have many important physical meanings and applications. Here are some common physical meanings:

Shape analysis and feature extraction : Laplacian coordinates can be used to describe local geometric features of 3D shapes. By calculating the Laplace coordinates of the vertex, the geometric structure information around the vertex, such as curvature, surface normal, etc., can be obtained. This information is useful for shape analysis, feature extraction, and model comparison.
Model smoothing and denoising : Laplacian coordinates can be used for model smoothing and denoising. By smoothing the Laplacian coordinates of the vertices, it is possible to reduce high-frequency noise in the shape and preserve the overall structure of the shape. This is used frequently in rendering and modeling in computer graphics.
Shape deformation and animation : Laplacian coordinates also play an important role in shape deformation and animation. By deforming the Laplace coordinates of vertices, local or overall deformation effects of the shape can be achieved. This is very common in computer animation, virtual reality, and game development.
Model editing and reconstruction : Using Laplacian coordinates, model editing and reconstruction can be performed. By modifying the Laplacian coordinates of the vertices, you can edit the shape of the model, such as stretching, compressing, twisting, and other operations. At the same time, Laplacian coordinates can also be used to reconstruct three-dimensional shapes. For example, a surface reconstruction algorithm is used to restore a three-dimensional model with continuity and smoothness from point cloud data.

In short, the Laplacian coordinates of vertices are widely used in the analysis, processing and editing of three-dimensional shapes. They can provide geometric feature information of shapes and support various shape operations and applications, including shape analysis, model smoothing, and shape deformation. , model editing, etc.