Skinning is a popular way to rig and deform characters for animation, to
compute reduced-order simulations, and to define features for geometry
processing. Methods built on skinning rely on weight functions that distribute
the influence of each degree of freedom across the mesh. Automatic skinning
methods generate these weight functions with minimal user input, usually by
solving a variational problem on a mesh whose boundary is the skinned surface.
This formulation necessitates tetrahedralizing the volume inside the surface,
which brings with it meshing artifacts, the possibility of tetrahedralization
failure, and the impossibility of generating weights for surfaces that are not
closed. We introduce a mesh-free and robust automatic skinning method that
generates high-quality skinning weights comparable to the current state of the
art without volumetric meshes. Our method reliably works even on open surfaces
and triangle soups where current methods fail. We achieve this through the use
of a Lagrangian representation for skinning weights, which circumvents the need
for finite elements while optimizing the biharmonic energy.

无网格皮肤化方法通过拉格朗日表示的蒙皮权重优化双调和能量，生成高质量蒙皮权重，适用于开放表面和三角形混乱点云，避免了有限元与体网格化的问题。

鲁棒双调和曲肌肤模型的几何场方法

Robust Biharmonic Skinning Using Geometric Fields

We present TetSphere splatting, an explicit, Lagrangian representation for
reconstructing 3D shapes with high-quality geometry. In contrast to
conventional object reconstruction methods which predominantly use Eulerian
representations, including both neural implicit (e.g., NeRF, NeuS) and explicit
representations (e.g., DMTet), and often struggle with high computational
demands and suboptimal mesh quality, TetSphere splatting utilizes an underused
but highly effective geometric primitive -- tetrahedral meshes. This approach
directly yields superior mesh quality without relying on neural networks or
post-processing. It deforms multiple initial tetrahedral spheres to accurately
reconstruct the 3D shape through a combination of differentiable rendering and
geometric energy optimization, resulting in significant computational
efficiency. Serving as a robust and versatile geometry representation,
Tet-Sphere splatting seamlessly integrates into diverse applications, including
single-view 3D reconstruction, image-/text-to-3D content generation.
Experimental results demonstrate that TetSphere splatting outperforms existing
representations, delivering faster optimization speed, enhanced mesh quality,
and reliable preservation of thin structures.

我们提出了 TetSphere 喷洒，一种用于重建高质量几何形状的显式的拉格朗日表示。与传统的物体重建方法不同，它主要使用欧拉表示，包括神经隐式方法（如 NeRF，NeuS）和显式表示方法（如 DMTet），通常在高计算需求和低优化网格质量方面存在问题。 TetSphere 喷洒利用了一个被低估但非常有效的几何原语 - 四面体网格。通过可微渲染和几何能量优化的组合，它将多个初始四面体球体变形以准确重建 3D 形状，从而实现了显著的计算效率。作为一种强大而灵活的几何表示方法，Tet-Sphere 喷洒无缝地集成到各种应用中，包括基于单视角的 3D 重建和图像 / 文本到 3D 内容的生成。实验结果表明，TetSphere 喷洒优于现有的表示方法，具有更快的优化速度，增强的网格质量和可靠的薄结构保留。

TetSphere Splatting: 用拉格朗日体网格表示高质量几何体

TetSphere Splatting: Representing High-Quality Geometry with Lagrangian  Volumetric Meshes

Incorporating neural networks for the solution of Ordinary Differential
Equations (ODEs) represents a pivotal research direction within computational
mathematics. Within neural network architectures, the integration of the
intrinsic structure of ODEs offers advantages such as enhanced predictive
capabilities and reduced data utilization. Among these structural ODE forms,
the Lagrangian representation stands out due to its significant physical
underpinnings. Building upon this framework, Bhattoo introduced the concept of
Lagrangian Neural Networks (LNNs). Then in this article, we introduce a
groundbreaking extension (Genralized Lagrangian Neural Networks) to Lagrangian
Neural Networks (LNNs), innovatively tailoring them for non-conservative
systems. By leveraging the foundational importance of the Lagrangian within
Lagrange's equations, we formulate the model based on the generalized
Lagrange's equation. This modification not only enhances prediction accuracy
but also guarantees Lagrangian representation in non-conservative systems.
Furthermore, we perform various experiments, encompassing 1-dimensional and
2-dimensional examples, along with an examination of the impact of network
parameters, which proved the superiority of Generalized Lagrangian Neural
Networks(GLNNs).

基于广义拉格朗日方程的变形不仅提高了预测准确性，而且在非保守系统中保证了拉格朗日表示，从而证明了广义拉格朗日神经网络（GLNNs）的优越性。